diff --git a/changelog b/changelog index 160ae63..c4acf60 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,4 @@ +20081122 tpd src/axiom-website put website under change control 20081122 tpd zips/gcl-2.6.8pre3.unixport.makefile.patch added 20081122 tpd zips/gcl-2.6.8pre3.unixport.init_gcl.lsp.in.patch added 20081122 tpd zips/gcl-2.6.8pre3.cmpnew.gcl_cmpflet.lsp.patch added diff --git a/src/axiom-website/CATS/index.html b/src/axiom-website/CATS/index.html new file mode 100644 index 0000000..b74f1c8 --- /dev/null +++ b/src/axiom-website/CATS/index.html @@ -0,0 +1,128 @@ + + + + (CATS) Computer Algebra Test Suite + + + +This work is part of the Computer Algebra Test Suite (CATS). +These files show the results that Axiom computes given the +set of integrals listed in +
+  Spiegel, Murray R. 
+  Mathematical Handbook of Formulas and Tables
+  Schaum's Outline Series McGraw-Hill 1968 
+
+

+Each integral is computed by Axiom and compared against the +published result. +

+Each Axiom result is differenced from the published result +and reduced to a constant (usually 0). +

+ + + Schaums 14.59-14.83  + source + pdf
+ Schaums 14.84-14.104  + source + pdf
+ Schaums 14.104-14.112  + source + pdf
+ Schaums 14.113-.119  + source + pdf
+ Schaums 14.120-14.124  + source + pdf
+ Schaums 14.125-14.143  + source + pdf
+ Schaums 14.144-14.162  + source + pdf
+ Schaums 14.163-14.181  + source + pdf
+ Schaums 14.182-14.209  + source + pdf
+ Schaums 14.210-14.236  + source + pdf
+ Schaums 14.237-14.264  + source + pdf
+ Schaums 14.265-14.279  + source + pdf
+ Schaums 14.280-14.298  + source + pdf
+ Schaums 14.299-14.310  + source + pdf
+ Schaums 14.311-14.324  + source + pdf
+ Schaums 14.325-14.338  + source + pdf
+ Schaums 14.339-14.368  + source + pdf
+ Schaums 14.369-14.398  + source + pdf
+ Schaums 14.399-14.428  + source + pdf
+ Schaums 14.429-14.439  + source + pdf
+ Schaums 14.440-14.450  + source + pdf
+ Schaums 14.451-14.460  + source + pdf
+ Schaums 14.461-14.470  + source + pdf
+ Schaums 14.471-14.508  + source + pdf
+ Schaums 14.509-14.524  + source + pdf
+ Schaums 14.525-14.539  + source + pdf
+ Schaums 14.540-14.561  + source + pdf
+ Schaums 14.562-14.589  + source + pdf
+ Schaums 14.590-14.603  + source + pdf
+ Schaums 14.604-14.614  + source + pdf
+ Schaums 14.615-14.625  + source + pdf
+ Schaums 14.626-14.635  + source + pdf
+ Schaums 14.636-14.645  + source + pdf
+ Schaums 14.646-14.677  + source + pdf
+ + \ No newline at end of file diff --git a/src/axiom-website/CATS/schaum1.input.pamphlet b/src/axiom-website/CATS/schaum1.input.pamphlet new file mode 100644 index 0000000..615d907 --- /dev/null +++ b/src/axiom-website/CATS/schaum1.input.pamphlet @@ -0,0 +1,1430 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum1.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.59~~~~~$\displaystyle +\int{\frac{dx}{ax+b}}$} +$$\int{\frac{1}{ax+b}}= +\frac{1}{a}~\ln(ax+b) +$$ +<<*>>= +)spool schaum1.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(1/(a*x+b),x) +--R +--R log(a x + b) +--R (1) ------------ +--R a +--R Type: Union(Expression Integer,...) +--E 1 + +--S 2 +bb:=1/a*log(a*x+b) +--R +--R log(a x + b) +--R (2) ------------ +--R a +--R Type: Expression Integer +--E + +--S 3 14:59 Schaums and Axiom agree +cc:=bb-aa +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ +\section{\cite{1}:14.60~~~~~$\displaystyle +\int{\frac{x~dx}{ax+b}}$} +$$\int{\frac{x}{ax+b}}= +\frac{x}{a}-\frac{b}{a^2}~\ln(ax+b) +$$ +<<*>>= +)clear all + +--S 4 +aa:=integrate(x/(a*x+b),x) +--R +--R +--R - b log(a x + b) + a x +--R (1) ---------------------- +--R 2 +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 5 +bb:=x/a-b/a^2*log(a*x+b) +--R +--R - b log(a x + b) + a x +--R (2) ---------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 6 14:60 Schaums and Axiom agree +cc:=bb-aa +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.61~~~~~$\displaystyle +\int{\frac{x^2~dx}{ax+b}}$} +$$\int{\frac{x^2}{ax+b}}= +\frac{(ax+b)^2}{2a^3}-\frac{2b(ax+b)}{a^3}+\frac{b^2}{a^3}~\ln(ax+b) +$$ +<<*>>= +)clear all + +--S 7 +aa:=integrate(x^2/(a*x+b),x) +--R +--R 2 2 2 +--R 2b log(a x + b) + a x - 2a b x +--R (1) ------------------------------- +--R 3 +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 8 +bb:=(a*x+b)^2/(2*a^3)-(2*b*(a*x+b))/a^3+b^2/a^3*log(a*x+b) +--R +--R 2 2 2 2 +--R 2b log(a x + b) + a x - 2a b x - 3b +--R (2) ------------------------------------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 9 +cc:=bb-aa +--R +--R 2 +--R 3b +--R (3) - --- +--R 3 +--R 2a +--R Type: Expression Integer +--E +@ +This factor is constant with respect to $x$ as shown by taking the +derivative. It is a constant of integration. +<<*>>= +--S 10 14:61 Schaums and Axiom differ by a constant +differentiate(cc,x) +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ +\section{\cite{1}:14.62~~~~~$\displaystyle +\int{\frac{x^3~dx}{ax+b}}$} +$$\int{\frac{x^3}{ax+b}}= +\frac{(ax+b)^3}{3a^4}-\frac{3b(ax+b)^2}{2a^4}+ +\frac{3b^2(ax+b)}{a^4}-\frac{b^3}{a^4}~\ln(ax+b) +$$ +<<*>>= +)clear all + +--S 11 +aa:=integrate(x^3/(a*x+b),x) +--R +--R 3 3 3 2 2 2 +--R - 6b log(a x + b) + 2a x - 3a b x + 6a b x +--R (1) -------------------------------------------- +--R 4 +--R 6a +--R Type: Union(Expression Integer,...) +--E +@ +and the book expression is: +<<*>>= +--S 12 +bb:=(a*x+b)^3/(3*a^4)-(3*b*(a*x+b)^2)/(2*a^4)+(3*b^2*(a*x+b))/a^4-(b^3/a^4)*log(a*x+b) +--R +--R 3 3 3 2 2 2 3 +--R - 6b log(a x + b) + 2a x - 3a b x + 6a b x + 11b +--R (2) --------------------------------------------------- +--R 4 +--R 6a +--R Type: Expression Integer +--E +@ + +The difference is a constant with respect to x: +<<*>>= +--S 13 +cc:=aa-bb +--R +--R 3 +--R 11b +--R (3) - ---- +--R 4 +--R 6a +--R Type: Expression Integer +--E +@ + +If we differentiate each expression we see that this is the integration +constant. +<<*>>= +--S 14 14:62 Schaums and Axiom differ by a constant +dd:=D(cc,x) +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.63~~~~~$\displaystyle +\int{\frac{dx}{x~(ax+b)}}$} +$$\int{\frac{1}{x~(ax+b)}}= +\frac{1}{b}~\ln\left(\frac{x}{ax+b}\right) +$$ +<<*>>= +)clear all + +--S 15 +aa:=integrate(1/(x*(a*x+b)),x) +--R +--R - log(a x + b) + log(x) +--R (1) ----------------------- +--R b +--R Type: Union(Expression Integer,...) +--E + +--S 16 +bb:=1/b*log(x/(a*x+b)) +--R +--R x +--R log(-------) +--R a x + b +--R (2) ------------ +--R b +--R Type: Expression Integer +--E + +--S 17 +cc:=aa-bb +--R +--R x +--R - log(a x + b) + log(x) - log(-------) +--R a x + b +--R (3) -------------------------------------- +--R b +--R Type: Expression Integer +--E +@ +but we know that $$\log(a)-\log(b)=\log(\frac{a}{b})$$ + +We can express this fact as a rule: +<<*>>= +--S 18 +logdiv:=rule(log(a)-log(b) == log(a/b)) +--R +--R a +--I (4) - log(b) + log(a) + %I == log(-) + %I +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E +@ +and use this rule to rewrite the logs into divisions: +<<*>>= +--S 19 14:63 Schaums and Axiom agree +dd:=logdiv cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ +so we can see the equivalence directly. + +\section{\cite{1}:14.64~~~~~$\displaystyle +\int{\frac{dx}{x^2~(ax+b)}}$} +$$\int{\frac{1}{x^2~(ax+b)}}= +-\frac{1}{bx}+\frac{a}{b^2}~\ln\left(\frac{ax+b}{x}\right) +$$ +<<*>>= +)clear all + +--S 20 +aa:=integrate(1/(x^2*(a*x+b)),x) +--R +--R a x log(a x + b) - a x log(x) - b +--R (1) --------------------------------- +--R 2 +--R b x +--R Type: Union(Expression Integer,...) +--E +@ + +The original form given in the book expands to: +<<*>>= +--S 21 +bb:=-1/(b*x)+a/b^2*log((a*x+b)/x) +--R +--R a x + b +--R a x log(-------) - b +--R x +--R (2) -------------------- +--R 2 +--R b x +--R Type: Expression Integer +--E + +--S 22 +cc:=aa-bb +--R +--R a x + b +--R a log(a x + b) - a log(x) - a log(-------) +--R x +--R (3) ------------------------------------------ +--R 2 +--R b +--R Type: Expression Integer +--E +@ + +We can define the following rule to expand log forms: +<<*>>= +--S 23 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E +@ +and apply it to the difference +<<*>>= +--S 24 14:64 Schaums and Axiom agree +divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.65~~~~~$\displaystyle +\int{\frac{dx}{x^3~(ax+b)}}$} +$$\int{\frac{1}{x^3~(ax+b)}}= +\frac{2ax-b}{2b^2x^2}+\frac{a^2}{b^3}~\ln\left(\frac{x}{ax+b}\right) +$$ +<<*>>= +)clear all +--S 25 +aa:=integrate(1/(x^3*(a*x+b)),x) +--R +--R 2 2 2 2 2 +--R - 2a x log(a x + b) + 2a x log(x) + 2a b x - b +--R (1) ----------------------------------------------- +--R 3 2 +--R 2b x +--R Type: Union(Expression Integer,...) +--E + +--S 26 +bb:=(2*a*x-b)/(2*b^2*x^2)+a^2/b^3*log(x/(a*x+b)) +--R +--R 2 2 x 2 +--R 2a x log(-------) + 2a b x - b +--R a x + b +--R (2) ------------------------------- +--R 3 2 +--R 2b x +--R Type: Expression Integer +--E + +--S 27 +cc:=aa-bb +--R +--R 2 2 2 x +--R - a log(a x + b) + a log(x) - a log(-------) +--R a x + b +--R (3) -------------------------------------------- +--R 3 +--R b +--R Type: Expression Integer +--E + +--S 28 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 29 14:65 Schaums and Axiom agree +dd:=divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.66~~~~~$\displaystyle +\int{\frac{dx}{(ax+b)^2}}$} +$$\int{\frac{1}{(ax+b)^2}}= +\frac{-1}{a~(ax+b)} +$$ +<<*>>= +)clear all + +--S 30 +aa:=integrate(1/(a*x+b)^2,x) +--R +--R 1 +--R (1) - --------- +--R 2 +--R a x + a b +--R Type: Union(Expression Integer,...) +--E + +--S 31 +bb:=-1/(a*(a*x+b)) +--R +--R 1 +--R (2) - --------- +--R 2 +--R a x + a b +--R Type: Fraction Polynomial Integer +--E + +--S 32 14:66 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.67~~~~~$\displaystyle +\int{\frac{x~dx}{(ax+b)^2}}$} +$$\int{\frac{x}{(ax+b)^2}}= +\frac{b}{a^2~(ax+b)}+\frac{1}{a^2}~\ln(ax+b) +$$ +<<*>>= +)clear all + +--S 33 +aa:=integrate(x/(a*x+b)^2,x) +--R +--R (a x + b)log(a x + b) + b +--R (1) ------------------------- +--R 3 2 +--R a x + a b +--R Type: Union(Expression Integer,...) +--E + +--S 34 +bb:=b/(a^2*(a*x+b))+1/a^2*log(a*x+b) +--R +--R (a x + b)log(a x + b) + b +--R (2) ------------------------- +--R 3 2 +--R a x + a b +--R Type: Expression Integer +--E + +--S 35 14:67 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.68~~~~~$\displaystyle +\int{\frac{x^2~dx}{(ax+b)^2}}$} +$$\int{\frac{x^2}{(ax+b)^2}}= +\frac{ax+b}{a^3}-\frac{b^2}{a^3~(ax+b)} +-\frac{2b}{a^3}~\ln(ax+b) +$$ +<<*>>= +)clear all + +--S 36 +aa:=integrate(x^2/(a*x+b)^2,x) +--R +--R 2 2 2 2 +--R (- 2a b x - 2b )log(a x + b) + a x + a b x - b +--R (1) ------------------------------------------------ +--R 4 3 +--R a x + a b +--R Type: Union(Expression Integer,...) +--E +@ +and the book expression expands into +<<*>>= +--S 37 +bb:=(a*x+b)/a^3-b^2/(a^3*(a*x+b))-((2*b)/a^3)*log(a*x+b) +--R +--R 2 2 2 +--R (- 2a b x - 2b )log(a x + b) + a x + 2a b x +--R (2) -------------------------------------------- +--R 4 3 +--R a x + a b +--R Type: Expression Integer +--E +@ + +These two expressions differ by the constant +<<*>>= +--S 38 +cc:=aa-bb +--R +--R b +--R (3) - -- +--R 3 +--R a +--R Type: Expression Integer +--E +@ + +That this expression is constant can be shown by differentiation: +<<*>>= +--S 39 14:68 Schaums and Axiom differ by a constant +D(cc,x) +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.69~~~~~$\displaystyle +\int{\frac{x^3~dx}{(ax+b)^2}}$} +$$\int{\frac{x^3}{(ax+b)^2}}= +\frac{(ax+b)^2}{2a^4}-\frac{3b(ax+b)}{a^4}+\frac{b^3}{a^4(ax+b)} ++\frac{3b^2}{a^4}~\ln(ax+b) +$$ +<<*>>= +)clear all + +--S 40 +aa:=integrate(x^3/(a*x+b)^2,x) +--R +--R 2 3 3 3 2 2 2 3 +--R (6a b x + 6b )log(a x + b) + a x - 3a b x - 4a b x + 2b +--R (1) ---------------------------------------------------------- +--R 5 4 +--R 2a x + 2a b +--R Type: Union(Expression Integer,...) +--E + +--S 41 +bb:=(a*x+b)^2/(2*a^4)-(3*b*(a*x+b))/a^4+b^3/(a^4*(a*x+b))+(3*b^2/a^4)*log(a*x+b) +--R +--R 2 3 3 3 2 2 2 3 +--R (6a b x + 6b )log(a x + b) + a x - 3a b x - 9a b x - 3b +--R (2) ---------------------------------------------------------- +--R 5 4 +--R 2a x + 2a b +--R Type: Expression Integer +--E + +--S 42 +cc:=aa-bb +--R +--R 2 +--R 5b +--R (3) --- +--R 4 +--R 2a +--R Type: Expression Integer +--E + +--S 43 14:69 Schaums and Axiom differ by a constant +dd:=D(cc,x) +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ +\section{\cite{1}:14.70~~~~~$\displaystyle +\int{\frac{dx}{x~(ax+b)^2}}$} +$$\int{\frac{1}{x~(ax+b)^2}}= +\frac{1}{b~(ax+b)}+\frac{1}{b^2}~\ln\left(\frac{x}{ax+b}\right) +$$ +<<*>>= +)clear all + +--S 44 +aa:=integrate(1/(x*(a*x+b)^2),x) +--R +--R (- a x - b)log(a x + b) + (a x + b)log(x) + b +--R (1) --------------------------------------------- +--R 2 3 +--R a b x + b +--R Type: Union(Expression Integer,...) +--E +@ +and the book says: +<<*>>= +--S 45 +bb:=(1/(b*(a*x+b))+(1/b^2)*log(x/(a*x+b))) +--R +--R x +--R (a x + b)log(-------) + b +--R a x + b +--R (2) ------------------------- +--R 2 3 +--R a b x + b +--R Type: Expression Integer +--E + +--S 46 +cc:=aa-bb +--R +--R x +--R - log(a x + b) + log(x) - log(-------) +--R a x + b +--R (3) -------------------------------------- +--R 2 +--R b +--R Type: Expression Integer +--E +@ +So we look at the divlog rule again: +<<*>>= +--S 47 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E +@ + +we apply it: +<<*>>= +--S 48 14:70 Schaums and Axiom agree +dd:=divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.71~~~~~$\displaystyle +\int{\frac{dx}{x^2~(ax+b)^2}}$} +$$\int{\frac{1}{x^2~(ax+b)^2}}= +\frac{-a}{b^2~(ax+b)}-\frac{1}{b^2~x}+ +\frac{2a}{b^3}~\ln\left(\frac{ax+b}{x}\right) +$$ +<<*>>= +)clear all + +--S 49 +aa:=integrate(1/(x^2*(a*x+b)^2),x) +--R +--R 2 2 2 2 2 +--R (2a x + 2a b x)log(a x + b) + (- 2a x - 2a b x)log(x) - 2a b x - b +--R (1) --------------------------------------------------------------------- +--R 3 2 4 +--R a b x + b x +--R Type: Union(Expression Integer,...) +--E +@ +and the book says: +<<*>>= +--S 50 +bb:=(-a/(b^2*(a*x+b)))-(1/(b^2*x))+((2*a)/b^3)*log((a*x+b)/x) +--R +--R 2 2 a x + b 2 +--R (2a x + 2a b x)log(-------) - 2a b x - b +--R x +--R (2) ------------------------------------------ +--R 3 2 4 +--R a b x + b x +--R Type: Expression Integer +--E + +--S 51 +cc:=aa-bb +--R +--R a x + b +--R 2a log(a x + b) - 2a log(x) - 2a log(-------) +--R x +--R (3) --------------------------------------------- +--R 3 +--R b +--R Type: Expression Integer +--E +@ +which calls for our divlog rule: +<<*>>= +--S 52 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E +@ +which we use to transform the result: +<<*>>= +--S 53 14:71 Schaums and Axiom agree +dd:=divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ +\section{\cite{1}:14.72~~~~~$\displaystyle +\int{\frac{dx}{x^3~(ax+b)^2}}$} +$$\int{\frac{1}{x^3~(ax+b)^2}}= +-\frac{(ax+b)^2}{2b^4x^2}+\frac{3a(ax+b)}{b^4x}- +\frac{a^3x}{b^4(ax+b)}-\frac{3a^2}{b^4}~\ln\left(\frac{ax+b}{x}\right) +$$ +<<*>>= +)clear all + +--S 54 +aa:=integrate(1/(x^3*(a*x+b)^2),x) +--R +--R (1) +--R 3 3 2 2 3 3 2 2 2 2 +--R (- 6a x - 6a b x )log(a x + b) + (6a x + 6a b x )log(x) + 6a b x +--R + +--R 2 3 +--R 3a b x - b +--R / +--R 4 3 5 2 +--R 2a b x + 2b x +--R Type: Union(Expression Integer,...) +--E + +--S 55 +bb:=-(a*x+b)^2/(2*b^4*x^2)+(3*a*(a*x+b))/(b^4*x)-(a^3*x)/(b^4*(a*x+b))-((3*a^2)/b^4)*log((a*x+b)/x) +--R +--R 3 3 2 2 a x + b 3 3 2 2 2 3 +--R (- 6a x - 6a b x )log(-------) + 3a x + 9a b x + 3a b x - b +--R x +--R (2) --------------------------------------------------------------- +--R 4 3 5 2 +--R 2a b x + 2b x +--R Type: Expression Integer +--E + +--S 56 +cc:=aa-bb +--R +--R 2 2 2 a x + b 2 +--R - 6a log(a x + b) + 6a log(x) + 6a log(-------) - 3a +--R x +--R (3) ----------------------------------------------------- +--R 4 +--R 2b +--R Type: Expression Integer +--E + +--S 57 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 58 +dd:=divlog cc +--R +--R 2 +--R 3a +--R (5) - --- +--R 4 +--R 2b +--R Type: Expression Integer +--E + +--S 59 14:72 Schaums and Axiom differ by a constant +ee:=D(dd,x) +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.73~~~~~$\displaystyle +\int{\frac{dx}{(ax+b)^3}}$} +$$\int{\frac{1}{(ax+b)^3}}= +\frac{-1}{2a(ax+b)^2} +$$ +<<*>>= +)clear all + +--S 60 +aa:=integrate(1/(a*x+b)^3,x) +--R +--R 1 +--R (1) - ---------------------- +--R 3 2 2 2 +--R 2a x + 4a b x + 2a b +--R Type: Union(Expression Integer,...) +--E + +--S 61 +bb:=-1/(2*(a*x+b)^2) +--R +--R 1 +--R (2) - -------------------- +--R 2 2 2 +--R 2a x + 4a b x + 2b +--R Type: Fraction Polynomial Integer +--E + +--S 62 +cc:=aa-bb +--R +--R a - 1 +--R (3) ---------------------- +--R 3 2 2 2 +--R 2a x + 4a b x + 2a b +--R Type: Expression Integer +--E + +--S 63 +dd:=aa/bb +--R +--R 1 +--R (4) - +--R a +--R Type: Expression Integer +--E + +--S 64 14:73 Schaums and Axiom differ by a constant +ee:=D(dd,x) +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.74~~~~~$\displaystyle +\int{\frac{x~dx}{(ax+b)^3}}$} +$$\int{\frac{x}{(ax+b)^3}}= +\frac{-1}{a^2(ax+b)}+\frac{b}{2a^2(ax+b)^2} +$$ +<<*>>= +)clear all + +--S 65 +aa:=integrate(x/(a*x+b)^3,x) +--R +--R - 2a x - b +--R (1) ---------------------- +--R 4 2 3 2 2 +--R 2a x + 4a b x + 2a b +--R Type: Union(Expression Integer,...) +--E + +--S 66 +bb:=-1/(a^2*(a*x+b))+b/(2*a^2*(a*x+b)^2) +--R +--R - 2a x - b +--R (2) ---------------------- +--R 4 2 3 2 2 +--R 2a x + 4a b x + 2a b +--R Type: Fraction Polynomial Integer +--E + +--S 67 14:74 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.75~~~~~$\displaystyle +\int{\frac{x^2~dx}{(ax+b)^3}}$} +$$\int{\frac{x^2}{(ax+b)^3}}= +\frac{2b}{a^3(ax+b)}-\frac{b^2}{2a^3(ax+b)^2}+ +\frac{1}{a^3}~\ln(ax+b) +$$ +<<*>>= +)clear all + +--S 68 +aa:=integrate(x^2/(a*x+b)^3,x) +--R +--R 2 2 2 2 +--R (2a x + 4a b x + 2b )log(a x + b) + 4a b x + 3b +--R (1) ------------------------------------------------- +--R 5 2 4 3 2 +--R 2a x + 4a b x + 2a b +--R Type: Union(Expression Integer,...) +--E + +--S 69 +bb:=(2*b)/(a^3*(a*x+b))-(b^2)/(2*a^3*(a*x+b)^2)+1/a^3*log(a*x+b) +--R +--R 2 2 2 2 +--R (2a x + 4a b x + 2b )log(a x + b) + 4a b x + 3b +--R (2) ------------------------------------------------- +--R 5 2 4 3 2 +--R 2a x + 4a b x + 2a b +--R Type: Expression Integer +--E + +--S 70 14:75 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.76~~~~~$\displaystyle +\int{\frac{x^3~dx}{(ax+b)^3}}$} +$$\int{\frac{x^3}{(ax+b)^3}}= +\frac{x}{a^3}-\frac{3b^2}{a^4(ax+b)}+\frac{b^3}{2a^4(ax+b)^2}- +\frac{3b}{a^4}~\ln(ax+b) +$$ +<<*>>= +)clear all +--S 71 +aa:=integrate(x^3/(a*x+b)^3,x) +--R +--R (1) +--R 2 2 2 3 3 3 2 2 2 3 +--R (- 6a b x - 12a b x - 6b )log(a x + b) + 2a x + 4a b x - 4a b x - 5b +--R ------------------------------------------------------------------------ +--R 6 2 5 4 2 +--R 2a x + 4a b x + 2a b +--R Type: Union(Expression Integer,...) +--E + +--S 72 +bb:=(x/a^3)-(3*b^2)/(a^4*(a*x+b))+b^3/(2*a^4*(a*x+b)^2)-(3*b)/a^4*log(a*x+b) +--R +--R (2) +--R 2 2 2 3 3 3 2 2 2 3 +--R (- 6a b x - 12a b x - 6b )log(a x + b) + 2a x + 4a b x - 4a b x - 5b +--R ------------------------------------------------------------------------ +--R 6 2 5 4 2 +--R 2a x + 4a b x + 2a b +--R Type: Expression Integer +--E + +--S 73 14:76 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.77~~~~~$\displaystyle +\int{\frac{dx}{x(ax+b)^3}}$} +$$\int{\frac{1}{x(ax+b)^3}}= +\frac{3}{2b(ax+b)^2}+\frac{2ax}{2b^2(ax+b)^2}- +\frac{1}{b^3}*\ln\left(\frac{ax+b}{x}\right) +$$ + +<<*>>= +)clear all + +--S 74 +aa:=integrate(1/(x*(a*x+b)^3),x) +--R +--R (1) +--R 2 2 2 2 2 2 +--R (- 2a x - 4a b x - 2b )log(a x + b) + (2a x + 4a b x + 2b )log(x) +--R + +--R 2 +--R 2a b x + 3b +--R / +--R 2 3 2 4 5 +--R 2a b x + 4a b x + 2b +--R Type: Union(Expression Integer,...) +--E + +--S 75 +bb:=(a^2*x^2)/(2*b^3*(a*x+b)^2)-(2*a*x)/(b^3*(a*x+b))-(1/b^3)*log((a*x+b)/x) +--R +--R 2 2 2 a x + b 2 2 +--R (- 2a x - 4a b x - 2b )log(-------) - 3a x - 4a b x +--R x +--R (2) ----------------------------------------------------- +--R 2 3 2 4 5 +--R 2a b x + 4a b x + 2b +--R Type: Expression Integer +--E + +--S 76 +cc:=aa-bb +--R +--R a x + b +--R - 2log(a x + b) + 2log(x) + 2log(-------) + 3 +--R x +--R (3) --------------------------------------------- +--R 3 +--R 2b +--R Type: Expression Integer +--E + +--S 77 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 78 +dd:=divlog cc +--R +--R 3 +--R (5) --- +--R 3 +--R 2b +--R Type: Expression Integer +--E + +--S 79 14:77 Schaums and Axiom differ by a constant +ee:=D(dd,x) +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.78~~~~~$\displaystyle +\int{\frac{dx}{x^2(ax+b)^3}}$} +$$\int{\frac{1}{x^2(ax+b)^3}}= +\frac{-a}{2b^2(ax+b)^2}-\frac{2a}{b^3(ax+b)}- +\frac{1}{b^3x}+\frac{3a}{b^4}~\ln\left(\frac{ax+b}{x}\right) +$$ +<<*>>= +)clear all + +--S 80 +aa:=integrate(1/(x^2*(a*x+b)^3),x) +--R +--R (1) +--R 3 3 2 2 2 +--R (6a x + 12a b x + 6a b x)log(a x + b) +--R + +--R 3 3 2 2 2 2 2 2 3 +--R (- 6a x - 12a b x - 6a b x)log(x) - 6a b x - 9a b x - 2b +--R / +--R 2 4 3 5 2 6 +--R 2a b x + 4a b x + 2b x +--R Type: Union(Expression Integer,...) +--E + +--S 81 +bb:=-a/(2*b^2*(a*x+b)^2)-(2*a)/(b^3*(a*x+b))-1/(b^3*x)+((3*a)/b^4)*log((a*x+b)/x) +--R +--R 3 3 2 2 2 a x + b 2 2 2 3 +--R (6a x + 12a b x + 6a b x)log(-------) - 6a b x - 9a b x - 2b +--R x +--R (2) ---------------------------------------------------------------- +--R 2 4 3 5 2 6 +--R 2a b x + 4a b x + 2b x +--R Type: Expression Integer +--E + +--S 82 +cc:=aa-bb +--R +--R a x + b +--R 3a log(a x + b) - 3a log(x) - 3a log(-------) +--R x +--R (3) --------------------------------------------- +--R 4 +--R b +--R Type: Expression Integer +--E + +--S 83 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 84 14:78 Schaums and Axiom agree +dd:=divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.79~~~~~$\displaystyle +\int{\frac{dx}{x^3(ax+b)^3}}$} +$$\int{\frac{1}{x^3(ax+b)^3}}= +-\frac{1}{2bx^2(ax+b)^2}+ +\frac{2a}{b^2x(ax+b)^2}+ +\frac{9a^2}{b^3(ax+b)^2}+ +\frac{6a^3x}{b^4(ax+b)^2}- +\frac{6a^2}{b^5}~\ln\left(\frac{ax+b}{x}\right)$$ + +<<*>>= +)clear all + +--S 85 +aa:=integrate(1/(x^3*(a*x+b)^3),x) +--R +--R (1) +--R 4 4 3 3 2 2 2 +--R (- 12a x - 24a b x - 12a b x )log(a x + b) +--R + +--R 4 4 3 3 2 2 2 3 3 2 2 2 3 4 +--R (12a x + 24a b x + 12a b x )log(x) + 12a b x + 18a b x + 4a b x - b +--R / +--R 2 5 4 6 3 7 2 +--R 2a b x + 4a b x + 2b x +--R Type: Union(Expression Integer,...) +--E + +--S 86 +bb:=-1/(2*b*x^2*(a*x+b)^2)_ + +(2*a)/(b^2*x*(a*x+b)^2)_ + +(9*a^2)/(b^3*(a*x+b)^2)_ + +(6*a^3*x)/(b^4*(a*x+b)^2)_ + +(-6*a^2)/b^5*log((a*x+b)/x) +--R +--R (2) +--R 4 4 3 3 2 2 2 a x + b 3 3 2 2 2 +--R (- 12a x - 24a b x - 12a b x )log(-------) + 12a b x + 18a b x +--R x +--R + +--R 3 4 +--R 4a b x - b +--R / +--R 2 5 4 6 3 7 2 +--R 2a b x + 4a b x + 2b x +--R Type: Expression Integer +--E + +--S 87 +cc:=aa-bb +--R +--R 2 2 2 a x + b +--R - 6a log(a x + b) + 6a log(x) + 6a log(-------) +--R x +--R (3) ----------------------------------------------- +--R 5 +--R b +--R Type: Expression Integer +--E + +--S 88 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 89 14:79 Schaums and Axiom agree +dd:=divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.80~~~~~$\displaystyle +\int{(ax+b)^n~dx}$} +$$\int{(ax+b)^n}= +\frac{(ax+b)^{n+1}}{(n+1)a}{\rm\ provided\ }n \ne -1 +$$ +<<*>>= +)clear all +--S 90 +aa:=integrate((a*x+b)^n,x) +--R +--R n log(a x + b) +--R (a x + b)%e +--R (1) ------------------------- +--R a n + a +--R Type: Union(Expression Integer,...) +--E + +--S 91 +bb:=(a*x+b)^(n+1)/((n+1)*a) +--R +--R n + 1 +--R (a x + b) +--R (2) -------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 92 +cc:=aa-bb +--R +--R n log(a x + b) n + 1 +--R (a x + b)%e - (a x + b) +--R (3) ------------------------------------------ +--R a n + a +--R Type: Expression Integer +--E +@ +This messy formula can be simplified using the explog rule: +<<*>>= +--S 93 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 94 +dd:=explog cc +--R +--R n + 1 n +--R - (a x + b) + (a x + b)(a x + b) +--R (5) -------------------------------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 95 14:80 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.81~~~~~$\displaystyle +\int{x(ax+b)^n~dx}$} +$$\int{x(ax+b)^n}= +\frac{(ax+b)^{n+2}}{(n+2)a^2}-\frac{b(ax+b)^{n+1}}{(n+1)a^2} +{\rm\ provided\ }n \ne -1,-2 +$$ +<<*>>= +)clear all +--S 96 +aa:=integrate(x*(a*x+b)^n,x) +--R +--R 2 2 2 2 n log(a x + b) +--R ((a n + a )x + a b n x - b )%e +--R (1) --------------------------------------------- +--R 2 2 2 2 +--R a n + 3a n + 2a +--R Type: Union(Expression Integer,...) +--E + +--S 97 +bb:=((a*x+b)^(n+2))/((n+2)*a^2)-(b*(a*x+b)^(n+1))/((n+1)*a^2) +--R +--R n + 2 n + 1 +--R (n + 1)(a x + b) + (- b n - 2b)(a x + b) +--R (2) -------------------------------------------------- +--R 2 2 2 2 +--R a n + 3a n + 2a +--R Type: Expression Integer +--E + +--S 98 +cc:=aa-bb +--R +--R (3) +--R 2 2 2 2 n log(a x + b) n + 2 +--R ((a n + a )x + a b n x - b )%e + (- n - 1)(a x + b) +--R + +--R n + 1 +--R (b n + 2b)(a x + b) +--R / +--R 2 2 2 2 +--R a n + 3a n + 2a +--R Type: Expression Integer +--E + +--S 99 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 100 +dd:=explog cc +--R +--R (5) +--R n + 2 n + 1 +--R (- n - 1)(a x + b) + (b n + 2b)(a x + b) +--R + +--R 2 2 2 2 n +--R ((a n + a )x + a b n x - b )(a x + b) +--R / +--R 2 2 2 2 +--R a n + 3a n + 2a +--R Type: Expression Integer +--E + +--S 101 +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ +\section{\cite{1}:14.82~~~~~$\displaystyle +\int{x^2(ax+b)^n~dx}$} +$$\int{x^2(ax+b)^n}= +\frac{(ax+b)^{n+2}}{(n+3)a^3}- +\frac{2b(ax+b)^{n+2}}{(n+2)a^3}+ +\frac{b^2(ax+b)^{n+1}}{(n+1)a^3} +{\rm\ provided\ }n \ne -1,-2,-3 +$$ + +<<*>>= +)clear all +--S 102 +aa:=integrate(x^2*(a*x+b)^n,x) +--R +--R (1) +--R 3 2 3 3 3 2 2 2 2 2 3 n log(a x + b) +--R ((a n + 3a n + 2a )x + (a b n + a b n)x - 2a b n x + 2b )%e +--R ----------------------------------------------------------------------------- +--R 3 3 3 2 3 3 +--R a n + 6a n + 11a n + 6a +--R Type: Union(Expression Integer,...) +--E + +--S 103 +bb:=(a*x+b)^(n+3)/((n+3)*a^3)-(2*b*(a*x+b)^(n+2))/((n+2)*a^3)+(b^2*(a*x+b)^(n+1))/((n+1)*a^3) +--R +--R (2) +--R 2 n + 3 2 n + 2 +--R (n + 3n + 2)(a x + b) + (- 2b n - 8b n - 6b)(a x + b) +--R + +--R 2 2 2 2 n + 1 +--R (b n + 5b n + 6b )(a x + b) +--R / +--R 3 3 3 2 3 3 +--R a n + 6a n + 11a n + 6a +--R Type: Expression Integer +--E + +--S 104 +cc:=aa-bb +--R +--R (3) +--R 3 2 3 3 3 2 2 2 2 2 3 +--R ((a n + 3a n + 2a )x + (a b n + a b n)x - 2a b n x + 2b ) +--R * +--R n log(a x + b) +--R %e +--R + +--R 2 n + 3 2 n + 2 +--R (- n - 3n - 2)(a x + b) + (2b n + 8b n + 6b)(a x + b) +--R + +--R 2 2 2 2 n + 1 +--R (- b n - 5b n - 6b )(a x + b) +--R / +--R 3 3 3 2 3 3 +--R a n + 6a n + 11a n + 6a +--R Type: Expression Integer +--E + +--S 105 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 106 +dd:=explog cc +--R +--R (5) +--R 2 n + 3 2 n + 2 +--R (- n - 3n - 2)(a x + b) + (2b n + 8b n + 6b)(a x + b) +--R + +--R 2 2 2 2 n + 1 +--R (- b n - 5b n - 6b )(a x + b) +--R + +--R 3 2 3 3 3 2 2 2 2 2 3 n +--R ((a n + 3a n + 2a )x + (a b n + a b n)x - 2a b n x + 2b )(a x + b) +--R / +--R 3 3 3 2 3 3 +--R a n + 6a n + 11a n + 6a +--R Type: Expression Integer +--E + +--S 107 14:82 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ +\section{\cite{1}:14.83~~~~~$\displaystyle +\int{x^m(ax+b)^n}~dx$} +$$\int{x^m(ax+b)^n} +\left\{ +\begin{array}{l} +\displaystyle +\frac{x^{m+1}(ax+b)^n}{m+n+1} ++\frac{nb}{m+n+1}\int{x^m(ax+b)^{n-1}}\\ +\\ +\displaystyle +\frac{x^{m+1}(ax+b)^{n+1}}{(m+n+1)a} +-\frac{mb}{(m+n+1)a}\int{x^{m-1}(ax+b)^n}\\ +\\ +\displaystyle +\frac{-x^{m+1}(ax+b)^{n+1}}{(n+1)b} ++\frac{m+n+2}{(n+1)b}\int{x^m(ax+b)^{n+1}}\\ +\end{array} +\right. +$$ + +<<*>>= +--S 108 14:83 Axiom cannot do this integration +aa:=integrate(x^m*(a*x+b)^n,x) +--R +--R x +--R ++ m n +--I (1) | %U (b + %U a) d%U +--R ++ +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp60-61 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum1.input.pdf b/src/axiom-website/CATS/schaum1.input.pdf new file mode 100644 index 0000000..1c316b5 --- /dev/null +++ b/src/axiom-website/CATS/schaum1.input.pdf @@ -0,0 +1,3029 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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+\section{\cite{1}:14.210~~~~~$\displaystyle\int{\frac{dx}{\sqrt{x^2-a^2}}}$} +$$\int{\frac{1}{\sqrt{x^2-a^2}}}=\ln\left(x+\sqrt{x^2-a^2}\right)$$ +<<*>>= +)spool schaum10.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(1/(sqrt(x^2-a^2)),x) +--R +--R +--R +-------+ +--R | 2 2 +--R (1) - log(\|x - a - x) +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +--R | 2 2 +--R (2) log(\|x - a + x) +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R (3) - log(\|x - a + x) - log(\|x - a - x) +--R Type: Expression Integer +--E + +--S 4 +logmul1:=rule(c*log(a)+c*log(b) == c*log(a*b)) +--R +--I (4) c log(b) + c log(a) + %I == c log(a b) + %I +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 5 14:210 Schaums and Axiom differ by a constant +dd:=logmul1 cc +--R +--R 2 +--R (5) - log(- a ) +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.xxx~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{x^2-a^2}}}$} +$$\int{\frac{x}{\sqrt{x^2-a^2}}}=\sqrt{x^2-a^2}$$ +<<*>>= +)clear all + +--S 6 +aa:=integrate(x/(sqrt(x^2-a^2)),x) +--R +--R +--R +-------+ +--R | 2 2 2 2 +--R - x\|x - a + x - a +--R (1) ----------------------- +--R +-------+ +--R | 2 2 +--R \|x - a - x +--R Type: Union(Expression Integer,...) +--E + +--S 7 +bb:=sqrt(x^2-a^2) +--R +--R +-------+ +--R | 2 2 +--R (2) \|x - a +--R Type: Expression Integer +--E + +--S 8 14:xxx Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.211~~~~~$\displaystyle +\int{\frac{x^2~dx}{\sqrt{x^2-a^2}}}$} +$$\int{\frac{x^2}{\sqrt{x^2-a^2}}}= +\frac{x\sqrt{x^2-a^2}}{2}+\frac{a^2}{2}\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 9 +aa:=integrate(x^2/sqrt(x^2-a^2),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 2 | 2 2 2 2 4 | 2 2 +--R (- 2a x\|x - a + 2a x - a )log(\|x - a - x) +--R + +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (- 2x + a x)\|x - a + 2x - 2a x +--R / +--R +-------+ +--R | 2 2 2 2 +--R 4x\|x - a - 4x + 2a +--R Type: Union(Expression Integer,...) +--E + +--S 10 +bb:=(x*sqrt(x^2-a^2))/2+a^2/2*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 2 | 2 2 | 2 2 +--R a log(\|x - a + x) + x\|x - a +--R (2) ----------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 11 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R - a log(\|x - a + x) - a log(\|x - a - x) +--R (3) ----------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 12 14:211 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 2 +--R a log(- a ) +--R (4) - ----------- +--R 2 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.212~~~~~$\displaystyle +\int{\frac{x^3~dx}{\sqrt{x^2-a^2}}}$} +$$\int{\frac{x^3}{\sqrt{x^2-a^2}}}= +\frac{(x^2-a^2)^{3/2}}{3}+a^2\sqrt{x^2-a^2} +$$ +<<*>>= +)clear all + +--S 13 +aa:=integrate(x^3/sqrt(x^2-a^2),x) +--R +--R +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x - 5a x + 6a x)\|x - a + 4x + 3a x - 9a x + 2a +--R (1) ------------------------------------------------------------ +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (12x - 3a )\|x - a - 12x + 9a x +--R Type: Union(Expression Integer,...) +--E + +--S 14 +bb:=(x^2-a^2)^(3/2)/3+a^2*sqrt(x^2-a^2) +--R +--R +-------+ +--R 2 2 | 2 2 +--R (x + 2a )\|x - a +--R (2) -------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 15 14:212 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.213~~~~~$\displaystyle\int{\frac{dx}{x\sqrt{x^2-a^2}}}$} +$$\int{\frac{1}{x\sqrt{x^2-a^2}}}= +\frac{1}{a}\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 16 +aa:=integrate(1/(x*sqrt(x^2-a^2)),x) +--R +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x +--R 2atan(--------------) +--R a +--R (1) --------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 17 +bb:=1/a*asec(x/a) +--R +--R x +--R asec(-) +--R a +--R (2) ------- +--R a +--R Type: Expression Integer +--E + +--S 18 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R 2atan(--------------) - asec(-) +--R a a +--R (3) ------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 19 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (4) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 20 +dd:=asecrule cc +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x \|x - a - x +--R - 2%i log(------------------) + 4atan(--------------) - %pi +--R x a +--R (5) ----------------------------------------------------------- +--R 2a +--R Type: Expression Complex Integer +--E + +--S 21 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 22 +ee:=atanrule dd +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R - 2%i log(------------------) - 2%i log(-----------------------) - %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R (7) ---------------------------------------------------------------------- +--R 2a +--R Type: Expression Complex Integer +--E + +--S 23 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2%i log(\|x - a - x + %i a) - 2%i log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R - 2%i log(x |------- + %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R | 2 +--R \| x +--R / +--R 2a +--R Type: Expression Complex Integer +--E + +--S 24 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 2%i log(\|x - a + %i a) + 2%i log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R - 2%i log(\|x - a - x - %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R / +--R 2a +--R Type: Expression Complex Integer +--E + +--S 25 14:213 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R %pi +--R (10) - --- +--R 2a +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.214~~~~~$\displaystyle +\int{\frac{dx}{x^2\sqrt{x^2-a^2}}}$} +$$\int{\frac{1}{x^2\sqrt{x^2-a^2}}}= +\frac{\sqrt{x^2-a^2}}{a^2x} +$$ +<<*>>= +)clear all + +--S 26 +aa:=integrate(1/(x^2*sqrt(x^2-a^2)),x) +--R +--R +--R 1 +--R (1) - ---------------- +--R +-------+ +--R | 2 2 2 +--R x\|x - a - x +--R Type: Union(Expression Integer,...) +--E + +--S 27 +bb:=sqrt(x^2-a^2)/(a^2*x) +--R +--R +-------+ +--R | 2 2 +--R \|x - a +--R (2) ---------- +--R 2 +--R a x +--R Type: Expression Integer +--E + +--S 28 14:214 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 1 +--R (3) -- +--R 2 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.215~~~~~$\displaystyle\int{\frac{dx}{x^3\sqrt{x^2-a^2}}}$} +$$\int{\frac{1}{x^3\sqrt{x^2-a^2}}}= +-\frac{\sqrt{x^2-a^2}}{2a^2x^2}+\frac{1}{2a^3} +\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 29 +aa:=integrate(1/(x^3*sqrt(x^2-a^2)),x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 3 | 2 2 4 2 2 \|x - a - x +--R (4x \|x - a - 4x + 2a x )atan(--------------) +--R a +--R + +--R +-------+ +--R 2 3 | 2 2 3 3 +--R (- 2a x + a )\|x - a + 2a x - 2a x +--R / +--R +-------+ +--R 3 3 | 2 2 3 4 5 2 +--R 4a x \|x - a - 4a x + 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 30 +bb:=sqrt(x^2-a^2)/(2*a^2*x^2)+1/(2*a^3)*asec(x/a) +--R +--R +-------+ +--R | 2 2 2 x +--R a\|x - a + x asec(-) +--R a +--R (2) ----------------------- +--R 3 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 31 +cc:=aa-bb +--R +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R 2atan(--------------) - asec(-) +--R a a +--R (3) ------------------------------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 32 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 33 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R - %i log(-----------------------) - asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R (5) ------------------------------------------- +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 34 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 35 +ee:=asecrule dd +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R - 2%i log(------------------) - 2%i log(-----------------------) - %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R (7) ---------------------------------------------------------------------- +--R 3 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 36 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2%i log(\|x - a - x + %i a) - 2%i log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R - 2%i log(x |------- + %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R | 2 +--R \| x +--R / +--R 3 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 37 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 2%i log(\|x - a + %i a) + 2%i log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R - 2%i log(\|x - a - x - %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R / +--R 3 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 38 14:215 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R %pi +--R (10) - --- +--R 3 +--R 4a +--R Type: Expression Complex Integer +--E + +@ + +\section{\cite{1}:14.216~~~~~$\displaystyle\int{\sqrt{x^2-a^2}}~dx$} +$$\int{\sqrt{x^2-a^2}}= +\frac{x\sqrt{x^2-a^2}}{2}-\frac{a^2}{2}\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 39 +aa:=integrate(sqrt(x^2-a^2),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 2 | 2 2 2 2 4 | 2 2 +--R (2a x\|x - a - 2a x + a )log(\|x - a - x) +--R + +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (- 2x + a x)\|x - a + 2x - 2a x +--R / +--R +-------+ +--R | 2 2 2 2 +--R 4x\|x - a - 4x + 2a +--R Type: Union(Expression Integer,...) +--E + +--S 40 +bb:=(x*sqrt(x^2-a^2))/2-a^2/2*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 2 | 2 2 | 2 2 +--R - a log(\|x - a + x) + x\|x - a +--R (2) ------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 41 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R a log(\|x - a + x) + a log(\|x - a - x) +--R (3) --------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 42 14:216 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 2 +--R a log(- a ) +--R (4) ----------- +--R 2 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.217~~~~~$\displaystyle\int{x\sqrt{x^2-a^2}}~dx$} +$$\int{x\sqrt{x^2-a^2}}= +\frac{(x^2-a^2)^{3/2}}{3} +$$ +<<*>>= +)clear all + +--S 43 +aa:=integrate(x*sqrt(x^2-a^2),x) +--R +--R +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x + 7a x - 3a x)\|x - a + 4x - 9a x + 6a x - a +--R (1) ----------------------------------------------------------- +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (12x - 3a )\|x - a - 12x + 9a x +--R Type: Union(Expression Integer,...) +--E + +--S 44 +bb:=(x^2-a^2)^(3/2)/3 +--R +--R +-------+ +--R 2 2 | 2 2 +--R (x - a )\|x - a +--R (2) ------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 45 14:217 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.218~~~~~$\displaystyle +\int{x^2\sqrt{x^2-a^2}}~dx$} +$$\int{x^2\sqrt{x^2-a^2}}= +\frac{x(x^2-a^2)^{3/2}}{4}+\frac{a^2x\sqrt{x^2-a^2}}{8}- +\frac{a^4}{8}\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 46 +aa:=integrate(x^2*sqrt(x^2-a^2),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 4 3 6 | 2 2 4 4 6 2 8 | 2 2 +--R ((8a x - 4a x)\|x - a - 8a x + 8a x - a )log(\|x - a - x) +--R + +--R +-------+ +--R 7 2 5 4 3 6 | 2 2 8 2 6 4 4 6 2 +--R (- 16x + 24a x - 10a x + a x)\|x - a + 16x - 32a x + 20a x - 4a x +--R / +--R +-------+ +--R 3 2 | 2 2 4 2 2 4 +--R (64x - 32a x)\|x - a - 64x + 64a x - 8a +--R Type: Union(Expression Integer,...) +--E + +--S 47 +bb:=(x*(x^2-a^2)^(3/2))/4+(a^2*x*sqrt(x^2-a^2))/8-a^4/8*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 4 | 2 2 3 2 | 2 2 +--R - a log(\|x - a + x) + (2x - a x)\|x - a +--R (2) ----------------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 48 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 4 | 2 2 4 | 2 2 +--R a log(\|x - a + x) + a log(\|x - a - x) +--R (3) --------------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 49 14:218 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 4 2 +--R a log(- a ) +--R (4) ----------- +--R 8 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.219~~~~~$\displaystyle +\int{x^3\sqrt{x^2-a^2}}~dx$} +$$\int{x^3\sqrt{x^2-a^2}}= +\frac{(x^2-a^2)^{5/2}}{5}+\frac{a^2(x^2-a^2)^{3/2}}{3} +$$ +<<*>>= +)clear all + +--S 50 +aa:=integrate(x^3*sqrt(x^2-a^2),x) +--R +--R +--R (1) +--R +-------+ +--R 9 2 7 4 5 6 3 8 | 2 2 10 2 8 +--R (- 48x + 76a x - 3a x - 35a x + 10a x)\|x - a + 48x - 100a x +--R + +--R 4 6 6 4 8 2 10 +--R 35a x + 40a x - 25a x + 2a +--R / +--R +-------+ +--R 4 2 2 4 | 2 2 5 2 3 4 +--R (240x - 180a x + 15a )\|x - a - 240x + 300a x - 75a x +--R Type: Union(Expression Integer,...) +--E + +--S 51 +bb:=(x^2-a^2)^(5/2)/5+(a^2*(x^2-a^2)^(3/2))/3 +--R +--R +-------+ +--R 4 2 2 4 | 2 2 +--R (3x - a x - 2a )\|x - a +--R (2) ---------------------------- +--R 15 +--R Type: Expression Integer +--E + +--S 52 14:219 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.220~~~~~$\displaystyle +\int{\frac{\sqrt{x^2-a^2}}{x}}~dx$} +$$\int{\frac{\sqrt{x^2-a^2}}{x}}= +\sqrt{x^2-a^2}-a\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 53 +aa:=integrate(sqrt(x^2-a^2)/x,x) +--R +--R +--R +-------+ +--R +-------+ | 2 2 +-------+ +--R | 2 2 \|x - a - x | 2 2 2 2 +--R (- 2a\|x - a + 2a x)atan(--------------) - x\|x - a + x - a +--R a +--R (1) ------------------------------------------------------------------- +--R +-------+ +--R | 2 2 +--R \|x - a - x +--R Type: Union(Expression Integer,...) +--E + +--S 54 +bb:=sqrt(x^2-a^2)-a*asec(x/a) +--R +--R +-------+ +--R | 2 2 x +--R (2) \|x - a - a asec(-) +--R a +--R Type: Expression Integer +--E + +--S 55 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R (3) - 2a atan(--------------) + a asec(-) +--R a a +--R Type: Expression Integer +--E + +--S 56 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 57 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R (5) %i a log(-----------------------) + a asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R Type: Expression Complex Integer +--E + +--S 58 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 59 +ee:=asecrule dd +--R +--R (7) +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R 2%i a log(------------------) + 2%i a log(-----------------------) + a %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R -------------------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 60 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 2%i a log(\|x - a - x + %i a) + 2%i a log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R 2%i a log(x |------- + %i a) - 2%i a log(x) + 2%i a log(- 1) + a %pi +--R | 2 +--R \| x +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 61 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2%i a log(\|x - a + %i a) - 2%i a log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R 2%i a log(\|x - a - x - %i a) - 2%i a log(x) + 2%i a log(- 1) + a %pi +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 62 14:220 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R a %pi +--R (10) ----- +--R 2 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.221~~~~~$\displaystyle +\int{\frac{\sqrt{x^2-a^2}}{x^2}}~dx$} +$$\int{\frac{\sqrt{x^2-a^2}}{x^2}}= +-\frac{\sqrt{x^2-a^2}}{x}+\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 63 +aa:=integrate(sqrt(x^2-a^2)/x^2,x) +--R +--R +--R +-------+ +-------+ +--R | 2 2 2 | 2 2 2 +--R (- x\|x - a + x )log(\|x - a - x) + a +--R (1) -------------------------------------------- +--R +-------+ +--R | 2 2 2 +--R x\|x - a - x +--R Type: Union(Expression Integer,...) +--E + +--S 64 +bb:=-sqrt(x^2-a^2)/x+log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R x log(\|x - a + x) - \|x - a +--R (2) ---------------------------------- +--R x +--R Type: Expression Integer +--E + +--S 65 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R (3) - log(\|x - a + x) - log(\|x - a - x) - 1 +--R Type: Expression Integer +--E + +--S 66 14:221 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 +--R (4) - log(- a ) - 1 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.222~~~~~$\displaystyle +\int{\frac{\sqrt{x^2-a^2}}{x^3}}~dx$} +$$\int{\frac{\sqrt{x^2-a^2}}{x^3}}= +-\frac{\sqrt{x^2-a^2}}{2x^2}+\frac{1}{2a} +\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 67 +aa:=integrate(sqrt(x^2-a^2)/x^3,x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 3 | 2 2 4 2 2 \|x - a - x +--R (4x \|x - a - 4x + 2a x )atan(--------------) +--R a +--R + +--R +-------+ +--R 2 3 | 2 2 3 3 +--R (2a x - a )\|x - a - 2a x + 2a x +--R / +--R +-------+ +--R 3 | 2 2 4 3 2 +--R 4a x \|x - a - 4a x + 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 68 +bb:=-sqrt(x^2-a^2)/(2*x^2)+1/(2*a)*asec(x/a) +--R +--R +-------+ +--R | 2 2 2 x +--R - a\|x - a + x asec(-) +--R a +--R (2) ------------------------- +--R 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 69 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R 2atan(--------------) - asec(-) +--R a a +--R (3) ------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 70 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (4) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 71 +dd:=asecrule cc +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x \|x - a - x +--R - 2%i log(------------------) + 4atan(--------------) - %pi +--R x a +--R (5) ----------------------------------------------------------- +--R 4a +--R Type: Expression Complex Integer +--E + +--S 72 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 73 +ee:=atanrule dd +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R - 2%i log(------------------) - 2%i log(-----------------------) - %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R (7) ---------------------------------------------------------------------- +--R 4a +--R Type: Expression Complex Integer +--E + +--S 74 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2%i log(\|x - a - x + %i a) - 2%i log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R - 2%i log(x |------- + %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R | 2 +--R \| x +--R / +--R 4a +--R Type: Expression Complex Integer +--E + +--S 75 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 2%i log(\|x - a + %i a) + 2%i log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R - 2%i log(\|x - a - x - %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R / +--R 4a +--R Type: Expression Complex Integer +--E + +--S 76 14:222 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R %pi +--R (10) - --- +--R 4a +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.223~~~~~$\displaystyle\int{\frac{dx}{(x^2-a^2)^{3/2}}}$} +$$\int{\frac{1}{(x^2-a^2)^{3/2}}}= +-\frac{x}{a^2\sqrt{x^2-a^2}} +$$ +<<*>>= +)clear all + +--S 77 +aa:=integrate(1/(x^2-a^2)^(3/2),x) +--R +--R +--R 1 +--R (1) - --------------------- +--R +-------+ +--R | 2 2 2 2 +--R x\|x - a - x + a +--R Type: Union(Expression Integer,...) +--E + +--S 78 +bb:=-x/(a^2*sqrt(x^2-a^2)) +--R +--R x +--R (2) - ------------ +--R +-------+ +--R 2 | 2 2 +--R a \|x - a +--R Type: Expression Integer +--E + +--S 79 14:223 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 1 +--R (3) - -- +--R 2 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.224~~~~~$\displaystyle +\int{\frac{x~dx}{(x^2-a^2)^{3/2}}}$} +$$\int{\frac{x}{(x^2-a^2)^{3/2}}}= +\frac{-1}{\sqrt{x^2-a^2}} +$$ +<<*>>= +)clear all + +--S 80 +aa:=integrate(x/(x^2-a^2)^(3/2),x) +--R +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x +--R (1) --------------------- +--R +-------+ +--R | 2 2 2 2 +--R x\|x - a - x + a +--R Type: Union(Expression Integer,...) +--E + +--S 81 +bb:=-1/sqrt(x^2-a^2) +--R +--R 1 +--R (2) - ---------- +--R +-------+ +--R | 2 2 +--R \|x - a +--R Type: Expression Integer +--E + +--S 82 14:224 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.225~~~~~$\displaystyle +\int{\frac{x^2dx}{(x^2-a^2)^{3/2}}}$} +$$\int{\frac{x^2}{(x^2-a^2)^{3/2}}}= +\frac{-x}{\sqrt{x^2-a^2}}+\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 83 +aa:=integrate(x^2/(x^2-a^2)^(3/2),x) +--R +--R +--R +-------+ +-------+ +--R | 2 2 2 2 | 2 2 2 +--R (- x\|x - a + x - a )log(\|x - a - x) - a +--R (1) ------------------------------------------------- +--R +-------+ +--R | 2 2 2 2 +--R x\|x - a - x + a +--R Type: Union(Expression Integer,...) +--E + +--S 84 +bb:=-x/sqrt(x^2-a^2)+log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R \|x - a log(\|x - a + x) - x +--R (2) --------------------------------- +--R +-------+ +--R | 2 2 +--R \|x - a +--R Type: Expression Integer +--E + +--S 85 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R (3) - log(\|x - a + x) - log(\|x - a - x) - 1 +--R Type: Expression Integer +--E + +--S 86 14:225 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 +--R (4) - log(- a ) - 1 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.226~~~~~$\displaystyle +\int{\frac{x^3dx}{(x^2-a^2)^{3/2}}}$} +$$\int{\frac{x^3}{(x^2-a^2)^{3/2}}}= +\sqrt{x^2-a^2}-\frac{a^2}{\sqrt{x^2-a^2}} +$$ +<<*>>= +)clear all + +--S 87 +aa:=integrate(x^3/(x^2-a^2)^(3/2),x) +--R +--R +--R +-------+ +--R 3 2 | 2 2 4 2 2 4 +--R (- 2x + 4a x)\|x - a + 2x - 5a x + 2a +--R (1) -------------------------------------------- +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (2x - a )\|x - a - 2x + 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 88 +bb:=sqrt(x^2-a^2)-a^2/sqrt(x^2-a^2) +--R +--R 2 2 +--R x - 2a +--R (2) ---------- +--R +-------+ +--R | 2 2 +--R \|x - a +--R Type: Expression Integer +--E + +--S 89 14:226 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.227~~~~~$\displaystyle +\int{\frac{dx}{x(x^2-a^2)^{3/2}}}$} +$$\int{\frac{1}{x(x^2-a^2)^{3/2}}}= +\frac{-1}{a^2\sqrt{x^2-a^2}}- +\frac{1}{a^3}\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 90 +aa:=integrate(1/(x*(x^2-a^2)^(3/2)),x) +--R +--R +--R +-------+ +--R +-------+ | 2 2 +-------+ +--R | 2 2 2 2 \|x - a - x | 2 2 +--R (- 2x\|x - a + 2x - 2a )atan(--------------) + a\|x - a - a x +--R a +--R (1) -------------------------------------------------------------------- +--R +-------+ +--R 3 | 2 2 3 2 5 +--R a x\|x - a - a x + a +--R Type: Union(Expression Integer,...) +--E + +--S 91 +bb:=-1/(a^2*sqrt(x^2-a^2))-1/a^3*asec(x/a) +--R +--R +-------+ +--R x | 2 2 +--R - asec(-)\|x - a - a +--R a +--R (2) ----------------------- +--R +-------+ +--R 3 | 2 2 +--R a \|x - a +--R Type: Expression Integer +--E + +--S 92 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R - 2atan(--------------) + asec(-) +--R a a +--R (3) --------------------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 93 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 94 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R %i log(-----------------------) + asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R (5) ----------------------------------------- +--R 3 +--R a +--R Type: Expression Complex Integer +--E + +--S 95 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 96 +ee:=asecrule dd +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R 2%i log(------------------) + 2%i log(-----------------------) + %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R (7) -------------------------------------------------------------------- +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 97 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 2%i log(\|x - a - x + %i a) + 2%i log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R 2%i log(x |------- + %i a) - 2%i log(x) + 2%i log(- 1) + %pi +--R | 2 +--R \| x +--R / +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 98 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2%i log(\|x - a + %i a) - 2%i log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R 2%i log(\|x - a - x - %i a) - 2%i log(x) + 2%i log(- 1) + %pi +--R / +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 99 14:227 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R %pi +--R (10) --- +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + +@ + +\section{\cite{1}:14.228~~~~~$\displaystyle +\int{\frac{dx}{x^2(x^2-a^2)^{3/2}}}$} +$$\int{\frac{1}{x^2(x^2-a^2)^{3/2}}}= +-\frac{\sqrt{x^2-a^2}}{a^4x}-\frac{x}{a^4\sqrt{x^2-a^2}} +$$ +<<*>>= +)clear all + +--S 100 +aa:=integrate(1/(x^2*(x^2-a^2)^(3/2)),x) +--R +--R +--R 1 +--R (1) - ----------------------------------- +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (2x - a x)\|x - a - 2x + 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 101 +bb:=-sqrt(x^2-a^2)/(a^4*x)-x/(a^4*sqrt(x^2-a^2)) +--R +--R 2 2 +--R - 2x + a +--R (2) ------------- +--R +-------+ +--R 4 | 2 2 +--R a x\|x - a +--R Type: Expression Integer +--E + +--S 102 14:228 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 2 +--R (3) - -- +--R 4 +--R a +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.229~~~~~$\displaystyle +\int{\frac{dx}{x^3(x^2-a^2)^{3/2}}}$} +$$\int{\frac{1}{x^3(x^2-a^2)^{3/2}}}= +\frac{1}{2a^2x^2\sqrt{x^2-a^2}}- +\frac{3}{2a^4\sqrt{x^2-a^2}}- +\frac{3}{2a^5}\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 103 +aa:=integrate(1/(x^3*(x^2-a^2)^(3/2)),x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 5 2 3 | 2 2 6 2 4 4 2 \|x - a - x +--R ((- 24x + 18a x )\|x - a + 24x - 30a x + 6a x )atan(--------------) +--R a +--R + +--R +-------+ +--R 4 3 2 5 | 2 2 5 3 3 5 +--R (12a x - 7a x + a )\|x - a - 12a x + 13a x - 3a x +--R / +--R +-------+ +--R 5 5 7 3 | 2 2 5 6 7 4 9 2 +--R (8a x - 6a x )\|x - a - 8a x + 10a x - 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 104 +bb:=1/(2*a^2*x^2*sqrt(x^2-a^2))-3/(2*a^4*sqrt(x^2-a^2))-3/(2*a^5)*asec(x/a) +--R +--R +-------+ +--R 2 x | 2 2 2 3 +--R - 3x asec(-)\|x - a - 3a x + a +--R a +--R (2) ----------------------------------- +--R +-------+ +--R 5 2 | 2 2 +--R 2a x \|x - a +--R Type: Expression Integer +--E + +--S 105 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R - 6atan(--------------) + 3asec(-) +--R a a +--R (3) ---------------------------------- +--R 5 +--R 2a +--R Type: Expression Integer +--E + +--S 106 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 107 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R 3%i log(-----------------------) + 3asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R (5) ------------------------------------------- +--R 5 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 108 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 109 +ee:=asecrule dd +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R 6%i log(------------------) + 6%i log(-----------------------) + 3%pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R (7) --------------------------------------------------------------------- +--R 5 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 110 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 6%i log(\|x - a - x + %i a) + 6%i log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R 6%i log(x |------- + %i a) - 6%i log(x) + 6%i log(- 1) + 3%pi +--R | 2 +--R \| x +--R / +--R 5 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 111 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 6%i log(\|x - a + %i a) - 6%i log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R 6%i log(\|x - a - x - %i a) - 6%i log(x) + 6%i log(- 1) + 3%pi +--R / +--R 5 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 112 14:229 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R 3%pi +--R (10) ---- +--R 5 +--R 4a +--R Type: Expression Complex Integer +--E +@ +\section{\cite{1}:14.230~~~~~$\displaystyle\int{(x^2-a^2)^{3/2}}~dx$} +$$\int{(x^2-a^2)^{3/2}}= +\frac{x(x^2-a^2)^{3/2}}{4}-\frac{3a^2x\sqrt{x^2-a^2}}{8}+ +\frac{3}{8}a^4\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 113 +aa:=integrate((x^2-a^2)^(3/2),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 4 3 6 | 2 2 4 4 6 2 8 | 2 2 +--R ((- 24a x + 12a x)\|x - a + 24a x - 24a x + 3a )log(\|x - a - x) +--R + +--R +-------+ +--R 7 2 5 4 3 6 | 2 2 8 2 6 4 4 +--R (- 16x + 56a x - 42a x + 5a x)\|x - a + 16x - 64a x + 68a x +--R + +--R 6 2 +--R - 20a x +--R / +--R +-------+ +--R 3 2 | 2 2 4 2 2 4 +--R (64x - 32a x)\|x - a - 64x + 64a x - 8a +--R Type: Union(Expression Integer,...) +--E + +--S 114 +bb:=(x*(x^2-a^2)^(3/2))/4-(3*a^2*x*sqrt(x^2-a^2))/8+3/8*a^4*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 4 | 2 2 3 2 | 2 2 +--R 3a log(\|x - a + x) + (2x - 5a x)\|x - a +--R (2) ----------------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 115 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 4 | 2 2 4 | 2 2 +--R - 3a log(\|x - a + x) - 3a log(\|x - a - x) +--R (3) ------------------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 116 14:230 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 4 2 +--R 3a log(- a ) +--R (4) - ------------ +--R 8 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.231~~~~~$\displaystyle\int{x(x^2-a^2)^{3/2}}~dx$} +$$\int{x(x^2-a^2)^{3/2}}=\frac{(x^2-a^2)^{5/2}}{5}$$ +<<*>>= +)clear all + +--S 117 +aa:=integrate(x*(x^2-a^2)^(3/2),x) +--R +--R +--R (1) +--R +-------+ +--R 9 2 7 4 5 6 3 8 | 2 2 10 2 8 +--R (- 16x + 52a x - 61a x + 30a x - 5a x)\|x - a + 16x - 60a x +--R + +--R 4 6 6 4 8 2 10 +--R 85a x - 55a x + 15a x - a +--R / +--R +-------+ +--R 4 2 2 4 | 2 2 5 2 3 4 +--R (80x - 60a x + 5a )\|x - a - 80x + 100a x - 25a x +--R Type: Union(Expression Integer,...) +--E + +--S 118 +bb:=(x^2-a^2)^(5/2)/5 +--R +--R +-------+ +--R 4 2 2 4 | 2 2 +--R (x - 2a x + a )\|x - a +--R (2) --------------------------- +--R 5 +--R Type: Expression Integer +--E + +--S 119 14:231 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.232~~~~~$\displaystyle\int{x^2(x^2-a^2)^{3/2}}~dx$} +$$\int{x^2(x^2-a^2)^{3/2}}= +\frac{x(x^2-a^2)^{5/2}}{6}+\frac{a^2x(x^2-a^2)^{3/2}}{24}- +\frac{a^4x\sqrt{x^2-a^2}}{16}+ +\frac{a^6}{16}\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 120 +aa:=integrate(x^2*(x^2-a^2)^(3/2),x) +--R +--R +--R (1) +--R +-------+ +--R 6 5 8 3 10 | 2 2 6 6 8 4 10 2 +--R (- 96a x + 96a x - 18a x)\|x - a + 96a x - 144a x + 54a x +--R + +--R 12 +--R - 3a +--R * +--R +-------+ +--R | 2 2 +--R log(\|x - a - x) +--R + +--R +-------+ +--R 11 2 9 4 7 6 5 8 3 10 | 2 2 +--R (- 256x + 832a x - 912a x + 404a x - 68a x + 3a x)\|x - a +--R + +--R 12 2 10 4 8 6 6 8 4 10 2 +--R 256x - 960a x + 1296a x - 772a x + 198a x - 18a x +--R / +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (1536x - 1536a x + 288a x)\|x - a - 1536x + 2304a x - 864a x + 48a +--R Type: Union(Expression Integer,...) +--E + +--S 121 +bb:=(x*(x^2-a^2)^(5/2))/6+(a^2*x*(x^2-a^2)^(3/2))/24-(a^4*x*sqrt(x^2-a^2))/16+a^6/16*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 6 | 2 2 5 2 3 4 | 2 2 +--R 3a log(\|x - a + x) + (8x - 14a x + 3a x)\|x - a +--R (2) -------------------------------------------------------- +--R 48 +--R Type: Expression Integer +--E + +--S 122 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 6 | 2 2 6 | 2 2 +--R - a log(\|x - a + x) - a log(\|x - a - x) +--R (3) ----------------------------------------------- +--R 16 +--R Type: Expression Integer +--E + +--S 123 14:232 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 6 2 +--R a log(- a ) +--R (4) - ----------- +--R 16 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.233~~~~~$\displaystyle\int{x^3(x^2-a^2)^{3/2}}~dx$} +$$\int{x^3(x^2-a^2)^{3/2}}= +\frac{(x^2-a^2)^{7/2}}{7}+\frac{a^2(x^2-a^2)^{5/2}}{5} +$$ +<<*>>= +)clear all + +--S 124 +aa:=integrate(x^3*(x^2-a^2)^(3/2),x) +--R +--R +--R (1) +--R 13 2 11 4 9 6 7 8 5 10 3 +--R - 320x + 1072a x - 1240a x + 467a x + 112a x - 105a x +--R + +--R 12 +--R 14a x +--R * +--R +-------+ +--R | 2 2 +--R \|x - a +--R + +--R 14 2 12 4 10 6 8 8 6 10 4 12 2 +--R 320x - 1232a x + 1736a x - 973a x + 21a x + 175a x - 49a x +--R + +--R 14 +--R 2a +--R / +--R +-------+ +--R 6 2 4 4 2 6 | 2 2 7 2 5 +--R (2240x - 2800a x + 840a x - 35a )\|x - a - 2240x + 3920a x +--R + +--R 4 3 6 +--R - 1960a x + 245a x +--R Type: Union(Expression Integer,...) +--E + +--S 125 +bb:=(x^2-a^2)^(7/2)/7+(a^2*(x^2-a^2)^(5/2))/5 +--R +--R +-------+ +--R 6 2 4 4 2 6 | 2 2 +--R (5x - 8a x + a x + 2a )\|x - a +--R (2) ------------------------------------ +--R 35 +--R Type: Expression Integer +--E + +--S 126 14:233 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.234~~~~~$\displaystyle +\int{\frac{(x^2-a^2)^{3/2}}{x}}~dx$} +$$\int{\frac{(x^2-a^2)^{3/2}}{x}}= +\frac{(x^2-a^2)^{3/2}}{3}-a^2\sqrt{x^2-a^2}+ +a^3\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 127 +aa:=integrate((x^2-a^2)^(3/2)/x,x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 3 2 5 | 2 2 3 3 5 \|x - a - x +--R ((24a x - 6a )\|x - a - 24a x + 18a x)atan(--------------) +--R a +--R + +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x + 19a x - 12a x)\|x - a + 4x - 21a x + 21a x - 4a +--R / +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (12x - 3a )\|x - a - 12x + 9a x +--R Type: Union(Expression Integer,...) +--E + +--S 128 +bb:=(x^2-a^2)^(3/2)/3-a^2*sqrt(x^2-a^2)+a^3*asec(x/a) +--R +--R +-------+ +--R 2 2 | 2 2 3 x +--R (x - 4a )\|x - a + 3a asec(-) +--R a +--R (2) --------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 129 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R 3 \|x - a - x 3 x +--R (3) 2a atan(--------------) - a asec(-) +--R a a +--R Type: Expression Integer +--E + +--S 130 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (4) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 131 +dd:=asecrule cc +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R 3 \| x 3 \|x - a - x 3 +--R - 2%i a log(------------------) + 4a atan(--------------) - a %pi +--R x a +--R (5) ----------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 132 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 133 +ee:=atanrule dd +--R +--R (7) +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R 3 \| x 3 - \|x - a + x + %i a 3 +--R - 2%i a log(------------------) - 2%i a log(-----------------------) - a %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R ---------------------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 134 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R 3 | 2 2 3 | 2 2 +--R 2%i a log(\|x - a - x + %i a) - 2%i a log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R 3 |x - a 3 3 3 +--R - 2%i a log(x |------- + %i a) + 2%i a log(x) - 2%i a log(- 1) - a %pi +--R | 2 +--R \| x +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 135 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 3 | 2 2 3 | 2 2 +--R - 2%i a log(\|x - a + %i a) + 2%i a log(\|x - a - x + %i a) +--R + +--R +-------+ +--R 3 | 2 2 3 3 3 +--R - 2%i a log(\|x - a - x - %i a) + 2%i a log(x) - 2%i a log(- 1) - a %pi +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 136 14:234 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R 3 +--R a %pi +--R (10) - ----- +--R 2 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.235~~~~~$\displaystyle +\int{\frac{(x^2-a^2)^{3/2}}{x^2}}~dx$} +$$\int{\frac{(x^2-a^2)^{3/2}}{x^2}}= +-\frac{(x^2-a^2)^{3/2}}{x}+\frac{3x\sqrt{x^2-a^2}}{2}- +\frac{3}{2}a^2\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 137 +aa:=integrate((x^2-a^2)^{3/2}/x^2,x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 2 3 4 | 2 2 2 4 4 2 | 2 2 +--R ((12a x - 3a x)\|x - a - 12a x + 9a x )log(\|x - a - x) +--R + +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x + 3a x + 4a x)\|x - a + 4x - 5a x - 3a x + 2a +--R / +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (8x - 2a x)\|x - a - 8x + 6a x +--R Type: Union(Expression Integer,...) +--E + +--S 138 +bb:=-(x^2-a^2)^(3/2)/x+3*x*sqrt(x^2-a^2)/2-3/2*a^2*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 2 | 2 2 +--R - 3a x log(\|x - a + x) + (x + 2a )\|x - a +--R (2) ------------------------------------------------- +--R 2x +--R Type: Expression Integer +--E + +--S 139 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 2 +--R 3a log(\|x - a + x) + 3a log(\|x - a - x) + 2a +--R (3) ----------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 140 14:235 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 2 2 +--R 3a log(- a ) + 2a +--R (4) ------------------ +--R 2 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.236~~~~~$\displaystyle +\int{\frac{(x^2-a^2)^{3/2}}{x^3}}~dx$} +$$\int{\frac{(x^2-a^2)^{3/2}}{x^3}}= +-\frac{(x^2-a^2)^{3/2}}{2x^2}+\frac{3}{2}\sqrt{x^2-a^2}- +\frac{3}{2}a\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 141 +aa:=integrate((x^2-a^2)^(3/2)/x^3,x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 4 3 2 | 2 2 5 3 3 \|x - a - x +--R ((- 24a x + 6a x )\|x - a + 24a x - 18a x )atan(--------------) +--R a +--R + +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 8x + 2a x + 3a x)\|x - a + 8x - 6a x - 3a x + a +--R / +--R +-------+ +--R 4 2 2 | 2 2 5 2 3 +--R (8x - 2a x )\|x - a - 8x + 6a x +--R Type: Union(Expression Integer,...) +--E + +--S 142 +bb:=-(x^2-a^2)^(3/2)/(2*x^2)+(3*sqrt(x^2-a^2))/2-3/2*a*asec(x/a) +--R +--R +-------+ +--R 2 2 | 2 2 2 x +--R (2x + a )\|x - a - 3a x asec(-) +--R a +--R (2) ----------------------------------- +--R 2 +--R 2x +--R Type: Expression Integer +--E + +--S 143 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R - 6a atan(--------------) + 3a asec(-) +--R a a +--R (3) -------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 144 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 145 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R 3%i a log(-----------------------) + 3a asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R (5) ----------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 146 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 147 +ee:=asecrule dd +--R +--R (7) +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R 6%i a log(------------------) + 6%i a log(-----------------------) + 3a %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R --------------------------------------------------------------------------- +--R 4 +--R Type: Expression Complex Integer +--E + +--S 148 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 6%i a log(\|x - a - x + %i a) + 6%i a log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R 6%i a log(x |------- + %i a) - 6%i a log(x) + 6%i a log(- 1) + 3a %pi +--R | 2 +--R \| x +--R / +--R 4 +--R Type: Expression Complex Integer +--E + +--S 149 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 6%i a log(\|x - a + %i a) - 6%i a log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R 6%i a log(\|x - a - x - %i a) - 6%i a log(x) + 6%i a log(- 1) + 3a %pi +--R / +--R 4 +--R Type: Expression Complex Integer +--E + +--S 150 14:236 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R 3a %pi +--R (10) ------ +--R 4 +--R Type: Expression Complex Integer +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp68-69 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum10.input.pdf b/src/axiom-website/CATS/schaum10.input.pdf new file mode 100644 index 0000000..8271532 --- /dev/null +++ b/src/axiom-website/CATS/schaum10.input.pdf @@ -0,0 +1,3941 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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+\section{\cite{1}:14.237~~~~~$\displaystyle\int{\frac{dx}{\sqrt{a^2-x^2}}}$} +$$\int{\frac{1}{\sqrt{a^2-x^2}}}=\ln\left(x+\sqrt{a^2-x^2}\right)$$ +<<*>>= +)spool schaum11.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(1/(sqrt(a^2-x^2)),x) +--R +--R +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R (1) - 2atan(----------------) +--R x +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=asin(x/a) +--R +--R x +--R (2) asin(-) +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R \|- x + a - a x +--R (3) - 2atan(----------------) - asin(-) +--R x a +--R Type: Expression Integer +--E + +--S 4 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 5 +dd:=atanrule cc +--R +--R +---------+ +--R | 2 2 +--R - \|- x + a + %i x + a x +--R (5) %i log(-------------------------) - asin(-) +--R +---------+ a +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 6 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (6) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 7 +ee:=asinrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R \| a - \|- x + a + %i x + a +--R (7) - %i log(--------------------) + %i log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 8 +ff:=rootSimp ee +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R %i\|x - a - %i x - \|x - a + x - %i a +--R (8) - %i log(-------------------) + %i log(-----------------------) +--R a +-------+ +--R | 2 2 +--R \|x - a + x + %i a +--R Type: Expression Complex Integer +--E + +--S 9 14:238 Schaums and Axiom agree +gg:=complexNormalize ff +--R +--R (9) 0 +--R Type: Expression Complex Integer +--E + +@ + +\section{\cite{1}:14.238~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{a^2-x^2}}}$} +$$\int{\frac{x}{\sqrt{a^2-x^2}}}=\sqrt{a^2-x^2}$$ +<<*>>= +)clear all + +--S 10 +aa:=integrate(x/(sqrt(a^2-x^2)),x) +--R +--R +--R 2 +--R x +--R (1) ---------------- +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R Type: Union(Expression Integer,...) +--E + +--S 11 +bb:=-sqrt(a^2-x^2) +--R +--R +---------+ +--R | 2 2 +--R (2) - \|- x + a +--R Type: Expression Integer +--E + +--S 12 14:238 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R (3) - a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.239~~~~~$\displaystyle +\int{\frac{x^2~dx}{\sqrt{a^2-x^2}}}$} +$$\int{\frac{x^2}{\sqrt{a^2-x^2}}}= +\frac{x\sqrt{a^2-x^2}}{2}+\frac{a^2}{2}\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 13 +aa:=integrate(x^2/sqrt(a^2-x^2),x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 3 | 2 2 2 2 4 \|- x + a - a +--R (- 4a \|- x + a - 2a x + 4a )atan(----------------) +--R x +--R + +--R +---------+ +--R 3 2 | 2 2 3 3 +--R (- x + 2a x)\|- x + a + 2a x - 2a x +--R / +--R +---------+ +--R | 2 2 2 2 +--R 4a\|- x + a + 2x - 4a +--R Type: Union(Expression Integer,...) +--E + +--S 14 +bb:=-(x*sqrt(a^2-x^2))/2+a^2/2*asin(x/a) +--R +--R +---------+ +--R | 2 2 2 x +--R - x\|- x + a + a asin(-) +--R a +--R (2) --------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 15 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 2 \|- x + a - a 2 x +--R - 2a atan(----------------) - a asin(-) +--R x a +--R (3) --------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 16 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 17 +dd:=atanrule cc +--R +--R +---------+ +--R | 2 2 +--R 2 - \|- x + a + %i x + a 2 x +--R %i a log(-------------------------) - a asin(-) +--R +---------+ a +--R | 2 2 +--R \|- x + a + %i x - a +--R (5) ----------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 18 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (6) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 19 +ee:=asinrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 2 \| a 2 - \|- x + a + %i x + a +--R - %i a log(--------------------) + %i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R (7) ---------------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 20 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 2 |- x + a 2 | 2 2 +--R - %i a log(a |--------- - %i x) - %i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 2 | 2 2 2 2 +--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 21 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 2 | 2 2 2 2 +--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 22 14:239 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E + +@ + +\section{\cite{1}:14.240~~~~~$\displaystyle +\int{\frac{x^3~dx}{\sqrt{a^2-x^2}}}$} +$$\int{\frac{x^3}{\sqrt{a^2-x^2}}}= +\frac{(a^2-x^2)^{3/2}}{3}+a^2\sqrt{a^2-x^2} +$$ +<<*>>= +)clear all + +--S 23 +aa:=integrate(x^3/sqrt(a^2-x^2),x) +--R +--R +--R +---------+ +--R 4 | 2 2 6 2 4 +--R 3a x \|- x + a + x - 3a x +--R (1) --------------------------------------- +--R +---------+ +--R 2 2 | 2 2 2 3 +--R (3x - 12a )\|- x + a - 9a x + 12a +--R Type: Union(Expression Integer,...) +--E + +--S 24 +bb:=(a^2-x^2)^(3/2)/3-a^2*sqrt(a^2-x^2) +--R +--R +---------+ +--R 2 2 | 2 2 +--R (- x - 2a )\|- x + a +--R (2) ------------------------ +--R 3 +--R Type: Expression Integer +--E + +--S 25 14:240 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 3 +--R 2a +--R (3) - --- +--R 3 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.241~~~~~$\displaystyle\int{\frac{dx}{x\sqrt{a^2-x^2}}}$} +$$\int{\frac{1}{x\sqrt{a^2-x^2}}}= +\frac{1}{a}\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 26 +aa:=integrate(1/(x*sqrt(a^2-x^2)),x) +--R +--R +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R log(----------------) +--R x +--R (1) --------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 27 +bb:=-1/a*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +--R \|- x + a + a +--R log(----------------) +--R x +--R (2) - --------------------- +--R a +--R Type: Expression Integer +--E + +--S 28 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R log(----------------) + log(----------------) +--R x x +--R (3) --------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 29 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R log(\|- x + a + a) + log(\|- x + a - a) - 2log(x) +--R (4) ------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 30 +ee:=complexNormalize dd +--R +--R x +--R 2log(-------) +--R +----+ +--R | 2 +--R \|- x +--R (5) - ------------- +--R a +--R Type: Expression Integer +--E + +--S 31 14:241 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R 2log(\|- 1 ) +--R (6) ------------ +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.242~~~~~$\displaystyle +\int{\frac{dx}{x^2\sqrt{a^2-x^2}}}$} +$$\int{\frac{1}{x^2\sqrt{a^2-x^2}}}= +\frac{\sqrt{a^2-x^2}}{a^2x} +$$ +<<*>>= +)clear all + +--S 32 +aa:=integrate(1/(x^2*sqrt(a^2-x^2)),x) +--R +--R +--R +---------+ +--R | 2 2 2 2 +--R a\|- x + a + x - a +--R (1) ----------------------- +--R +---------+ +--R 2 | 2 2 3 +--R a x\|- x + a - a x +--R Type: Union(Expression Integer,...) +--E + +--S 33 +bb:=-sqrt(a^2-x^2)/(a^2*x) +--R +--R +---------+ +--R | 2 2 +--R \|- x + a +--R (2) - ------------ +--R 2 +--R a x +--R Type: Expression Integer +--E + +--S 34 14:242 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.243~~~~~$\displaystyle\int{\frac{dx}{x^3\sqrt{a^2-x^2}}}$} +$$\int{\frac{1}{x^3\sqrt{a^2-x^2}}}= +-\frac{\sqrt{a^2-x^2}}{2a^2x^2}+\frac{1}{2a^3} +\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 35 +aa:=integrate(1/(x^3*sqrt(a^2-x^2)),x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 2 | 2 2 4 2 2 \|- x + a - a +--R (2a x \|- x + a + x - 2a x )log(----------------) +--R x +--R + +--R +---------+ +--R 2 3 | 2 2 2 2 4 +--R (- a x + 2a )\|- x + a + 2a x - 2a +--R / +--R +---------+ +--R 4 2 | 2 2 3 4 5 2 +--R 4a x \|- x + a + 2a x - 4a x +--R Type: Union(Expression Integer,...) +--E + +--S 36 +bb:=-sqrt(a^2-x^2)/(2*a^2*x^2)-1/(2*a^3)*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 \|- x + a + a | 2 2 +--R - x log(----------------) - a\|- x + a +--R x +--R (2) ----------------------------------------- +--R 3 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 37 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R log(----------------) + log(----------------) +--R x x +--R (3) --------------------------------------------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 38 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R log(\|- x + a + a) + log(\|- x + a - a) - 2log(x) +--R (4) ------------------------------------------------------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 39 +ee:=complexNormalize dd +--R +--R x +--R log(-------) +--R +----+ +--R | 2 +--R \|- x +--R (5) - ------------ +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 40 14:243 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R log(\|- 1 ) +--R (6) ----------- +--R 3 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.244~~~~~$\displaystyle\int{\sqrt{a^2-x^2}}~dx$} +$$\int{\sqrt{a^2-x^2}}= +\frac{x\sqrt{a^2-x^2}}{2}-\frac{a^2}{2}\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 41 +aa:=integrate(sqrt(a^2-x^2),x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 3 | 2 2 2 2 4 \|- x + a - a +--R (- 4a \|- x + a - 2a x + 4a )atan(----------------) +--R x +--R + +--R +---------+ +--R 3 2 | 2 2 3 3 +--R (x - 2a x)\|- x + a - 2a x + 2a x +--R / +--R +---------+ +--R | 2 2 2 2 +--R 4a\|- x + a + 2x - 4a +--R Type: Union(Expression Integer,...) +--E + +--S 42 +bb:=(x*sqrt(a^2-x^2))/2+a^2/2*asin(x/a) +--R +--R +---------+ +--R | 2 2 2 x +--R x\|- x + a + a asin(-) +--R a +--R (2) ------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 43 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 2 \|- x + a - a 2 x +--R - 2a atan(----------------) - a asin(-) +--R x a +--R (3) --------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 44 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (4) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 45 +dd:=asinrule cc +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 2 \| a 2 \|- x + a - a +--R - %i a log(--------------------) - 2a atan(----------------) +--R a x +--R (5) ------------------------------------------------------------ +--R 2 +--R Type: Expression Complex Integer +--E + +--S 46 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 47 +ee:=atanrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 2 \| a 2 - \|- x + a + %i x + a +--R - %i a log(--------------------) + %i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R (7) ---------------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 48 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 2 |- x + a 2 | 2 2 +--R - %i a log(a |--------- - %i x) - %i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 2 | 2 2 2 2 +--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 49 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 2 | 2 2 2 2 +--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 50 14:244 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.245~~~~~$\displaystyle\int{x\sqrt{a^2-x^2}}~dx$} +$$\int{x\sqrt{a^2-x^2}}= +\frac{(a^2-x^2)^{3/2}}{3} +$$ +<<*>>= +)clear all + +--S 51 +aa:=integrate(x*sqrt(a^2-x^2),x) +--R +--R +--R +---------+ +--R 4 3 2 | 2 2 6 2 4 4 2 +--R (- 3a x + 6a x )\|- x + a - x + 6a x - 6a x +--R (1) -------------------------------------------------- +--R +---------+ +--R 2 2 | 2 2 2 3 +--R (3x - 12a )\|- x + a - 9a x + 12a +--R Type: Union(Expression Integer,...) +--E + +--S 52 +bb:=-(a^2-x^2)^(3/2)/3 +--R +--R +---------+ +--R 2 2 | 2 2 +--R (x - a )\|- x + a +--R (2) --------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 53 14:245 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 3 +--R a +--R (3) - -- +--R 3 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.246~~~~~$\displaystyle +\int{x^2\sqrt{a^2-x^2}}~dx$} +$$\int{x^2\sqrt{a^2-x^2}}= +\frac{x(a^2-x^2)^{3/2}}{4}+\frac{a^2x\sqrt{a^2-x^2}}{8}- +\frac{a^4}{8}\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 54 +aa:=integrate(x^2*sqrt(a^2-x^2),x) +--R +--R +--R (1) +--R +---------+ +--R 5 2 7 | 2 2 4 4 6 2 8 +--R ((- 8a x + 16a )\|- x + a - 2a x + 16a x - 16a ) +--R * +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R atan(----------------) +--R x +--R + +--R +---------+ +--R 7 2 5 4 3 6 | 2 2 7 3 5 5 3 7 +--R (2x - 17a x + 24a x - 8a x)\|- x + a - 8a x + 28a x - 28a x + 8a x +--R / +--R +---------+ +--R 2 3 | 2 2 4 2 2 4 +--R (32a x - 64a )\|- x + a + 8x - 64a x + 64a +--R Type: Union(Expression Integer,...) +--E + +--S 55 +bb:=-(x*(a^2-x^2)^(3/2))/4+(a^2*x*sqrt(a^2-x^2))/8+a^4/8*asin(x/a) +--R +--R +---------+ +--R 3 2 | 2 2 4 x +--R (2x - a x)\|- x + a + a asin(-) +--R a +--R (2) ----------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 56 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 4 \|- x + a - a 4 x +--R - 2a atan(----------------) - a asin(-) +--R x a +--R (3) --------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 57 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 58 +dd:=atanrule cc +--R +--R +---------+ +--R | 2 2 +--R 4 - \|- x + a + %i x + a 4 x +--R %i a log(-------------------------) - a asin(-) +--R +---------+ a +--R | 2 2 +--R \|- x + a + %i x - a +--R (5) ----------------------------------------------- +--R 8 +--R Type: Expression Complex Integer +--E + +--S 59 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (6) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 60 +ee:=asinrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 4 \| a 4 - \|- x + a + %i x + a +--R - %i a log(--------------------) + %i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R (7) ---------------------------------------------------------------------- +--R 8 +--R Type: Expression Complex Integer +--E + +--S 61 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 4 |- x + a 4 | 2 2 +--R - %i a log(a |--------- - %i x) - %i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 4 | 2 2 4 4 +--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 8 +--R Type: Expression Complex Integer +--E + +--S 62 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 4 | 2 2 4 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 4 | 2 2 4 4 +--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 8 +--R Type: Expression Complex Integer +--E + +--S 63 14:246 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.247~~~~~$\displaystyle +\int{x^3\sqrt{a^2-x^2}}~dx$} +$$\int{x^3\sqrt{a^2-x^2}}= +\frac{(a^2-x^2)^{5/2}}{5}+\frac{a^2(a^2-x^2)^{3/2}}{3} +$$ +<<*>>= +)clear all + +--S 64 +aa:=integrate(x^3*sqrt(a^2-x^2),x) +--R +--R +--R (1) +--R +---------+ +--R 8 3 6 5 4 | 2 2 10 2 8 4 6 6 4 +--R (- 15a x + 65a x - 60a x )\|- x + a - 3x + 40a x - 95a x + 60a x +--R -------------------------------------------------------------------------- +--R +---------+ +--R 4 2 2 4 | 2 2 4 3 2 5 +--R (15x - 180a x + 240a )\|- x + a - 75a x + 300a x - 240a +--R Type: Union(Expression Integer,...) +--E + +--S 65 +bb:=(a^2-x^2)^(5/2)/5-(a^2*(a^2-x^2)^(3/2))/3 +--R +--R +---------+ +--R 4 2 2 4 | 2 2 +--R (3x - a x - 2a )\|- x + a +--R (2) ------------------------------ +--R 15 +--R Type: Expression Integer +--E + +--S 66 14:247 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 5 +--R 2a +--R (3) - --- +--R 15 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.248~~~~~$\displaystyle +\int{\frac{\sqrt{a^2-x^2}}{x}}~dx$} +$$\int{\frac{\sqrt{a^2-x^2}}{x}}= +\sqrt{a^2-x^2}-a\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 67 +aa:=integrate(sqrt(a^2-x^2)/x,x) +--R +--R +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 2 \|- x + a - a 2 +--R (a\|- x + a - a )log(----------------) - x +--R x +--R (1) ---------------------------------------------- +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R Type: Union(Expression Integer,...) +--E + +--S 68 +bb:=sqrt(a^2-x^2)-a*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R \|- x + a + a | 2 2 +--R (2) - a log(----------------) + \|- x + a +--R x +--R Type: Expression Integer +--E + +--S 69 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R (3) a log(----------------) + a log(----------------) + a +--R x x +--R Type: Expression Integer +--E + +--S 70 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R (4) a log(\|- x + a + a) + a log(\|- x + a - a) - 2a log(x) + a +--R Type: Expression Integer +--E + +--S 71 +ee:=complexNormalize dd +--R +--R x +--R (5) - 2a log(-------) + a +--R +----+ +--R | 2 +--R \|- x +--R Type: Expression Integer +--E + +--S 72 14:248 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R (6) 2a log(\|- 1 ) + a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.249~~~~~$\displaystyle +\int{\frac{\sqrt{a^2-x^2}}{x^2}}~dx$} +$$\int{\frac{\sqrt{a^2-x^2}}{x^2}}= +-\frac{\sqrt{a^2-x^2}}{x}+\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 73 +aa:=integrate(sqrt(a^2-x^2)/x^2,x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +---------+ +--R | 2 2 \|- x + a - a | 2 2 2 2 +--R (2x\|- x + a - 2a x)atan(----------------) + a\|- x + a + x - a +--R x +--R ----------------------------------------------------------------------- +--R +---------+ +--R | 2 2 +--R x\|- x + a - a x +--R Type: Union(Expression Integer,...) +--E + +--S 74 +bb:=-sqrt(a^2-x^2)/x-asin(x/a) +--R +--R +---------+ +--R | 2 2 x +--R - \|- x + a - x asin(-) +--R a +--R (2) -------------------------- +--R x +--R Type: Expression Integer +--E + +--S 75 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R \|- x + a - a x +--R (3) 2atan(----------------) + asin(-) +--R x a +--R Type: Expression Integer +--E + +--S 76 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (4) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 77 +dd:=asinrule cc +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R \| a \|- x + a - a +--R (5) %i log(--------------------) + 2atan(----------------) +--R a x +--R Type: Expression Complex Integer +--E + +--S 78 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 79 +ee:=atanrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R \| a - \|- x + a + %i x + a +--R (7) %i log(--------------------) - %i log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 80 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R |- x + a | 2 2 +--R %i log(a |--------- - %i x) + %i log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R | 2 2 +--R - %i log(\|- x + a - %i x - a) - %i log(a) - %i log(- 1) +--R Type: Expression Complex Integer +--E + +--S 81 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R %i log(%i\|x - a + %i x - a) + %i log(%i\|x - a - %i x) +--R + +--R +-------+ +--R | 2 2 +--R - %i log(%i\|x - a - %i x - a) - %i log(a) - %i log(- 1) +--R Type: Expression Complex Integer +--E + +--S 82 14:249 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.250~~~~~$\displaystyle +\int{\frac{\sqrt{a^2-x^2}}{x^3}}~dx$} +$$\int{\frac{\sqrt{a^2-x^2}}{x^3}}= +-\frac{\sqrt{a^2-x^2}}{2x^2}+\frac{1}{2a} +\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 83 +aa:=integrate(sqrt(a^2-x^2)/x^3,x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 2 | 2 2 4 2 2 \|- x + a - a +--R (- 2a x \|- x + a - x + 2a x )log(----------------) +--R x +--R + +--R +---------+ +--R 2 3 | 2 2 2 2 4 +--R (- a x + 2a )\|- x + a + 2a x - 2a +--R / +--R +---------+ +--R 2 2 | 2 2 4 3 2 +--R 4a x \|- x + a + 2a x - 4a x +--R Type: Union(Expression Integer,...) +--E + +--S 84 +bb:=-sqrt(a^2-x^2)/(2*x^2)+1/(2*a)*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 \|- x + a + a | 2 2 +--R x log(----------------) - a\|- x + a +--R x +--R (2) --------------------------------------- +--R 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 85 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R - log(----------------) - log(----------------) +--R x x +--R (3) ----------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 86 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R - log(\|- x + a + a) - log(\|- x + a - a) + 2log(x) +--R (4) --------------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 87 +ee:=complexNormalize dd +--R +--R x +--R log(-------) +--R +----+ +--R | 2 +--R \|- x +--R (5) ------------ +--R a +--R Type: Expression Integer +--E + +--S 88 14:250 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R log(\|- 1 ) +--R (6) - ----------- +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.251~~~~~$\displaystyle\int{\frac{dx}{(a^2-x^2)^{3/2}}}$} +$$\int{\frac{1}{(a^2-x^2)^{3/2}}}= +-\frac{x}{a^2\sqrt{a^2-x^2}} +$$ +<<*>>= +)clear all + +--S 89 +aa:=integrate(1/(a^2-x^2)^(3/2),x) +--R +--R +--R +---------+ +--R | 2 2 +--R - x\|- x + a + a x +--R (1) -------------------------- +--R +---------+ +--R 3 | 2 2 2 2 4 +--R a \|- x + a + a x - a +--R Type: Union(Expression Integer,...) +--E + +--S 90 +bb:=x/(a^2*sqrt(a^2-x^2)) +--R +--R x +--R (2) -------------- +--R +---------+ +--R 2 | 2 2 +--R a \|- x + a +--R Type: Expression Integer +--E + +--S 91 14:251 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.252~~~~~$\displaystyle +\int{\frac{x~dx}{(a^2-x^2)^{3/2}}}$} +$$\int{\frac{x}{(a^2-x^2)^{3/2}}}= +\frac{-1}{\sqrt{a^2-x^2}} +$$ +<<*>>= +)clear all + +--S 92 +aa:=integrate(x/(a^2-x^2)^(3/2),x) +--R +--R +--R 2 +--R x +--R (1) -------------------------- +--R +---------+ +--R 2 | 2 2 2 3 +--R a \|- x + a + a x - a +--R Type: Union(Expression Integer,...) +--E + +--S 93 +bb:=1/sqrt(a^2-x^2) +--R +--R 1 +--R (2) ------------ +--R +---------+ +--R | 2 2 +--R \|- x + a +--R Type: Expression Integer +--E + +--S 94 14:252 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 1 +--R (3) - +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.253~~~~~$\displaystyle +\int{\frac{x^2dx}{(a^2-x^2)^{3/2}}}$} +$$\int{\frac{x^2}{(a^2-x^2)^{3/2}}}= +\frac{-x}{\sqrt{a^2-x^2}}+\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 95 +aa:=integrate(x^2/(a^2-x^2)^(3/2),x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +---------+ +--R | 2 2 2 2 \|- x + a - a | 2 2 +--R (2a\|- x + a + 2x - 2a )atan(----------------) - x\|- x + a + a x +--R x +--R ------------------------------------------------------------------------ +--R +---------+ +--R | 2 2 2 2 +--R a\|- x + a + x - a +--R Type: Union(Expression Integer,...) +--E + +--S 96 +bb:=x/sqrt(a^2-x^2)-asin(x/a) +--R +--R +---------+ +--R x | 2 2 +--R - asin(-)\|- x + a + x +--R a +--R (2) ------------------------- +--R +---------+ +--R | 2 2 +--R \|- x + a +--R Type: Expression Integer +--E + +--S 97 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R \|- x + a - a x +--R (3) 2atan(----------------) + asin(-) +--R x a +--R Type: Expression Integer +--E + +--S 98 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 99 +dd:=atanrule cc +--R +--R +---------+ +--R | 2 2 +--R - \|- x + a + %i x + a x +--R (5) - %i log(-------------------------) + asin(-) +--R +---------+ a +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 100 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (6) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 101 +ee:=asinrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R \| a - \|- x + a + %i x + a +--R (7) %i log(--------------------) - %i log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 102 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R |- x + a | 2 2 +--R %i log(a |--------- - %i x) + %i log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R | 2 2 +--R - %i log(\|- x + a - %i x - a) - %i log(a) - %i log(- 1) +--R Type: Expression Complex Integer +--E + +--S 103 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R %i log(%i\|x - a + %i x - a) + %i log(%i\|x - a - %i x) +--R + +--R +-------+ +--R | 2 2 +--R - %i log(%i\|x - a - %i x - a) - %i log(a) - %i log(- 1) +--R Type: Expression Complex Integer +--E + +--S 104 14:253 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.254~~~~~$\displaystyle +\int{\frac{x^3dx}{(a^2-x^2)^{3/2}}}$} +$$\int{\frac{x^3}{(a^2-x^2)^{3/2}}}= +\sqrt{a^2-x^2}-\frac{a^2}{\sqrt{a^2-x^2}} +$$ +<<*>>= +)clear all + +--S 105 +aa:=integrate(x^3/(a^2-x^2)^(3/2),x) +--R +--R +--R 4 +--R x +--R (1) - ------------------------------------ +--R +---------+ +--R 2 2 | 2 2 2 3 +--R (x - 2a )\|- x + a - 2a x + 2a +--R Type: Union(Expression Integer,...) +--E + +--S 106 +bb:=sqrt(a^2-x^2)+a^2/sqrt(a^2-x^2) +--R +--R 2 2 +--R - x + 2a +--R (2) ------------ +--R +---------+ +--R | 2 2 +--R \|- x + a +--R Type: Expression Integer +--E + +--S 107 14:254 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R (3) 2a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.255~~~~~$\displaystyle +\int{\frac{dx}{x(a^2-x^2)^{3/2}}}$} +$$\int{\frac{1}{x(a^2-x^2)^{3/2}}}= +\frac{-1}{a^2\sqrt{a^2-x^2}}- +\frac{1}{a^3}\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 108 +aa:=integrate(1/(x*(a^2-x^2)^(3/2)),x) +--R +--R +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 2 2 \|- x + a - a 2 +--R (a\|- x + a + x - a )log(----------------) + x +--R x +--R (1) --------------------------------------------------- +--R +---------+ +--R 4 | 2 2 3 2 5 +--R a \|- x + a + a x - a +--R Type: Union(Expression Integer,...) +--E + +--S 109 +bb:=1/(a^2*sqrt(a^2-x^2))-1/a^3*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 \|- x + a + a +--R - \|- x + a log(----------------) + a +--R x +--R (2) --------------------------------------- +--R +---------+ +--R 3 | 2 2 +--R a \|- x + a +--R Type: Expression Integer +--E + +--S 110 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R log(----------------) + log(----------------) + 1 +--R x x +--R (3) ------------------------------------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 111 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R log(\|- x + a + a) + log(\|- x + a - a) - 2log(x) + 1 +--R (4) ----------------------------------------------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 112 +ee:=complexNormalize dd +--R +--R x +--R - 2log(-------) + 1 +--R +----+ +--R | 2 +--R \|- x +--R (5) ------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 113 14:255 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R 2log(\|- 1 ) + 1 +--R (6) ---------------- +--R 3 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.256~~~~~$\displaystyle +\int{\frac{dx}{x^2(a^2-x^2)^{3/2}}}$} +$$\int{\frac{1}{x^2(a^2-x^2)^{3/2}}}= +-\frac{\sqrt{a^2-x^2}}{a^4x}-\frac{x}{a^4\sqrt{a^2-x^2}} +$$ +<<*>>= +)clear all + +--S 114 +aa:=integrate(1/(x^2*(a^2-x^2)^(3/2)),x) +--R +--R +--R +---------+ +--R 2 3 | 2 2 4 2 2 4 +--R (4a x - 2a )\|- x + a + 2x - 5a x + 2a +--R (1) --------------------------------------------- +--R +---------+ +--R 4 3 6 | 2 2 5 3 7 +--R (a x - 2a x)\|- x + a - 2a x + 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 115 +bb:=-sqrt(a^2-x^2)/(a^4*x)+x/(a^4*sqrt(a^2-x^2)) +--R +--R 2 2 +--R 2x - a +--R (2) --------------- +--R +---------+ +--R 4 | 2 2 +--R a x\|- x + a +--R Type: Expression Integer +--E + +--S 116 14:256 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.257~~~~~$\displaystyle +\int{\frac{dx}{x^3(a^2-x^2)^{3/2}}}$} +$$\int{\frac{1}{x^3(a^2-x^2)^{3/2}}}= +\frac{1}{2a^2x^2\sqrt{a^2-x^2}}- +\frac{3}{2a^4\sqrt{a^2-x^2}}- +\frac{3}{2a^5}\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 117 +aa:=integrate(1/(x^3*(a^2-x^2)^(3/2)),x) +--R +--R +--R (1) +--R +---------+ +--R 4 3 2 | 2 2 6 2 4 4 2 +--R ((9a x - 12a x )\|- x + a + 3x - 15a x + 12a x ) +--R * +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R log(----------------) +--R x +--R + +--R +---------+ +--R 4 3 2 5 | 2 2 6 2 4 4 2 6 +--R (3a x + 5a x - 4a )\|- x + a + 2x - a x - 7a x + 4a +--R / +--R +---------+ +--R 6 4 8 2 | 2 2 5 6 7 4 9 2 +--R (6a x - 8a x )\|- x + a + 2a x - 10a x + 8a x +--R Type: Union(Expression Integer,...) +--E + +--S 118 +bb:=-1/(2*a^2*x^2*sqrt(a^2-x^2))+3/(2*a^4*sqrt(a^2-x^2))-3/(2*a^5)*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R +---------+ | 2 2 +--R 2 | 2 2 \|- x + a + a 2 3 +--R - 3x \|- x + a log(----------------) + 3a x - a +--R x +--R (2) --------------------------------------------------- +--R +---------+ +--R 5 2 | 2 2 +--R 2a x \|- x + a +--R Type: Expression Integer +--E + +--S 119 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R 3log(----------------) + 3log(----------------) + 2 +--R x x +--R (3) --------------------------------------------------- +--R 5 +--R 2a +--R Type: Expression Integer +--E + +--S 120 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R 3log(\|- x + a + a) + 3log(\|- x + a - a) - 6log(x) + 2 +--R (4) ------------------------------------------------------------- +--R 5 +--R 2a +--R Type: Expression Integer +--E + +--S 121 +ee:=complexNormalize dd +--R +--R x +--R - 3log(-------) + 1 +--R +----+ +--R | 2 +--R \|- x +--R (5) ------------------- +--R 5 +--R a +--R Type: Expression Integer +--E + +--S 122 14:257 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R 3log(\|- 1 ) + 1 +--R (6) ---------------- +--R 5 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.258~~~~~$\displaystyle\int{(a^2-x^2)^{3/2}}~dx$} +$$\int{(a^2-x^2)^{3/2}}= +\frac{x(a^2-x^2)^{3/2}}{4}-\frac{3a^2x\sqrt{a^2-x^2}}{8}+ +\frac{3}{8}a^4\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 123 +aa:=integrate((a^2-x^2)^(3/2),x) +--R +--R +--R (1) +--R +---------+ +--R 5 2 7 | 2 2 4 4 6 2 8 +--R ((- 24a x + 48a )\|- x + a - 6a x + 48a x - 48a ) +--R * +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R atan(----------------) +--R x +--R + +--R +---------+ +--R 7 2 5 4 3 6 | 2 2 7 3 5 5 3 +--R (- 2x + 21a x - 56a x + 40a x)\|- x + a + 8a x - 44a x + 76a x +--R + +--R 7 +--R - 40a x +--R / +--R +---------+ +--R 2 3 | 2 2 4 2 2 4 +--R (32a x - 64a )\|- x + a + 8x - 64a x + 64a +--R Type: Union(Expression Integer,...) +--E + +--S 124 +bb:=(x*(a^2-x^2)^(3/2))/4+(3*a^2*x*sqrt(a^2-x^2))/8+3/8*a^4*asin(x/a) +--R +--R +---------+ +--R 3 2 | 2 2 4 x +--R (- 2x + 5a x)\|- x + a + 3a asin(-) +--R a +--R (2) --------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 125 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 4 \|- x + a - a 4 x +--R - 6a atan(----------------) - 3a asin(-) +--R x a +--R (3) ---------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 126 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (4) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 127 +ee:=asinrule cc +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 4 \| a 4 \|- x + a - a +--R - 3%i a log(--------------------) - 6a atan(----------------) +--R a x +--R (5) ------------------------------------------------------------- +--R 8 +--R Type: Expression Complex Integer +--E + +--S 128 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 129 +ff:=atanrule ee +--R +--R (7) +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 4 \| a 4 - \|- x + a + %i x + a +--R - 3%i a log(--------------------) + 3%i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R ------------------------------------------------------------------------ +--R 8 +--R Type: Expression Complex Integer +--E + +--S 130 +gg:=expandLog ff +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 4 |- x + a 4 | 2 2 +--R - 3%i a log(a |--------- - %i x) - 3%i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 4 | 2 2 4 4 +--R 3%i a log(\|- x + a - %i x - a) + 3%i a log(a) + 3%i a log(- 1) +--R / +--R 8 +--R Type: Expression Complex Integer +--E + +--S 131 +hh:=rootSimp gg +--R +--R (9) +--R +-------+ +-------+ +--R 4 | 2 2 4 | 2 2 +--R - 3%i a log(%i\|x - a + %i x - a) - 3%i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 4 | 2 2 4 4 +--R 3%i a log(%i\|x - a - %i x - a) + 3%i a log(a) + 3%i a log(- 1) +--R / +--R 8 +--R Type: Expression Complex Integer +--E + +--S 132 14:258 Schaums and Axiom agree +ii:=complexNormalize hh +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.259~~~~~$\displaystyle\int{x(a^2-x^2)^{3/2}}~dx$} +$$\int{x(a^2-x^2)^{3/2}}=\frac{(a^2-x^2)^{5/2}}{5}$$ +<<*>>= +)clear all + +--S 133 +aa:=integrate(x*(a^2-x^2)^(3/2),x) +--R +--R +--R (1) +--R +---------+ +--R 8 3 6 5 4 7 2 | 2 2 10 2 8 4 6 +--R (5a x - 30a x + 60a x - 40a x )\|- x + a + x - 15a x + 55a x +--R + +--R 6 4 8 2 +--R - 80a x + 40a x +--R / +--R +---------+ +--R 4 2 2 4 | 2 2 4 3 2 5 +--R (5x - 60a x + 80a )\|- x + a - 25a x + 100a x - 80a +--R Type: Union(Expression Integer,...) +--E + +--S 134 +bb:=-(a^2-x^2)^(5/2)/5 +--R +--R +---------+ +--R 4 2 2 4 | 2 2 +--R (- x + 2a x - a )\|- x + a +--R (2) ------------------------------- +--R 5 +--R Type: Expression Integer +--E + +--S 135 14:259 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 5 +--R a +--R (3) - -- +--R 5 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.260~~~~~$\displaystyle\int{x^2(a^2-x^2)^{3/2}}~dx$} +$$\int{x^2(a^2-x^2)^{3/2}}= +\frac{x(a^2-x^2)^{5/2}}{6}+\frac{a^2x(a^2-x^2)^{3/2}}{24}- +\frac{a^4x\sqrt{a^2-x^2}}{16}+ +\frac{a^6}{16}\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 136 +aa:=integrate(x^2*(a^2-x^2)^(3/2),x) +--R +--R +--R (1) +--R +---------+ +--R 7 4 9 2 11 | 2 2 6 6 8 4 +--R (- 36a x + 192a x - 192a )\|- x + a - 6a x + 108a x +--R + +--R 10 2 12 +--R - 288a x + 192a +--R * +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R atan(----------------) +--R x +--R + +--R +---------+ +--R 11 2 9 4 7 6 5 8 3 10 | 2 2 +--R (- 8x + 158a x - 639a x + 982a x - 592a x + 96a x)\|- x + a +--R + +--R 11 3 9 5 7 7 5 9 3 11 +--R 48a x - 388a x + 1062a x - 1266a x + 640a x - 96a x +--R / +--R +---------+ +--R 4 3 2 5 | 2 2 6 2 4 4 2 +--R (288a x - 1536a x + 1536a )\|- x + a + 48x - 864a x + 2304a x +--R + +--R 6 +--R - 1536a +--R Type: Union(Expression Integer,...) +--E + +--S 137 +bb:=-(x*(a^2-x^2)^(5/2))/6+(a^2*x*(a^2-x^2)^(3/2))/24+(a^4*x*sqrt(a^2-x^2))/16+a^6/16*asin(x/a) +--R +--R +---------+ +--R 5 2 3 4 | 2 2 6 x +--R (- 8x + 14a x - 3a x)\|- x + a + 3a asin(-) +--R a +--R (2) ------------------------------------------------ +--R 48 +--R Type: Expression Integer +--E + +--S 138 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 6 \|- x + a - a 6 x +--R - 2a atan(----------------) - a asin(-) +--R x a +--R (3) --------------------------------------- +--R 16 +--R Type: Expression Integer +--E + +--S 139 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 140 +dd:=atanrule cc +--R +--R +---------+ +--R | 2 2 +--R 6 - \|- x + a + %i x + a 6 x +--R %i a log(-------------------------) - a asin(-) +--R +---------+ a +--R | 2 2 +--R \|- x + a + %i x - a +--R (5) ----------------------------------------------- +--R 16 +--R Type: Expression Complex Integer +--E + +--S 141 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (6) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 142 +ee:=asinrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 6 \| a 6 - \|- x + a + %i x + a +--R - %i a log(--------------------) + %i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R (7) ---------------------------------------------------------------------- +--R 16 +--R Type: Expression Complex Integer +--E + +--S 143 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 6 |- x + a 6 | 2 2 +--R - %i a log(a |--------- - %i x) - %i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 6 | 2 2 6 6 +--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 16 +--R Type: Expression Complex Integer +--E + +--S 144 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 6 | 2 2 6 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 6 | 2 2 6 6 +--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 16 +--R Type: Expression Complex Integer +--E + +--S 145 14:260 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.261~~~~~$\displaystyle\int{x^3(a^2-x^2)^{3/2}}~dx$} +$$\int{x^3(a^2-x^2)^{3/2}}= +\frac{(a^2-x^2)^{7/2}}{7}+\frac{a^2(a^2-x^2)^{5/2}}{5} +$$ +<<*>>= +)clear all + +--S 146 +aa:=integrate(x^3*(a^2-x^2)^(3/2),x) +--R +--R +--R (1) +--R +---------+ +--R 12 3 10 5 8 7 6 9 4 | 2 2 14 +--R (35a x - 336a x + 1015a x - 1260a x + 560a x )\|- x + a + 5x +--R + +--R 2 12 4 10 6 8 8 6 10 4 +--R - 133a x + 721a x - 1575a x + 1540a x - 560a x +--R / +--R +---------+ +--R 6 2 4 4 2 6 | 2 2 6 3 4 +--R (35x - 840a x + 2800a x - 2240a )\|- x + a - 245a x + 1960a x +--R + +--R 5 2 7 +--R - 3920a x + 2240a +--R Type: Union(Expression Integer,...) +--E + +--S 147 +bb:=(a^2-x^2)^(7/2)/7-(a^2*(a^2-x^2)^(5/2))/5 +--R +--R +---------+ +--R 6 2 4 4 2 6 | 2 2 +--R (- 5x + 8a x - a x - 2a )\|- x + a +--R (2) ---------------------------------------- +--R 35 +--R Type: Expression Integer +--E + +--S 148 14:261 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 7 +--R 2a +--R (3) - --- +--R 35 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.262~~~~~$\displaystyle +\int{\frac{(a^2-x^2)^{3/2}}{x}}~dx$} +$$\int{\frac{(a^2-x^2)^{3/2}}{x}}= +\frac{(a^2-x^2)^{3/2}}{3}-a^2\sqrt{a^2-x^2}+ +a^3\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 149 +aa:=integrate((a^2-x^2)^(3/2)/x,x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 3 2 5 | 2 2 4 2 6 \|- x + a - a +--R ((3a x - 12a )\|- x + a - 9a x + 12a )log(----------------) +--R x +--R + +--R +---------+ +--R 4 3 2 | 2 2 6 2 4 4 2 +--R (3a x - 12a x )\|- x + a + x - 9a x + 12a x +--R / +--R +---------+ +--R 2 2 | 2 2 2 3 +--R (3x - 12a )\|- x + a - 9a x + 12a +--R Type: Union(Expression Integer,...) +--E + +--S 150 +bb:=(a^2-x^2)^(3/2)/3+a^2*sqrt(a^2-x^2)-a^3*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R 3 \|- x + a + a 2 2 | 2 2 +--R - 3a log(----------------) + (- x + 4a )\|- x + a +--R x +--R (2) ----------------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 151 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R 3 \|- x + a + a 3 \|- x + a - a 3 +--R 3a log(----------------) + 3a log(----------------) + 4a +--R x x +--R (3) --------------------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 152 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R 3 | 2 2 3 | 2 2 3 3 +--R 3a log(\|- x + a + a) + 3a log(\|- x + a - a) - 6a log(x) + 4a +--R (4) --------------------------------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 153 +ee:=complexNormalize dd +--R +--R 3 x 3 +--R - 6a log(-------) + 4a +--R +----+ +--R | 2 +--R \|- x +--R (5) ----------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 154 14:262 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R 3 +---+ 3 +--R 6a log(\|- 1 ) + 4a +--R (6) -------------------- +--R 3 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.263~~~~~$\displaystyle +\int{\frac{(a^2-x^2)^{3/2}}{x^2}}~dx$} +$$\int{\frac{(a^2-x^2)^{3/2}}{x^2}}= +-\frac{(a^2-x^2)^{3/2}}{x}+\frac{3x\sqrt{a^2-x^2}}{2}- +\frac{3}{2}a^2\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 155 +aa:=integrate((a^2-x^2)^{3/2}/x^2,x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 2 3 4 | 2 2 3 3 5 \|- x + a - a +--R ((6a x - 24a x)\|- x + a - 18a x + 24a x)atan(----------------) +--R x +--R + +--R +---------+ +--R 4 3 2 5 | 2 2 6 2 4 4 2 6 +--R (3a x + 2a x - 8a )\|- x + a + x - 3a x - 6a x + 8a +--R / +--R +---------+ +--R 3 2 | 2 2 3 3 +--R (2x - 8a x)\|- x + a - 6a x + 8a x +--R Type: Union(Expression Integer,...) +--E + +--S 156 +bb:=-(a^2-x^2)^(3/2)/x-(3*x*sqrt(a^2-x^2))/2-3/2*a^2*asin(x/a) +--R +--R +---------+ +--R 2 2 | 2 2 2 x +--R (- x - 2a )\|- x + a - 3a x asin(-) +--R a +--R (2) --------------------------------------- +--R 2x +--R Type: Expression Integer +--E + +--S 157 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 2 \|- x + a - a 2 x +--R 6a atan(----------------) + 3a asin(-) +--R x a +--R (3) -------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 158 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (4) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 159 +dd:=asinrule cc +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 2 \| a 2 \|- x + a - a +--R 3%i a log(--------------------) + 6a atan(----------------) +--R a x +--R (5) ----------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 160 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 161 +ee:=atanrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 2 \| a 2 - \|- x + a + %i x + a +--R 3%i a log(--------------------) - 3%i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R (7) ---------------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 162 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 2 |- x + a 2 | 2 2 +--R 3%i a log(a |--------- - %i x) + 3%i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 2 | 2 2 2 2 +--R - 3%i a log(\|- x + a - %i x - a) - 3%i a log(a) - 3%i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 163 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R 3%i a log(%i\|x - a + %i x - a) + 3%i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 2 | 2 2 2 2 +--R - 3%i a log(%i\|x - a - %i x - a) - 3%i a log(a) - 3%i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 164 14:263 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.264~~~~~$\displaystyle +\int{\frac{(a^2-x^2)^{3/2}}{x^3}}~dx$} +$$\int{\frac{(a^2-x^2)^{3/2}}{x^3}}= +-\frac{(a^2-x^2)^{3/2}}{2x^2}+\frac{3}{2}\sqrt{a^2-x^2}- +\frac{3}{2}a\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 165 +aa:=integrate((a^2-x^2)^(3/2)/x^3,x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 4 3 2 | 2 2 2 4 4 2 \|- x + a - a +--R ((- 3a x + 12a x )\|- x + a + 9a x - 12a x )log(----------------) +--R x +--R + +--R +---------+ +--R 4 3 2 5 | 2 2 6 2 4 4 2 6 +--R (4a x + 3a x - 4a )\|- x + a + 2x - 3a x - 5a x + 4a +--R / +--R +---------+ +--R 4 2 2 | 2 2 4 3 2 +--R (2x - 8a x )\|- x + a - 6a x + 8a x +--R Type: Union(Expression Integer,...) +--E + +--S 166 +bb:=-(a^2-x^2)^(3/2)/(2*x^2)-(3*sqrt(a^2-x^2))/2+3/2*a*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 \|- x + a + a 2 2 | 2 2 +--R 3a x log(----------------) + (- 2x - a )\|- x + a +--R x +--R (2) ----------------------------------------------------- +--R 2 +--R 2x +--R Type: Expression Integer +--E + +--S 167 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R - 3a log(----------------) - 3a log(----------------) - 2a +--R x x +--R (3) ---------------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 168 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R - 3a log(\|- x + a + a) - 3a log(\|- x + a - a) + 6a log(x) - 2a +--R (4) ---------------------------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 169 +ee:=complexNormalize dd +--R +--R x +--R (5) 3a log(-------) - a +--R +----+ +--R | 2 +--R \|- x +--R Type: Expression Integer +--E + +--S 170 14:264 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R (6) - 3a log(\|- 1 ) - a +--R Type: Expression Integer +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp68-69 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum11.input.pdf b/src/axiom-website/CATS/schaum11.input.pdf new file mode 100644 index 0000000..cb7ac08 --- /dev/null +++ b/src/axiom-website/CATS/schaum11.input.pdf @@ -0,0 +1,3980 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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b/src/axiom-website/CATS/schaum12.input.pamphlet @@ -0,0 +1,2870 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum12.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.265~~~~~$\displaystyle +\int{\frac{dx}{ax^2+bx+c}}$} +$$\int{\frac{1}{ax^2+bx+c}}= +\left\{ +\begin{array}{l} +\displaystyle\frac{2}{\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}}\\ +\displaystyle\frac{1}{\sqrt{b^2-4ac}}\ln\left( +\frac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}}\right) +\end{array} +\right. +$$ +<<*>>= +)spool schaum12.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(1/(a*x^2+b*x+c),x) +--R +--R (1) +--R [ +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 2 +bb1:=2/sqrt(4*a*c-b^2)*atan((2*a*x+b)/sqrt(4*a*c-b^2)) +--R +--R +--R 2a x + b +--R 2atan(------------) +--R +---------+ +--R | 2 +--R \|4a c - b +--R (2) ------------------- +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Integer +--E + +--S 3 +bb2:=1/sqrt(b^2-4*a*c)*log((2*a*x+b-sqrt(b^2-4*a*c))/(2*a*x+b+sqrt(b^2-4*a*c))) +--R +--R +--R +-----------+ +--R | 2 +--R - \|- 4a c + b + 2a x + b +--R log(---------------------------) +--R +-----------+ +--R | 2 +--R \|- 4a c + b + 2a x + b +--R (3) -------------------------------- +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R Type: Expression Integer +--E + +--S 4 +cc1:=aa.1-bb1 +--R +--R (4) +--R +---------+ +--R | 2 +--R \|4a c - b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R | 2 2a x + b +--R - 2\|- 4a c + b atan(------------) +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +-----------+ +---------+ +--R | 2 | 2 +--R \|- 4a c + b \|4a c - b +--R Type: Expression Integer +--E + +--S 5 +cc2:=aa.1-bb2 +--R +--R (5) +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R | 2 +--R - \|- 4a c + b + 2a x + b +--R - log(---------------------------) +--R +-----------+ +--R | 2 +--R \|- 4a c + b + 2a x + b +--R / +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R Type: Expression Integer +--E + +--S 6 +cc3:=aa.2-bb1 +--R +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b 2a x + b +--R 2atan(----------------------) - 2atan(------------) +--R 2 +---------+ +--R 4a c - b | 2 +--R \|4a c - b +--R (6) --------------------------------------------------- +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Integer +--E + +--S 7 +cc4:=aa.2-bb2 +--R +--R (7) +--R +-----------+ +--R +---------+ | 2 +--R | 2 - \|- 4a c + b + 2a x + b +--R - \|4a c - b log(---------------------------) +--R +-----------+ +--R | 2 +--R \|- 4a c + b + 2a x + b +--R + +--R +---------+ +--R +-----------+ | 2 +--R | 2 (2a x + b)\|4a c - b +--R 2\|- 4a c + b atan(----------------------) +--R 2 +--R 4a c - b +--R / +--R +-----------+ +---------+ +--R | 2 | 2 +--R \|- 4a c + b \|4a c - b +--R Type: Expression Integer +--E + +--S 8 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (8) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 9 +dd3:=atanrule cc3 +--R +--R (9) +--R +---------+ +--R | 2 +--R \|4a c - b + 2%i a x + %i b +--R %i log(-----------------------------) +--R +---------+ +--R | 2 +--R \|4a c - b - 2%i a x - %i b +--R + +--R +---------+ +--R | 2 2 +--R (- 2a x - b)\|4a c - b + 4%i a c - %i b +--R - %i log(------------------------------------------) +--R +---------+ +--R | 2 2 +--R (2a x + b)\|4a c - b + 4%i a c - %i b +--R / +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Complex Integer +--E + +--S 10 +ee3:=expandLog dd3 +--R +--R (10) +--R +---------+ +--R | 2 2 +--R %i log((2a x + b)\|4a c - b + 4%i a c - %i b ) +--R + +--R +---------+ +--R | 2 2 +--R - %i log((2a x + b)\|4a c - b - 4%i a c + %i b ) +--R + +--R +---------+ +--R | 2 +--R %i log(\|4a c - b + 2%i a x + %i b) +--R + +--R +---------+ +--R | 2 +--R - %i log(\|4a c - b - 2%i a x - %i b) - %i log(- 1) +--R / +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Complex Integer +--E + +--S 11 14:265 Schaums and Axiom agree +ff3:=complexNormalize ee3 +--R +--R (11) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.266~~~~~$\displaystyle +\int{\frac{x~dx}{ax^2+bx+c}}$} +$$\int{\frac{x}{ax^2+bx+c}}= +\frac{1}{2a}\ln(ax^2+bx+c)-\frac{b}{2a}\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 12 +aa:=integrate(x/(a*x^2+b*x+c),x) +--R +--R (1) +--R [ +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R log(a x + b x + c)\|- 4a c + b +--R / +--R +-----------+ +--R | 2 +--R 2a\|- 4a c + b +--R , +--R +---------+ +--R | 2 +---------+ +--R (2a x + b)\|4a c - b 2 | 2 +--R - 2b atan(----------------------) + log(a x + b x + c)\|4a c - b +--R 2 +--R 4a c - b +--R -------------------------------------------------------------------] +--R +---------+ +--R | 2 +--R 2a\|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 13 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--R [ +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 14 +bb1:=1/(2*a)*log(a*x^2+b*x+c)-b/(2*a)*t1.1 +--R +--R +--R (3) +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R log(a x + b x + c)\|- 4a c + b +--R / +--R +-----------+ +--R | 2 +--R 2a\|- 4a c + b +--R Type: Expression Integer +--E + +--S 15 +bb2:=1/(2*a)*log(a*x^2+b*x+c)-b/(2*a)*t1.2 +--R +--R +--R +---------+ +--R | 2 +---------+ +--R (2a x + b)\|4a c - b 2 | 2 +--R - 2b atan(----------------------) + log(a x + b x + c)\|4a c - b +--R 2 +--R 4a c - b +--R (4) ------------------------------------------------------------------- +--R +---------+ +--R | 2 +--R 2a\|4a c - b +--R Type: Expression Integer +--E + +--S 16 +cc1:=aa.1-bb1 +--R +--R (5) +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R 2a\|- 4a c + b +--R Type: Expression Integer +--E + +--S 17 +cc2:=aa.2-bb1 +--R +--R (6) +--R +---------+ +--R | 2 +--R b\|4a c - b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +---------+ +--R +-----------+ | 2 +--R | 2 (2a x + b)\|4a c - b +--R - 2b\|- 4a c + b atan(----------------------) +--R 2 +--R 4a c - b +--R / +--R +-----------+ +---------+ +--R | 2 | 2 +--R 2a\|- 4a c + b \|4a c - b +--R Type: Expression Integer +--E + +--S 18 +cc3:=aa.2-bb1 +--R +--R (7) +--R +---------+ +--R | 2 +--R b\|4a c - b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +---------+ +--R +-----------+ | 2 +--R | 2 (2a x + b)\|4a c - b +--R - 2b\|- 4a c + b atan(----------------------) +--R 2 +--R 4a c - b +--R / +--R +-----------+ +---------+ +--R | 2 | 2 +--R 2a\|- 4a c + b \|4a c - b +--R Type: Expression Integer +--E + +--S 19 14:266 Schaums and Axiom agree +cc4:=aa.2-bb2 +--R +--R (8) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.267~~~~~$\displaystyle +\int{\frac{x^2dx}{ax^2+bx+c}}$} +$$\int{\frac{x^2}{ax^2+bx+c}}= +\frac{x}{a}-\frac{b}{2a^2}\ln(ax^2+bx+c)+\frac{b^2-2ac}{2a^2} +\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 20 +aa:=integrate(x^2/(a*x^2+b*x+c),x) +--R +--R +--R (1) +--R [ +--R 2 +--R (2a c - b ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R (- b log(a x + b x + c) + 2a x)\|- 4a c + b +--R / +--R +-----------+ +--R 2 | 2 +--R 2a \|- 4a c + b +--R , +--R +--R +---------+ +--R | 2 +--R 2 (2a x + b)\|4a c - b +--R (- 4a c + 2b )atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R (- b log(a x + b x + c) + 2a x)\|4a c - b +--R / +--R +---------+ +--R 2 | 2 +--R 2a \|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 21 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--R [ +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 22 +bb1:=x/a-b/(2*a^2)*log(a*x^2+b*x+c)+(b^2-2*a*c)/(2*a^2)*t1.1 +--R +--R +--R (3) +--R 2 +--R (- 2a c + b ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R (- b log(a x + b x + c) + 2a x)\|- 4a c + b +--R / +--R +-----------+ +--R 2 | 2 +--R 2a \|- 4a c + b +--R Type: Expression Integer +--E + +--S 23 +bb2:=x/a-b/(2*a^2)*log(a*x^2+b*x+c)+(b^2-2*a*c)/(2*a^2)*t1.2 +--R +--R +--R (4) +--R +---------+ +--R | 2 +--R 2 (2a x + b)\|4a c - b +--R (- 4a c + 2b )atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R (- b log(a x + b x + c) + 2a x)\|4a c - b +--R / +--R +---------+ +--R 2 | 2 +--R 2a \|4a c - b +--R Type: Expression Integer +--E + +--S 24 +cc1:=bb1-aa.1 +--R +--R (5) +--R 2 +--R (- 2a c + b ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 +--R (- 2a c + b ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R 2 | 2 +--R 2a \|- 4a c + b +--R Type: Expression Integer +--E + +--S 25 14:267 Schaums and Axiom differ by a constant +dd1:=complexNormalize cc1 +--R +--R 2 3 2 2 +--R (- 2a c + b )log(- 16a c + 4a b ) +--R (6) --------------------------------- +--R +-----------+ +--R 2 | 2 +--R 2a \|- 4a c + b +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.268~~~~~$\displaystyle +\int{\frac{x^m~dx}{ax^2+bx+c}}$} +$$\int{\frac{x^m}{ax^2+bx+c}}= +\frac{x^{m-1}}{(m-1)a}-\frac{c}{a}\int{\frac{x^{m-2}}{ax^2+bx+c}}- +\frac{b}{a}\int{\frac{x^{m-1}}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 26 14:268 Axiom cannot compute this integral +aa:=integrate(x^m/(a*x^2+b*x+c),x) +--R +--R +--R x m +--I ++ %N +--I (1) | --------------- d%N +--R ++ 2 +--I c + %N b + %N a +--R Type: Union(Expression Integer,...) +--E + +@ + +\section{\cite{1}:14.269~~~~~$\displaystyle +\int{\frac{dx}{x(ax^2+bx+c)}}$} +$$\int{\frac{1}{x(ax^2+bx+c)}}= +\frac{1}{2c}\ln\left(\frac{x^2}{ax^2+bx+c}\right)- +\frac{b}{2c}\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 27 +aa:=integrate(1/(x*(a*x^2+b*x+c)),x) +--R +--R +--R (1) +--R [ +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R (- log(a x + b x + c) + 2log(x))\|- 4a c + b +--R / +--R +-----------+ +--R | 2 +--R 2c\|- 4a c + b +--R , +--R +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R - 2b atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R (- log(a x + b x + c) + 2log(x))\|4a c - b +--R / +--R +---------+ +--R | 2 +--R 2c\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 28 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--R [ +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 29 +bb1:=1/(2*c)*log(x^2/(a*x^2+b*x+c))-b/(2*c)*t1.1 +--R +--R +--R (3) +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 +-----------+ +--R x | 2 +--R log(--------------)\|- 4a c + b +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R 2c\|- 4a c + b +--R Type: Expression Integer +--E + +--S 30 +bb2:=1/(2*c)*log(x^2/(a*x^2+b*x+c))-b/(2*c)*t1.2 +--R +--R +--R +---------+ +--R | 2 2 +---------+ +--R (2a x + b)\|4a c - b x | 2 +--R - 2b atan(----------------------) + log(--------------)\|4a c - b +--R 2 2 +--R 4a c - b a x + b x + c +--R (4) ------------------------------------------------------------------- +--R +---------+ +--R | 2 +--R 2c\|4a c - b +--R Type: Expression Integer +--E + +--S 31 +cc1:=bb1-aa.1 +--R +--R (5) +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b +--R + +--R 2 2 3 +--R (- 8a c + 2a b )x - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 +-----------+ +--R 2 x | 2 +--R (log(a x + b x + c) - 2log(x) + log(--------------))\|- 4a c + b +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R 2c\|- 4a c + b +--R Type: Expression Integer +--E + +--S 32 +dd1:=expandLog cc1 +--R +--R (6) +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R + +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R + +--R 2 +--R 2b log(a x + b x + c) +--R / +--R +-----------+ +--R | 2 +--R 2c\|- 4a c + b +--R Type: Expression Integer +--E + +--S 33 14:269 Schaums and Axiom differ by a constant +ee1:=complexNormalize dd1 +--R +--R 3 2 2 +--R b log(- 16a c + 4a b ) +--R (7) - ---------------------- +--R +-----------+ +--R | 2 +--R 2c\|- 4a c + b +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.270~~~~~$\displaystyle +\int{\frac{dx}{x^2(ax^2+bx+c)}}$} +$$\int{\frac{1}{x^2(ax^2+bx+c)}}= +\frac{b}{2c^2}\ln\left(\frac{ax^2+bx+c}{x^2}\right)-\frac{1}{cx}+ +\frac{b^2-2ac}{2c^2}\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 34 +aa:=integrate(1/(x^2*(a*x^2+b*x+c)),x) +--R +--R +--R (1) +--R [ +--R 2 +--R (2a c - b )x +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R (b x log(a x + b x + c) - 2b x log(x) - 2c)\|- 4a c + b +--R / +--R +-----------+ +--R 2 | 2 +--R 2c x\|- 4a c + b +--R , +--R +--R +---------+ +--R | 2 +--R 2 (2a x + b)\|4a c - b +--R (- 4a c + 2b )x atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R (b x log(a x + b x + c) - 2b x log(x) - 2c)\|4a c - b +--R / +--R +---------+ +--R 2 | 2 +--R 2c x\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 35 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--R [ +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 36 +bb1:=b/(2*c^2)*log((a*x^2+b*x+c)/x^2)-1/(c*x)+(b^2-2*a*c)/(2*c^2)*t1.1 +--R +--R +--R (3) +--R 2 +--R (- 2a c + b )x +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 +-----------+ +--R a x + b x + c | 2 +--R (b x log(--------------) - 2c)\|- 4a c + b +--R 2 +--R x +--R / +--R +-----------+ +--R 2 | 2 +--R 2c x\|- 4a c + b +--R Type: Expression Integer +--E + +--S 37 +bb2:=b/(2*c^2)*log((a*x^2+b*x+c)/x^2)-1/(c*x)+(b^2-2*a*c)/(2*c^2)*t1.2 +--R +--R +--R (4) +--R +---------+ +--R | 2 +--R 2 (2a x + b)\|4a c - b +--R (- 4a c + 2b )x atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 +---------+ +--R a x + b x + c | 2 +--R (b x log(--------------) - 2c)\|4a c - b +--R 2 +--R x +--R / +--R +---------+ +--R 2 | 2 +--R 2c x\|4a c - b +--R Type: Expression Integer +--E + +--S 38 +cc1:=bb1-aa.1 +--R +--R (5) +--R 2 +--R (- 2a c + b ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 +--R (- 2a c + b ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 +-----------+ +--R 2 a x + b x + c | 2 +--R (- b log(a x + b x + c) + 2b log(x) + b log(--------------))\|- 4a c + b +--R 2 +--R x +--R / +--R +-----------+ +--R 2 | 2 +--R 2c \|- 4a c + b +--R Type: Expression Integer +--E + +--S 39 +dd1:=expandLog cc1 +--R +--R (6) +--R 2 +--R (- 2a c + b ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R + +--R 2 +--R (- 2a c + b ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R + +--R 2 2 +--R (4a c - 2b )log(a x + b x + c) +--R / +--R +-----------+ +--R 2 | 2 +--R 2c \|- 4a c + b +--R Type: Expression Integer +--E + +--S 40 14:270 Schaums and Axiom differ by a constant +ee1:=complexNormalize dd1 +--R +--R 2 3 2 2 +--R (- 2a c + b )log(- 16a c + 4a b ) +--R (7) --------------------------------- +--R +-----------+ +--R 2 | 2 +--R 2c \|- 4a c + b +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.271~~~~~$\displaystyle +\int{\frac{dx}{x^n(ax^2+bx+c)}}$} +$$\int{\frac{1}{x^n(ax^2+bx+c)}}= +-\frac{1}{(n-1)cx^{n-1}}- +\frac{b}{c}\int{\frac{1}{x^{n-1}(ax^2+bx+c)}}- +\frac{a}{c}\int{\frac{1}{x^{x-2}(ax^2+bx+c)}} +$$ +<<*>>= +)clear all + +--S 41 14:271 Axiom cannot compute this integral +aa:=integrate(1/(x^n*(a*x^2+b*x+c)),x) +--R +--R +--R x +--R ++ 1 +--I (1) | -------------------- d%N +--R ++ 2 n +--I (c + %N b + %N a)%N +--R Type: Union(Expression Integer,...) +--E + +@ + +\section{\cite{1}:14.272~~~~~$\displaystyle +\int{\frac{dx}{(ax^2+bx+c)^2}}$} +$$\int{\frac{1}{(ax^2+bx+c)^2}}= +\frac{2xa+b}{(4ac-b^2)(ax^2+bx+c)}+ +\frac{2a}{4ac-b^2}\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 42 +aa:=integrate(1/(a*x^2+b*x+c)^2,x) +--R +--R +--R (1) +--R [ +--R 2 2 +--R (2a x + 2a b x + 2a c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R | 2 +--R (2a x + b)\|- 4a c + b +--R / +--R +-----------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b +--R , +--R +---------+ +--R | 2 +---------+ +--R 2 2 (2a x + b)\|4a c - b | 2 +--R (4a x + 4a b x + 4a c)atan(----------------------) + (2a x + b)\|4a c - b +--R 2 +--R 4a c - b +--R ---------------------------------------------------------------------------- +--R +---------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 43 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--R [ +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 44 +bb1:=(2*a*x+b)/((4*a*c-b^2)*(a*x^2+b*x+c))+(2*a)/(4*a*c-b^2)*t1.1 +--R +--R (3) +--R 2 2 +--R (2a x + 2a b x + 2a c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R | 2 +--R (2a x + b)\|- 4a c + b +--R / +--R +-----------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 45 +bb2:=(2*a*x+b)/((4*a*c-b^2)*(a*x^2+b*x+c))+(2*a)/(4*a*c-b^2)*t1.2 +--R +--R (4) +--R +---------+ +--R | 2 +---------+ +--R 2 2 (2a x + b)\|4a c - b | 2 +--R (4a x + 4a b x + 4a c)atan(----------------------) + (2a x + b)\|4a c - b +--R 2 +--R 4a c - b +--R ---------------------------------------------------------------------------- +--R +---------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b +--R Type: Expression Integer +--E + +--S 46 +cc1:=aa.1-bb1 +--R +--R (5) 0 +--R Type: Expression Integer +--E + +--S 47 +cc2:=aa.2-bb1 +--R +--R (6) +--R - +--R +---------+ +--R | 2 +--R 2a\|4a c - b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +---------+ +--R +-----------+ | 2 +--R | 2 (2a x + b)\|4a c - b +--R 4a\|- 4a c + b atan(----------------------) +--R 2 +--R 4a c - b +--R / +--R +-----------+ +---------+ +--R 2 | 2 | 2 +--R (4a c - b )\|- 4a c + b \|4a c - b +--R Type: Expression Integer +--E + +--S 48 +cc3:=aa.1-bb2 +--R +--R (7) +--R +---------+ +--R | 2 +--R 2a\|4a c - b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +---------+ +--R +-----------+ | 2 +--R | 2 (2a x + b)\|4a c - b +--R - 4a\|- 4a c + b atan(----------------------) +--R 2 +--R 4a c - b +--R / +--R +-----------+ +---------+ +--R 2 | 2 | 2 +--R (4a c - b )\|- 4a c + b \|4a c - b +--R Type: Expression Integer +--E + +--S 49 14:272 Schaums and Axiom agree +cc4:=aa.2-bb2 +--R +--R (8) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.273~~~~~$\displaystyle +\int{\frac{x~dx}{(ax^2+bx+c)^2}}$} +$$\int{\frac{x}{(ax^2+bx+c)^2}}= +-\frac{bx+2c}{(4ac-b^2)(ax^2+bx+c)}- +\frac{b}{4ac-b^2}\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 50 +aa:=integrate(x/(a*x^2+b*x+c)^2,x) +--R +--R +--R (1) +--R [ +--R 2 2 +--R (a b x + b x + b c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R | 2 +--R (- b x - 2c)\|- 4a c + b +--R / +--R +-----------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b +--R , +--R +--R +---------+ +--R | 2 +--R 2 2 (2a x + b)\|4a c - b +--R (- 2a b x - 2b x - 2b c)atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R | 2 +--R (- b x - 2c)\|4a c - b +--R / +--R +---------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 51 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--R [ +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 52 +bb1:=-(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.1 +--R +--R +--R (3) +--R 2 2 +--R (- a b x - b x - b c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R | 2 +--R (- b x - 2c)\|- 4a c + b +--R / +--R +-----------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 53 +bb2:=-(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.2 +--R +--R +--R (4) +--R +---------+ +--R | 2 +--R 2 2 (2a x + b)\|4a c - b +--R (- 2a b x - 2b x - 2b c)atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R | 2 +--R (- b x - 2c)\|4a c - b +--R / +--R +---------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b +--R Type: Expression Integer +--E + +--S 54 +cc1:=bb1-aa.1 +--R +--R (5) +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b +--R + +--R 2 2 3 +--R (- 8a c + 2a b )x - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R 2 | 2 +--R (4a c - b )\|- 4a c + b +--R Type: Expression Integer +--E + +--S 55 +dd1:=expandLog cc1 +--R +--R (6) +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R + +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R + +--R 2 +--R 2b log(a x + b x + c) +--R / +--R +-----------+ +--R 2 | 2 +--R (4a c - b )\|- 4a c + b +--R Type: Expression Integer +--E + +--S 56 14:273 Schaums and Axiom differ by a constant +ee1:=complexNormalize dd1 +--R +--R 3 2 2 +--R b log(- 16a c + 4a b ) +--R (7) - ------------------------- +--R +-----------+ +--R 2 | 2 +--R (4a c - b )\|- 4a c + b +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.274~~~~~$\displaystyle +\int{\frac{x^2~dx}{(ax^2+bx+c)^2}}$} +$$\int{\frac{x^2}{(ax^2+bx+c)^2}}= +\frac{(b^2-2ac)x+bc}{a(4ac-b^2)(ax^2+bx+c)}+ +\frac{2c}{4ac-b^2}\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 57 +aa:=integrate(x^2/(a*x^2+b*x+c)^2,x) +--R +--R +--R (1) +--R [ +--R 2 2 2 +--R (2a c x + 2a b c x + 2a c ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R ((- 2a c + b )x + b c)\|- 4a c + b +--R / +--R +-----------+ +--R 3 2 2 2 2 3 2 2 2 | 2 +--R ((4a c - a b )x + (4a b c - a b )x + 4a c - a b c)\|- 4a c + b +--R , +--R +--R +---------+ +--R | 2 +--R 2 2 2 (2a x + b)\|4a c - b +--R (4a c x + 4a b c x + 4a c )atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R ((- 2a c + b )x + b c)\|4a c - b +--R / +--R +---------+ +--R 3 2 2 2 2 3 2 2 2 | 2 +--R ((4a c - a b )x + (4a b c - a b )x + 4a c - a b c)\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 58 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--R [ +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 59 +bb1:=((b^2-2*a*c)*x+b*c)/(a*(4*a*c-b^2)*(a*x^2+b*x+c))+(2*c)/(4*a*c-b^2)*t1.1 +--R +--R (3) +--R 2 2 2 +--R (2a c x + 2a b c x + 2a c ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R ((- 2a c + b )x + b c)\|- 4a c + b +--R / +--R +-----------+ +--R 3 2 2 2 2 3 2 2 2 | 2 +--R ((4a c - a b )x + (4a b c - a b )x + 4a c - a b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 60 +bb2:=((b^2-2*a*c)*x+b*c)/(a*(4*a*c-b^2)*(a*x^2+b*x+c))+(2*c)/(4*a*c-b^2)*t1.2 +--R +--R (4) +--R +---------+ +--R | 2 +--R 2 2 2 (2a x + b)\|4a c - b +--R (4a c x + 4a b c x + 4a c )atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R ((- 2a c + b )x + b c)\|4a c - b +--R / +--R +---------+ +--R 3 2 2 2 2 3 2 2 2 | 2 +--R ((4a c - a b )x + (4a b c - a b )x + 4a c - a b c)\|4a c - b +--R Type: Expression Integer +--E + +--S 61 14:274 Schaums and Axiom agree +cc1:=aa.1-bb1 +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.275~~~~~$\displaystyle +\int{\frac{x^m~dx}{(ax^2+bx+c)^n}}$} +$$ +\begin{array}{lr} +\displaystyle\int{\frac{x^m}{(ax^2+bx+c)^n}}= +&\displaystyle-\frac{x^{m-1}}{(2n-m-1)a(ax^2+bx+c)^{n-1}}\\ +&\\ +&\displaystyle+\frac{(m-1)c}{(2n-m-1)a}\int{\frac{x^{m-2}}{(ax^2+bx+2)^n}}\\ +&\\ +&\displaystyle-\frac{(n-m)b}{(2n-m-1)a}\int{\frac{x^{m-1}}{(ax^2+bx+c)^n}} +\end{array} +$$ +<<*>>= +)clear all + +--S 62 14:275 Axiom cannot compute this integral +aa:=integrate(x^m/(a*x^2+b*x+c)^n,x) +--R +--R +--R x m +--I ++ %N +--I (1) | ------------------ d%N +--R ++ 2 n +--I (c + %N b + %N a) +--R Type: Union(Expression Integer,...) +--E + +@ + +\section{\cite{1}:14.276~~~~~$\displaystyle +\int{\frac{x^{2n-1}~dx}{(ax^2+bx+c)^n}}$} +$$\begin{array}{lr} +\displaystyle\int{\frac{x^{2n-1}}{(ax^2+bx+c)^n}}= +&\displaystyle\frac{1}{a}\int{\frac{x^{2n-3}}{(ax^2+bx+c)^{n-1}}}\\ +&\\ +&\displaystyle-\frac{c}{a}\int{\frac{x^{2n-3}}{(ax^2+bx+c)^n}}\\ +&\\ +&\displaystyle-\frac{b}{a}\int{\frac{x^{2n-2}}{(ax^2+bx+c)^n}} +\end{array} +$$ +<<*>>= +)clear all + +--S 63 14:276 Axiom cannot compute this integral +aa:=integrate(x^(2*n-1)/(a*x^2+b*x+c)^n,x) +--R +--R +--R x 2n - 1 +--I ++ %N +--I (1) | ------------------ d%N +--R ++ 2 n +--I (c + %N b + %N a) +--R Type: Union(Expression Integer,...) +--E + +@ + +\section{\cite{1}:14.277~~~~~$\displaystyle +\int{\frac{dx}{x(ax^2+bx+c)^2}}$} +$$\begin{array}{lr} +\displaystyle\int{\frac{1}{x(ax^2+bx+c)^2}}= +&\displaystyle\frac{1}{2c(ax^2+bx+2)}\\ +&\\ +&\displaystyle-\frac{b}{2c}\int{\frac{1}{(ax^2+bx+c)^2}}\\ +&\\ +&\displaystyle+\frac{1}{c}\int{\frac{1}{x(ax^2+bx+c)}} +\end{array} +$$ +<<*>>= +)clear all + +--S 64 +aa:=integrate(1/(x*(a*x^2+b*x+c)^2),x) +--R +--R +--R (1) +--R [ +--R 2 3 2 2 4 2 3 +--R ((6a b c - a b )x + (6a b c - b )x + 6a b c - b c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) +--R + +--R 2 2 +--R - 2a b c x + 4a c - 2b c +--R * +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R / +--R +-----------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c + b +--R , +--R +--R 2 3 2 2 4 2 3 +--R ((- 12a b c + 2a b )x + (- 12a b c + 2b )x - 12a b c + 2b c) +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) +--R + +--R 2 2 +--R - 2a b c x + 4a c - 2b c +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +---------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 65 +t1:=integrate(1/(a*x^2+b*x+c)^2,x) +--R +--R +--R (2) +--R [ +--R 2 2 +--R (2a x + 2a b x + 2a c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R | 2 +--R (2a x + b)\|- 4a c + b +--R / +--R +-----------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b +--R , +--R +---------+ +--R | 2 +---------+ +--R 2 2 (2a x + b)\|4a c - b | 2 +--R (4a x + 4a b x + 4a c)atan(----------------------) + (2a x + b)\|4a c - b +--R 2 +--R 4a c - b +--R ---------------------------------------------------------------------------- +--R +---------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 66 +t2:=integrate(1/(x*(a*x^2+b*x+c)),x) +--R +--R +--R (3) +--R [ +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R (- log(a x + b x + c) + 2log(x))\|- 4a c + b +--R / +--R +-----------+ +--R | 2 +--R 2c\|- 4a c + b +--R , +--R +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R - 2b atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R (- log(a x + b x + c) + 2log(x))\|4a c - b +--R / +--R +---------+ +--R | 2 +--R 2c\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 67 +bb1:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.1+1/c*t2.1 +--R +--R +--R (4) +--R 2 2 2 2 +--R (- 2a b c x - 2a b c x - 2a b c ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 3 2 2 4 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + 4a b c - b c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) - 2a b c x +--R + +--R 2 2 +--R 4a c - 2b c +--R * +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R / +--R +-----------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c + b +--R Type: Expression Integer +--E + +--S 68 +bb2:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.2+1/c*t2.1 +--R +--R +--R (5) +--R +---------+ +--R 2 3 2 2 4 2 3 | 2 +--R ((4a b c - a b )x + (4a b c - b )x + 4a b c - b c)\|4a c - b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 2 2 2 | 2 +--R (- 4a b c x - 4a b c x - 4a b c )\|- 4a c + b +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) - 2a b c x +--R + +--R 2 2 +--R 4a c - 2b c +--R * +--R +-----------+ +---------+ +--R | 2 | 2 +--R \|- 4a c + b \|4a c - b +--R / +--R +-----------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c + b +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Integer +--E + +--S 69 +bb3:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.1+1/c*t2.2 +--R +--R +--R (6) +--R +---------+ +--R 2 2 2 2 | 2 +--R (- 2a b c x - 2a b c x - 2a b c )\|4a c - b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 3 2 2 4 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x - 8a b c + 2b c) +--R * +--R +---------+ +--R +-----------+ | 2 +--R | 2 (2a x + b)\|4a c - b +--R \|- 4a c + b atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) - 2a b c x +--R + +--R 2 2 +--R 4a c - 2b c +--R * +--R +-----------+ +---------+ +--R | 2 | 2 +--R \|- 4a c + b \|4a c - b +--R / +--R +-----------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c + b +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Integer +--E + +--S 70 +bb4:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.2+1/c*t2.2 +--R +--R +--R (7) +--R 2 3 2 2 4 2 3 +--R ((- 12a b c + 2a b )x + (- 12a b c + 2b )x - 12a b c + 2b c) +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) - 2a b c x +--R + +--R 2 2 +--R 4a c - 2b c +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +---------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|4a c - b +--R Type: Expression Integer +--E + +--S 71 +cc1:=aa.1-bb1 +--R +--R (8) +--R a b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R a b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R 2 2 | 2 +--R (4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 72 +dd1:=expandLog cc1 +--R +--R (9) +--R a b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R + +--R a b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R + +--R 2 +--R - 2a b log(a x + b x + c) +--R / +--R +-----------+ +--R 2 2 | 2 +--R (4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 73 14:277 Schaums and Axiom differ by a constant +ee1:=complexNormalize dd1 +--R +--R 3 2 2 +--R a b log(- 16a c + 4a b ) +--R (10) --------------------------- +--R +-----------+ +--R 2 2 | 2 +--R (4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.278~~~~~$\displaystyle +\int{\frac{dx}{x^2(ax^2+bx+c)^2}}$} +$$\begin{array}{lr} +\displaystyle\int{\frac{1}{x^2(ax^2+bx+c)^2}}= +&\displaystyle-\frac{1}{cx(ax^2+bx+c)}\\ +&\\ +&\displaystyle-\frac{3a}{c}\int{\frac{1}{(ax^2+bx+c)^2}}\\ +&\\ +&\displaystyle-\frac{2b}{c}\int{\frac{1}{x(ax^2+bx+c)^2}} +\end{array} +$$ +<<*>>= +)clear all + +--S 74 +aa:=integrate(1/(x^2*(a*x^2+b*x+c)^2),x) +--R +--R +--R (1) +--R [ +--R 3 2 2 2 4 3 2 2 3 5 2 +--R (6a c - 6a b c + a b )x + (6a b c - 6a b c + b )x +--R + +--R 2 3 2 2 4 +--R (6a c - 6a b c + b c)x +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 3 3 2 4 2 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 3 3 2 4 2 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b c)x) +--R * +--R log(x) +--R + +--R 2 2 2 2 2 3 3 2 2 +--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c +--R * +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R / +--R +-----------+ +--R 2 4 2 3 3 4 3 3 2 5 2 4 | 2 +--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|- 4a c + b +--R , +--R +--R 3 2 2 2 4 3 2 2 3 5 2 +--R (- 12a c + 12a b c - 2a b )x + (- 12a b c + 12a b c - 2b )x +--R + +--R 2 3 2 2 4 +--R (- 12a c + 12a b c - 2b c)x +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 3 3 2 4 2 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 3 3 2 4 2 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b c)x) +--R * +--R log(x) +--R + +--R 2 2 2 2 2 3 3 2 2 +--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +---------+ +--R 2 4 2 3 3 4 3 3 2 5 2 4 | 2 +--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 75 +t1:=integrate(1/(a*x^2+b*x+c)^2,x) +--R +--R +--R (2) +--R [ +--R 2 2 +--R (2a x + 2a b x + 2a c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R | 2 +--R (2a x + b)\|- 4a c + b +--R / +--R +-----------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b +--R , +--R +---------+ +--R | 2 +---------+ +--R 2 2 (2a x + b)\|4a c - b | 2 +--R (4a x + 4a b x + 4a c)atan(----------------------) + (2a x + b)\|4a c - b +--R 2 +--R 4a c - b +--R ---------------------------------------------------------------------------- +--R +---------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 76 +t2:=integrate(1/(x*(a*x^2+b*x+c)^2),x) +--R +--R +--R (3) +--R [ +--R 2 3 2 2 4 2 3 +--R ((6a b c - a b )x + (6a b c - b )x + 6a b c - b c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) +--R + +--R 2 2 +--R - 2a b c x + 4a c - 2b c +--R * +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R / +--R +-----------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c + b +--R , +--R +--R 2 3 2 2 4 2 3 +--R ((- 12a b c + 2a b )x + (- 12a b c + 2b )x - 12a b c + 2b c) +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) +--R + +--R 2 2 +--R - 2a b c x + 4a c - 2b c +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +---------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 77 +bb1:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.1-(2*b)/c*t2.1 +--R +--R +--R (4) +--R 3 2 3 2 2 2 2 3 +--R (- 6a c x - 6a b c x - 6a c x) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 2 4 3 3 5 2 2 2 4 +--R ((- 6a b c + a b )x + (- 6a b c + b )x + (- 6a b c + b c)x) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 3 3 2 4 2 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 3 3 2 4 2 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b c)x) +--R * +--R log(x) +--R + +--R 2 2 2 2 2 3 3 2 2 +--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c +--R * +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R / +--R +-----------+ +--R 2 4 2 3 3 4 3 3 2 5 2 4 | 2 +--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 78 +bb2:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.2-(2*b)/c*t2.1 +--R +--R +--R (5) +--R 2 2 4 3 3 5 2 2 2 4 +--R ((- 6a b c + a b )x + (- 6a b c + b )x + (- 6a b c + b c)x) +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 3 2 3 2 2 2 2 3 | 2 +--R (- 12a c x - 12a b c x - 12a c x)\|- 4a c + b +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 3 3 2 4 2 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 3 3 2 4 2 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b c)x) +--R * +--R log(x) +--R + +--R 2 2 2 2 2 3 3 2 2 +--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c +--R * +--R +-----------+ +---------+ +--R | 2 | 2 +--R \|- 4a c + b \|4a c - b +--R / +--R +-----------+ +--R 2 4 2 3 3 4 3 3 2 5 2 4 | 2 +--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|- 4a c + b +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Integer +--E + +--S 79 +bb3:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.1-(2*b)/c*t2.2 +--R +--R +--R (6) +--R +---------+ +--R 3 2 3 2 2 2 2 3 | 2 +--R (- 6a c x - 6a b c x - 6a c x)\|4a c - b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 2 4 3 3 5 2 2 2 4 +--R ((12a b c - 2a b )x + (12a b c - 2b )x + (12a b c - 2b c)x) +--R * +--R +---------+ +--R +-----------+ | 2 +--R | 2 (2a x + b)\|4a c - b +--R \|- 4a c + b atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 3 3 2 4 2 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 3 3 2 4 2 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b c)x) +--R * +--R log(x) +--R + +--R 2 2 2 2 2 3 3 2 2 +--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c +--R * +--R +-----------+ +---------+ +--R | 2 | 2 +--R \|- 4a c + b \|4a c - b +--R / +--R +-----------+ +--R 2 4 2 3 3 4 3 3 2 5 2 4 | 2 +--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|- 4a c + b +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Integer +--E + +--S 80 +bb4:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.2-(2*b)/c*t2.2 +--R +--R +--R (7) +--R 3 2 2 2 4 3 2 2 3 5 2 +--R (- 12a c + 12a b c - 2a b )x + (- 12a b c + 12a b c - 2b )x +--R + +--R 2 3 2 2 4 +--R (- 12a c + 12a b c - 2b c)x +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 3 3 2 4 2 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 3 3 2 4 2 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b c)x) +--R * +--R log(x) +--R + +--R 2 2 2 2 2 3 3 2 2 +--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +---------+ +--R 2 4 2 3 3 4 3 3 2 5 2 4 | 2 +--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|4a c - b +--R Type: Expression Integer +--E + +--S 81 +cc1:=aa.1-bb1 +--R +--R (8) +--R 2 +--R 6a +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 +--R 6a +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R 2 2 | 2 +--R (4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 82 +dd1:=expandLog cc1 +--R +--R (9) +--R 2 +--R 6a +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R + +--R 2 +--R 6a +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R + +--R 2 2 +--R - 12a log(a x + b x + c) +--R / +--R +-----------+ +--R 2 2 | 2 +--R (4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 83 14:278 Schaums and Axiom differ by a constant +ee1:=complexNormalize dd1 +--R +--R 2 3 2 2 +--R 6a log(- 16a c + 4a b ) +--R (10) --------------------------- +--R +-----------+ +--R 2 2 | 2 +--R (4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.279~~~~~$\displaystyle +\int{\frac{dx}{x^m(ax^2+bx+c)^n}}$} +$$\begin{array}{lr} +\displaystyle\int{\frac{1}{x^m(ax^2+bx+c)^n}}= +&\displaystyle-\frac{1}{(m-1)cx^{m-1}(ax^2+bx+c)^{n-1}}\\ +&\\ +&\displaystyle-\frac{(m+2n-3)a}{(m-1)c}\int{\frac{1}{x^{m-2}(ax^2+bx+c)^n}}\\ +&\\ +&\displaystyle-\frac{(m+n-2)b}{(m-1)c}\int{\frac{1}{x^{m-1}(ax^2+bx+c)^n}} +\end{array} +$$ +<<*>>= +)clear all + +--S 84 14:279 Axiom cannot compute this integral +aa:=integrate(1/(x^m*(a*x^2+b*x+c)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | --------------------- d%N +--R ++ m 2 n +--I %N (c + %N b + %N a) +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p71 +\end{thebibliography} +\end{document} diff --git 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n +0000152715 00000 n +0000152632 00000 n +0000155227 00000 n +0000153344 00000 n +0000153875 00000 n +0000154406 00000 n +0000155128 00000 n +0000155045 00000 n +0000157831 00000 n +0000155757 00000 n +0000156479 00000 n +0000156396 00000 n +0000157010 00000 n +0000157732 00000 n +0000157649 00000 n +0000158216 00000 n +0000158239 00000 n +0000158261 00000 n +trailer +<< +/Size 361 +/Root 2 0 R +/Info 1 0 R +>> +startxref +158382 +%%EOF diff --git a/src/axiom-website/CATS/schaum13.input.pamphlet b/src/axiom-website/CATS/schaum13.input.pamphlet new file mode 100644 index 0000000..678e41c --- /dev/null +++ b/src/axiom-website/CATS/schaum13.input.pamphlet @@ -0,0 +1,5196 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum13.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.280~~~~~$\displaystyle +\int{\frac{dx}{\sqrt{ax^2+bx+c}}}$} +$$\int{\frac{1}{\sqrt{ax^2+bx+c}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{\sqrt{a}}\ln\left(2\sqrt{a}\sqrt{ax^2+bx+c}+2ax+b\right)\\ +\\ +\displaystyle +-\frac{1}{\sqrt{-a}}\sin{-1}\left(\frac{2ax+b}{\sqrt{b^2-4ac}}\right)\\ +\\ +\displaystyle +\frac{1}{\sqrt{a}}\sinh^{-1}\left(\frac{2ax+b}{\sqrt{4ac-b^2}}\right) +\end{array} +\right. +$$ +<<*>>= +)spool schaum13.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (1) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 2 +bb1:=1/sqrt(a)*log(2*sqrt(a)*sqrt(a*x^2+b*x+c)+2*a*x+b) +--R +--R +--------------+ +--R +-+ | 2 +--R log(2\|a \|a x + b x + c + 2a x + b) +--R (2) -------------------------------------- +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 3 +bb2:=-1/sqrt(-a)*asin((2*a*x+b)/sqrt(b^2-4*a*c)) +--R +--R 2a x + b +--R asin(--------------) +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R (3) - -------------------- +--R +---+ +--R \|- a +--R Type: Expression Integer +--E + +--S 4 +bb3:=1/sqrt(a)*asinh((2*a*x+b)/sqrt(4*a*c-b^2)) +--R +--R 2a x + b +--R asinh(------------) +--R +---------+ +--R | 2 +--R \|4a c - b +--R (4) ------------------- +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 5 +cc1:=bb1-aa.1 +--R +--R (5) +--R +--------------+ +--R +-+ | 2 +--R log(2\|a \|a x + b x + c + 2a x + b) +--R + +--R - +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 6 +cc2:=bb1-aa.2 +--R +--R (6) +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a log(2\|a \|a x + b x + c + 2a x + b) +--R + +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R +-+ \|- a \|a x + b x + c - \|- a \|c +--R - 2\|a atan(------------------------------------) +--R a x +--R / +--R +---+ +-+ +--R \|- a \|a +--R Type: Expression Integer +--E + +--S 7 +cc3:=bb2-aa.1 +--R +--R (7) +--R - +--R +---+ +--R \|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +-+ 2a x + b +--R - \|a asin(--------------) +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R / +--R +---+ +-+ +--R \|- a \|a +--R Type: Expression Integer +--E + +--S 8 +cc4:=bb2-aa.2 +--R +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c 2a x + b +--R - 2atan(------------------------------------) - asin(--------------) +--R a x +-----------+ +--R | 2 +--R \|- 4a c + b +--R (8) -------------------------------------------------------------------- +--R +---+ +--R \|- a +--R Type: Expression Integer +--E + +--S 9 +cc5:=bb3-aa.1 +--R +--R (9) +--R - +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2a x + b +--R asinh(------------) +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 10 +cc6:=bb3-aa.2 +--R +--R (10) +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R +-+ \|- a \|a x + b x + c - \|- a \|c +---+ 2a x + b +--R - 2\|a atan(------------------------------------) + \|- a asinh(------------) +--R a x +---------+ +--R | 2 +--R \|4a c - b +--R ----------------------------------------------------------------------------- +--R +---+ +-+ +--R \|- a \|a +--R Type: Expression Integer +--E + +--S 11 +dd1:=simplifyLog cc1 +--R +--R (11) +--R log +--R +--------------+ +--R +-+ +-+ | 2 +--R ((4a x + 2b)\|c + (- 2b x - 4c)\|a )\|a x + b x + c +--R + +--R 2 +-+ +-+ 2 2 +--R (4a x + 4b x + 4c)\|a \|c - 2a b x + (- 4a c - b )x - 2b c +--R / +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 12 14:280 Schaums and Axiom differ by a constant +ee1:=ratDenom dd1 +--R +--R +-+ +-+ +--R +-+ - 2a\|c + b\|a +--R \|a log(----------------) +--R a +--R (12) ------------------------- +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.281~~~~~$\displaystyle +\int{\frac{x~dx}{\sqrt{ax^2+bx+c}}}$} +$$\int{\frac{x}{\sqrt{ax^2+bx+c}}}= +\frac{\sqrt{ax^2+bx+c}}{a}-\frac{b}{2a}\int{\frac{1}{\sqrt{ax^2+bx+c}}} +$$ +<<*>>= +)clear all + +--S 11 +aa:=integrate(x/sqrt(a*x^2+b*x+c),x) +--R +--R +--R (1) +--R [ +--R +--------------+ +--R +-+ | 2 2 +--R (2b\|c \|a x + b x + c - b x - 2b c) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 2 +-+ +-+ +--R - 2b x\|a \|a x + b x + c + (4a x + 2b x)\|a \|c +--R / +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R 4a\|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|a +--R , +--R +--R +--------------+ +--R +-+ | 2 2 +--R (- 2b\|c \|a x + b x + c + b x + 2b c) +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ | 2 2 +---+ +-+ +--R - b x\|- a \|a x + b x + c + (2a x + b x)\|- a \|c +--R / +--R +--------------+ +--R +---+ +-+ | 2 +---+ +--R 2a\|- a \|c \|a x + b x + c + (- a b x - 2a c)\|- a +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 12 +t1:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 13 +bb1:=sqrt(a*x^2+b*x+c)/a-b/(2*a)*t1.1 +--R +--R (3) +--R - +--R b +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 +--R 2\|a \|a x + b x + c +--R / +--R +-+ +--R 2a\|a +--R Type: Expression Integer +--E + +--S 14 +bb2:=sqrt(a*x^2+b*x+c)/a-b/(2*a)*t1.2 +--R +--R (4) +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--------------+ +--R \|- a \|a x + b x + c - \|- a \|c +---+ | 2 +--R - b atan(------------------------------------) + \|- a \|a x + b x + c +--R a x +--R ------------------------------------------------------------------------ +--R +---+ +--R a\|- a +--R Type: Expression Integer +--E + +--S 15 +cc1:=bb1-aa.1 +--R +--R (5) +--R +--------------+ +--R +-+ | 2 2 +--R (- 2b\|c \|a x + b x + c + b x + 2b c) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 2 +--R (- 2b\|c \|a x + b x + c + b x + 2b c) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 +-+ +-+ +--R - 4c\|a \|a x + b x + c + (2b x + 4c)\|a \|c +--R / +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R 4a\|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|a +--R Type: Expression Integer +--E + +--S 16 +cc2:=bb1-aa.2 +--R +--R (6) +--R +--------------+ +--R +---+ +-+ | 2 2 +---+ +--R (- 2b\|- a \|c \|a x + b x + c + (b x + 2b c)\|- a ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ +-+ | 2 2 +-+ +--R (4b\|a \|c \|a x + b x + c + (- 2b x - 4b c)\|a ) +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ +-+ | 2 +---+ +-+ +-+ +--R - 4c\|- a \|a \|a x + b x + c + (2b x + 4c)\|- a \|a \|c +--R / +--R +--------------+ +--R +---+ +-+ +-+ | 2 +---+ +-+ +--R 4a\|- a \|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|- a \|a +--R Type: Expression Integer +--E + +--S 17 +cc3:=bb2-aa.1 +--R +--R (7) +--R +--------------+ +--R +---+ +-+ | 2 2 +---+ +--R (- 2b\|- a \|c \|a x + b x + c + (b x + 2b c)\|- a ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ +-+ | 2 2 +-+ +--R (- 4b\|a \|c \|a x + b x + c + (2b x + 4b c)\|a ) +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ +-+ | 2 +---+ +-+ +-+ +--R - 4c\|- a \|a \|a x + b x + c + (2b x + 4c)\|- a \|a \|c +--R / +--R +--------------+ +--R +---+ +-+ +-+ | 2 +---+ +-+ +--R 4a\|- a \|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|- a \|a +--R Type: Expression Integer +--E + +--S 18 +cc4:=bb2-aa.2 +--R +--R +--------------+ +--R | 2 +-+ +--R - 2c\|a x + b x + c + (b x + 2c)\|c +--R (8) -------------------------------------- +--R +--------------+ +--R +-+ | 2 +--R 2a\|c \|a x + b x + c - a b x - 2a c +--R Type: Expression Integer +--E + +--S 19 14:281 Schaums and Axiom differ by a constant +dd1:=ratDenom cc4 +--R +--R +-+ +--R \|c +--R (9) - ---- +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.282~~~~~$\displaystyle +\int{\frac{x^2dx}{\sqrt{ax^2+bx+c}}}$} +$$\int{\frac{x^2}{\sqrt{ax^2+bx+c}}}= +\frac{2ax-3b}{4a^2}\sqrt{ax^2+bx+c}+\frac{3b^2-4ac}{8a^2} +\int{\frac{1}{\sqrt{ax^2+bx+c}}} +$$ +<<*>>= +)clear all + +--S 19 +aa:=integrate(x^2/sqrt(a*x^2+b*x+c),x) +--R +--R (1) +--R [ +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((16a b c - 12b )x + 32a c - 24b c)\|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 2 2 +--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c + 24b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 2 3 3 2 2 2 +-+ +--R ((- 16a c - 4a b )x + (- 8a b c + 6b )x + (- 32a c + 24b c)x)\|a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 4 2 2 3 3 2 +--R 16a b x + (32a c - 8a b )x + (24a b c - 18b )x +--R + +--R 2 2 +--R (32a c - 24b c)x +--R * +--R +-+ +-+ +--R \|a \|c +--R / +--R +--------------+ +--R 2 2 +-+ +-+ | 2 +--R (32a b x + 64a c)\|a \|c \|a x + b x + c +--R + +--R 3 2 2 2 2 2 2 +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|a +--R , +--R +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((- 16a b c + 12b )x - 32a c + 24b c)\|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 2 2 +--R (16a c - 8a b c - 3b )x + (32a b c - 24b c)x + 32a c - 24b c +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 2 2 3 3 2 2 2 +---+ +--R ((- 8a c - 2a b )x + (- 4a b c + 3b )x + (- 16a c + 12b c)x)\|- a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 4 2 2 3 3 2 2 2 +--R (8a b x + (16a c - 4a b )x + (12a b c - 9b )x + (16a c - 12b c)x) +--R * +--R +---+ +-+ +--R \|- a \|c +--R / +--R +--------------+ +--R 2 2 +---+ +-+ | 2 +--R (16a b x + 32a c)\|- a \|c \|a x + b x + c +--R + +--R 3 2 2 2 2 2 2 +---+ +--R ((- 16a c - 4a b )x - 32a b c x - 32a c )\|- a +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 20 +t1:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 21 +bb1:=(2*a*x-3*b)/(4*a^2)*sqrt(a*x^2+b*x+c)+(3*b^2-4*a*c)/(8*a^2)*t1.1 +--R +--R (3) +--R 2 +--R (- 4a c + 3b ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 +--R (4a x - 6b)\|a \|a x + b x + c +--R / +--R 2 +-+ +--R 8a \|a +--R Type: Expression Integer +--E + +--S 22 +bb2:=(2*a*x-3*b)/(4*a^2)*sqrt(a*x^2+b*x+c)+(3*b^2-4*a*c)/(8*a^2)*t1.2 +--R +--R (4) +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R 2 \|- a \|a x + b x + c - \|- a \|c +--R (- 4a c + 3b )atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ | 2 +--R (2a x - 3b)\|- a \|a x + b x + c +--R / +--R 2 +---+ +--R 4a \|- a +--R Type: Expression Integer +--E + +--S 23 +cc1:=aa.1-bb1 +--R +--R (5) +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((16a b c - 12b )x + 32a c - 24b c)\|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 2 2 +--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c + 24b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((16a b c - 12b )x + 32a c - 24b c)\|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 2 2 +--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c + 24b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R 2 2 +-+ | 2 +--R (- 24b c x - 48b c )\|a \|a x + b x + c +--R + +--R 3 2 2 2 +-+ +-+ +--R ((24a b c + 6b )x + 48b c x + 48b c )\|a \|c +--R / +--R +--------------+ +--R 2 2 +-+ +-+ | 2 +--R (32a b x + 64a c)\|a \|c \|a x + b x + c +--R + +--R 3 2 2 2 2 2 2 +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|a +--R Type: Expression Integer +--E + +--S 24 +cc2:=aa.2-bb1 +--R +--R (6) +--R +--------------+ +--R 3 2 2 +---+ +-+ | 2 +--R ((16a b c - 12b )x + 32a c - 24b c)\|- a \|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 +--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c +--R + +--R 2 2 +--R 24b c +--R * +--R +---+ +--R \|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R 3 2 2 +-+ +-+ | 2 +--R ((- 32a b c + 24b )x - 64a c + 48b c)\|a \|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 +--R (32a c - 16a b c - 6b )x + (64a b c - 48b c)x + 64a c +--R + +--R 2 2 +--R - 48b c +--R * +--R +-+ +--R \|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R 2 2 +---+ +-+ | 2 +--R (- 24b c x - 48b c )\|- a \|a \|a x + b x + c +--R + +--R 3 2 2 2 +---+ +-+ +-+ +--R ((24a b c + 6b )x + 48b c x + 48b c )\|- a \|a \|c +--R / +--R +--------------+ +--R 2 2 +---+ +-+ +-+ | 2 +--R (32a b x + 64a c)\|- a \|a \|c \|a x + b x + c +--R + +--R 3 2 2 2 2 2 2 +---+ +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|- a \|a +--R Type: Expression Integer +--E + +--S 25 +cc3:=aa.2-bb1 +--R +--R (7) +--R +--------------+ +--R 3 2 2 +---+ +-+ | 2 +--R ((16a b c - 12b )x + 32a c - 24b c)\|- a \|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 +--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c +--R + +--R 2 2 +--R 24b c +--R * +--R +---+ +--R \|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R 3 2 2 +-+ +-+ | 2 +--R ((- 32a b c + 24b )x - 64a c + 48b c)\|a \|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 +--R (32a c - 16a b c - 6b )x + (64a b c - 48b c)x + 64a c +--R + +--R 2 2 +--R - 48b c +--R * +--R +-+ +--R \|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R 2 2 +---+ +-+ | 2 +--R (- 24b c x - 48b c )\|- a \|a \|a x + b x + c +--R + +--R 3 2 2 2 +---+ +-+ +-+ +--R ((24a b c + 6b )x + 48b c x + 48b c )\|- a \|a \|c +--R / +--R +--------------+ +--R 2 2 +---+ +-+ +-+ | 2 +--R (32a b x + 64a c)\|- a \|a \|c \|a x + b x + c +--R + +--R 3 2 2 2 2 2 2 +---+ +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|- a \|a +--R Type: Expression Integer +--E + +--S 26 +cc4:=aa.2-bb2 +--R +--R (8) +--R +--------------+ +--R 2 2 | 2 +--R (- 12b c x - 24b c )\|a x + b x + c +--R + +--R 3 2 2 2 +-+ +--R ((12a b c + 3b )x + 24b c x + 24b c )\|c +--R / +--R +--------------+ +--R 2 2 +-+ | 2 3 2 2 2 2 +--R (16a b x + 32a c)\|c \|a x + b x + c + (- 16a c - 4a b )x - 32a b c x +--R + +--R 2 2 +--R - 32a c +--R Type: Expression Integer +--E + +--S 27 14:282 Schaums and Axiom differ by a constant +dd4:=ratDenom cc4 +--R +--R +-+ +--R 3b\|c +--R (9) - ------ +--R 2 +--R 4a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.283~~~~~$\displaystyle +\int{\frac{dx}{x\sqrt{ax^2+bx+c}}}$} +$$\int{\frac{1}{x\sqrt{ax^2+bx+c}}}= +\left\{ +\begin{array}{l} +\displaystyle +-\frac{1}{\sqrt{c}} +\ln\left(\frac{2\sqrt{c}{\sqrt{ax^2+bx+c}}+bx+2c}{x}\right)\\ +\\ +\displaystyle +\frac{1}{\sqrt{-c}}\sin^{-1}\left(\frac{bx+2c}{|x|\sqrt{b^2-4ac}}\right)\\ +\\ +\displaystyle +-\frac{1}{\sqrt{c}}\sinh^{-1}\left(\frac{bx+2c}{|x|\sqrt{4ac-b^2}}\right) +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 27 +aa:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) +--R +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R (1) -------------------------------------- +--R +-+ +--R \|c +--R Type: Union(Expression Integer,...) +--E + +--S 28 +bb1:=-1/sqrt(c)*log((2*sqrt(c)*sqrt(a*x^2+b*x+c)+b*x+2*c)/x) +--R +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c + b x + 2c +--R log(---------------------------------) +--R x +--R (2) - -------------------------------------- +--R +-+ +--R \|c +--R Type: Expression Integer +--E + +--S 29 +bb2:=1/sqrt(-c)*asin((b*x+2*c)/(x*sqrt(b^2-4*a*c))) +--R +--R b x + 2c +--R asin(---------------) +--R +-----------+ +--R | 2 +--R x\|- 4a c + b +--R (3) --------------------- +--R +---+ +--R \|- c +--R Type: Expression Integer +--E + +--S 30 +bb3:=-1/sqrt(c)*asinh((b*x+2*c)/(x*sqrt(4*a*c-b^2))) +--R +--R b x + 2c +--R asinh(-------------) +--R +---------+ +--R | 2 +--R x\|4a c - b +--R (4) - -------------------- +--R +-+ +--R \|c +--R Type: Expression Integer +--E + +--S 31 +cc1:=aa-bb1 +--R +--R (5) +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c + b x + 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R / +--R +-+ +--R \|c +--R Type: Expression Integer +--E + +--S 32 +cc2:=aa-bb2 +--R +--R (6) +--R +--------------+ +--R +-+ | 2 +--R +---+ 2\|c \|a x + b x + c - b x - 2c +-+ b x + 2c +--R \|- c log(---------------------------------) - \|c asin(---------------) +--R x +-----------+ +--R | 2 +--R x\|- 4a c + b +--R ------------------------------------------------------------------------ +--R +---+ +-+ +--R \|- c \|c +--R Type: Expression Integer +--E + +--S 33 +cc3:=aa-bb3 +--R +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c b x + 2c +--R log(---------------------------------) + asinh(-------------) +--R x +---------+ +--R | 2 +--R x\|4a c - b +--R (7) ------------------------------------------------------------- +--R +-+ +--R \|c +--R Type: Expression Integer +--E + +--S 34 +dd1:=expandLog cc1 +--R +--R (8) +--R +--------------+ +--R +-+ | 2 +--R log(2\|c \|a x + b x + c + b x + 2c) +--R + +--R +--------------+ +--R +-+ | 2 +--R log(2\|c \|a x + b x + c - b x - 2c) - 2log(x) +--R / +--R +-+ +--R \|c +--R Type: Expression Integer +--E + +--S 35 +ee1:=ratDenom dd1 +--R +--R (9) +--R +--------------+ +--R +-+ +-+ | 2 +--R \|c log(2\|c \|a x + b x + c + b x + 2c) +--R + +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R \|c log(2\|c \|a x + b x + c - b x - 2c) - 2log(x)\|c +--R / +--R c +--R Type: Expression Integer +--E + +--S 36 14:283 Schaums and Axiom differ by a constant +ff1:=complexNormalize ee1 +--R +--R 2 +-+ +--R log(4a c - b )\|c +--R (10) ------------------ +--R c +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.284~~~~~$\displaystyle +\int{\frac{dx}{x^2\sqrt{ax^2+bx+c}}}$} +$$\int{\frac{1}{x^2\sqrt{ax^2+bx+c}}}= +-\frac{\sqrt{ax^2+bx+c}}{cx}-\frac{b}{2c}\int{\frac{1}{x\sqrt{ax^2+bx+c}}} +$$ +<<*>>= +)clear all + +--S 37 +aa:=integrate(1/(x^2*sqrt(a*x^2+b*x+c)),x) +--R +--R (1) +--R +--------------+ +--R +-+ | 2 2 2 +--R (- 4b x\|c \|a x + b x + c + 2b x + 4b c x) +--R * +--R +--------------+ +--R | 2 +-+ +--R 2c\|a x + b x + c + (- b x - 2c)\|c +--R log(--------------------------------------) +--R 2c x +--R + +--R +--------------+ +--R +-+ | 2 2 2 2 +--R (2b x + 8c)\|c \|a x + b x + c + (- 8a c + b )x - 6b c x - 8c +--R / +--R +--------------+ +--R 2 | 2 2 2 +-+ +--R 8c x\|a x + b x + c + (- 4b c x - 8c x)\|c +--R Type: Union(Expression Integer,...) +--E + +--S 38 +t1:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) +--R +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R (2) -------------------------------------- +--R +-+ +--R \|c +--R Type: Union(Expression Integer,...) +--E + +--S 39 +bb:=-sqrt(a*x^2+b*x+c)/(c*x)-b/(2*c)*t1 +--R +--R +--------------+ +--R +-+ | 2 +--------------+ +--R 2\|c \|a x + b x + c - b x - 2c +-+ | 2 +--R - b x log(---------------------------------) - 2\|c \|a x + b x + c +--R x +--R (3) --------------------------------------------------------------------- +--R +-+ +--R 2c x\|c +--R Type: Expression Integer +--E + +--S 40 +cc:=aa-bb +--R +--R (4) +--R +--------------+ +--R | 2 2 +-+ +--R (4b c\|a x + b x + c + (- 2b x - 4b c)\|c ) +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R | 2 2 +-+ +--R (- 4b c\|a x + b x + c + (2b x + 4b c)\|c ) +--R * +--R +--------------+ +--R | 2 +-+ +--R 2c\|a x + b x + c + (- b x - 2c)\|c +--R log(--------------------------------------) +--R 2c x +--R + +--R +--------------+ +--R | 2 2 +-+ +--R - 2b c\|a x + b x + c + (b x + 2b c)\|c +--R / +--R +--------------+ +--R 2 +-+ | 2 2 3 +--R 8c \|c \|a x + b x + c - 4b c x - 8c +--R Type: Expression Integer +--E + +--S 41 +dd:=expandLog cc +--R +--R (5) +--R +--------------+ +--R | 2 2 +-+ +--R (4b c\|a x + b x + c + (- 2b x - 4b c)\|c ) +--R * +--R +--------------+ +--R +-+ | 2 +--R log(2\|c \|a x + b x + c - b x - 2c) +--R + +--R +--------------+ +--R | 2 2 +-+ +--R (- 4b c\|a x + b x + c + (2b x + 4b c)\|c ) +--R * +--R +--------------+ +--R | 2 +-+ +--R log(2c\|a x + b x + c + (- b x - 2c)\|c ) +--R + +--R +--------------+ +--R | 2 +--R (4b c log(c) + 4b c log(2) - 2b c)\|a x + b x + c +--R + +--R 2 2 2 +-+ +--R ((- 2b x - 4b c)log(c) + (- 2b x - 4b c)log(2) + b x + 2b c)\|c +--R / +--R +--------------+ +--R 2 +-+ | 2 2 3 +--R 8c \|c \|a x + b x + c - 4b c x - 8c +--R Type: Expression Integer +--E + +--S 42 +ee:=ratDenom dd +--R +--R (6) +--R +--------------+ +--R +-+ +-+ | 2 +--R 2b\|c log(2\|c \|a x + b x + c - b x - 2c) +--R + +--R +--------------+ +--R +-+ | 2 +-+ +--R - 2b\|c log(2c\|a x + b x + c + (- b x - 2c)\|c ) +--R + +--R +-+ +--R (2b log(c) + 2b log(2) - b)\|c +--R / +--R 2 +--R 4c +--R Type: Expression Integer +--E + +--S 43 14:284 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R +-+ +--R (b log(c) + 2b log(2) - b)\|c +--R (7) ------------------------------ +--R 2 +--R 4c +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.285~~~~~$\displaystyle +\int{\sqrt{ax^2+bx+c}}~dx$} +$$\int{\sqrt{ax^2+bx+c}}= +\frac{(2ax+b)\sqrt{ax^2+bx+c}}{4a}+ +\frac{4ac-b^2}{8a}\int{\frac{1}{\sqrt{ax^2+bx+c}}} +$$ +<<*>>= +)clear all + +--S 44 +aa:=integrate(sqrt(a*x^2+b*x+c),x) +--R +--R +--R (1) +--R [ +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((16a b c - 4b )x + 32a c - 8b c)\|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +--R (- 16a c + b )x + (- 32a b c + 8b c)x - 32a c + 8b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 2 3 3 2 2 2 +-+ +--R ((- 16a c - 4a b )x + (- 40a b c - 2b )x + (- 32a c - 8b c)x)\|a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 4 2 2 3 3 2 +--R 16a b x + (32a c + 24a b )x + (56a b c + 6b )x +--R + +--R 2 2 +--R (32a c + 8b c)x +--R * +--R +-+ +-+ +--R \|a \|c +--R / +--R +--------------+ +--R +-+ +-+ | 2 +--R (32a b x + 64a c)\|a \|c \|a x + b x + c +--R + +--R 2 2 2 2 +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|a +--R , +--R +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((16a b c - 4b )x + 32a c - 8b c)\|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +--R (- 16a c + b )x + (- 32a b c + 8b c)x - 32a c + 8b c +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 2 2 3 3 2 2 2 +---+ +--R ((- 8a c - 2a b )x + (- 20a b c - b )x + (- 16a c - 4b c)x)\|- a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 4 2 2 3 3 2 2 2 +--R (8a b x + (16a c + 12a b )x + (28a b c + 3b )x + (16a c + 4b c)x) +--R * +--R +---+ +-+ +--R \|- a \|c +--R / +--R +--------------+ +--R +---+ +-+ | 2 +--R (16a b x + 32a c)\|- a \|c \|a x + b x + c +--R + +--R 2 2 2 2 +---+ +--R ((- 16a c - 4a b )x - 32a b c x - 32a c )\|- a +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 45 +t1:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 46 +bb1:=((2*a*x+b)*sqrt(a*x^2+b*x+c))/(4*a)+(4*a*c-b^2)/(8*a)*t1.1 +--R +--R (3) +--R 2 +--R (4a c - b ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 +--R (4a x + 2b)\|a \|a x + b x + c +--R / +--R +-+ +--R 8a\|a +--R Type: Expression Integer +--E + +--S 47 +bb2:=((2*a*x+b)*sqrt(a*x^2+b*x+c))/(4*a)+(4*a*c-b^2)/(8*a)*t1.2 +--R +--R (4) +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R 2 \|- a \|a x + b x + c - \|- a \|c +--R (4a c - b )atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ | 2 +--R (2a x + b)\|- a \|a x + b x + c +--R / +--R +---+ +--R 4a\|- a +--R Type: Expression Integer +--E + +--S 48 +cc1:=aa.1-bb1 +--R +--R (5) +--R +--------------+ +--R 2 2 | 2 +--R (4b c x + 8b c )\|a x + b x + c +--R + +--R 3 2 2 2 +-+ +--R ((- 4a b c - b )x - 8b c x - 8b c )\|c +--R / +--R +--------------+ +--R +-+ | 2 2 2 2 +--R (16a b x + 32a c)\|c \|a x + b x + c + (- 16a c - 4a b )x - 32a b c x +--R + +--R 2 +--R - 32a c +--R Type: Expression Integer +--E + +--S 49 +cc2:=aa.2-bb1 +--R +--R (6) +--R +--------------+ +--R 3 2 2 +---+ +-+ | 2 +--R ((- 16a b c + 4b )x - 32a c + 8b c)\|- a \|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +---+ +--R ((16a c - b )x + (32a b c - 8b c)x + 32a c - 8b c )\|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R 3 2 2 +-+ +-+ | 2 +--R ((32a b c - 8b )x + 64a c - 16b c)\|a \|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +-+ +--R ((- 32a c + 2b )x + (- 64a b c + 16b c)x - 64a c + 16b c )\|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R 2 2 +---+ +-+ | 2 +--R (8b c x + 16b c )\|- a \|a \|a x + b x + c +--R + +--R 3 2 2 2 +---+ +-+ +-+ +--R ((- 8a b c - 2b )x - 16b c x - 16b c )\|- a \|a \|c +--R / +--R +--------------+ +--R +---+ +-+ +-+ | 2 +--R (32a b x + 64a c)\|- a \|a \|c \|a x + b x + c +--R + +--R 2 2 2 2 +---+ +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|- a \|a +--R Type: Expression Integer +--E + +--S 50 +cc3:=aa.1-bb2 +--R +--R (7) +--R +--------------+ +--R 3 2 2 +---+ +-+ | 2 +--R ((16a b c - 4b )x + 32a c - 8b c)\|- a \|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +---+ +--R ((- 16a c + b )x + (- 32a b c + 8b c)x - 32a c + 8b c )\|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R 3 2 2 +-+ +-+ | 2 +--R ((- 32a b c + 8b )x - 64a c + 16b c)\|a \|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +-+ +--R ((32a c - 2b )x + (64a b c - 16b c)x + 64a c - 16b c )\|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R 2 2 +---+ +-+ | 2 +--R (8b c x + 16b c )\|- a \|a \|a x + b x + c +--R + +--R 3 2 2 2 +---+ +-+ +-+ +--R ((- 8a b c - 2b )x - 16b c x - 16b c )\|- a \|a \|c +--R / +--R +--------------+ +--R +---+ +-+ +-+ | 2 +--R (32a b x + 64a c)\|- a \|a \|c \|a x + b x + c +--R + +--R 2 2 2 2 +---+ +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|- a \|a +--R Type: Expression Integer +--E + +--S 51 +cc4:=aa.2-bb2 +--R +--R (8) +--R +--------------+ +--R 2 2 | 2 +--R (4b c x + 8b c )\|a x + b x + c +--R + +--R 3 2 2 2 +-+ +--R ((- 4a b c - b )x - 8b c x - 8b c )\|c +--R / +--R +--------------+ +--R +-+ | 2 2 2 2 +--R (16a b x + 32a c)\|c \|a x + b x + c + (- 16a c - 4a b )x - 32a b c x +--R + +--R 2 +--R - 32a c +--R Type: Expression Integer +--E + +--S 52 14:285 Schaums and Axiom differ by a constant +dd4:=ratDenom cc4 +--R +--R +-+ +--R b\|c +--R (9) ----- +--R 4a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.286~~~~~$\displaystyle +\int{x\sqrt{ax^2+bx+c}}~dx$} +$$ +\begin{array}{rl} +\displaystyle +\int{x\sqrt{ax^2+bx+c}}=&\displaystyle\frac{(ax^2+bx+c)^{3/2}}{3a}\\ +&\\ +&\displaystyle-\frac{b(2ax+b)}{8a^2}\sqrt{ax^2+bx+c}\\ +&\\ +&\displaystyle-\frac{b(4ac-b^2)}{16a^2}\int{\frac{1}{\sqrt{ax^2+bx+c}}} +\end{array} +$$ +<<*>>= +)clear all + +--S 53 +aa:=integrate(x*sqrt(a*x^2+b*x+c),x) +--R +--R +--R (1) +--R [ +--R 2 2 3 5 2 2 2 4 +--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x +--R + +--R 3 3 2 +--R 384a b c - 96b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 +--R (- 144a b c + 24a b c + 3b )x +--R + +--R 2 3 3 2 5 2 2 3 4 2 +--R (- 288a b c - 144a b c + 54b c)x + (- 576a b c + 144b c )x +--R + +--R 4 3 3 +--R - 384a b c + 96b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 3 2 3 5 3 2 2 2 4 4 +--R (- 192a b c - 16a b )x + (- 384a c - 336a b c - 4a b )x +--R + +--R 2 2 3 5 3 +--R (- 1056a b c - 16a b c + 6b )x +--R + +--R 2 3 2 2 4 2 3 3 2 +--R (- 768a c - 288a b c + 72b c)x + (- 384a b c + 96b c )x +--R * +--R +--------------+ +--R +-+ | 2 +--R \|a \|a x + b x + c +--R + +--R 4 3 2 6 3 2 3 5 +--R (128a c + 96a b )x + (672a b c + 120a b )x +--R + +--R 3 2 2 2 4 4 2 2 3 5 3 +--R (768a c + 816a b c - 12a b )x + (1632a b c + 64a b c - 30b )x +--R + +--R 2 3 2 2 4 2 3 3 2 +--R (768a c + 480a b c - 120b c)x + (384a b c - 96b c )x +--R * +--R +-+ +-+ +--R \|a \|c +--R / +--R 3 2 2 2 2 2 2 +-+ +-+ +--R ((384a c + 288a b )x + 1536a b c x + 1536a c )\|a \|c +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 2 2 2 +--R (- 576a b c - 48a b )x + (- 1152a c - 864a b c)x - 2304a b c x +--R + +--R 2 3 +--R - 1536a c +--R * +--R +-+ +--R \|a +--R , +--R +--R 2 2 3 5 2 2 2 4 +--R (- 96a b c - 48a b c + 18b )x + (- 384a b c + 96b c)x +--R + +--R 3 3 2 +--R - 384a b c + 96b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 2 3 3 2 5 2 +--R (144a b c - 24a b c - 3b )x + (288a b c + 144a b c - 54b c)x +--R + +--R 2 3 4 2 4 3 3 +--R (576a b c - 144b c )x + 384a b c - 96b c +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 3 2 3 5 3 2 2 2 4 4 +--R (- 96a b c - 8a b )x + (- 192a c - 168a b c - 2a b )x +--R + +--R 2 2 3 5 3 2 3 2 2 4 2 +--R (- 528a b c - 8a b c + 3b )x + (- 384a c - 144a b c + 36b c)x +--R + +--R 3 3 2 +--R (- 192a b c + 48b c )x +--R * +--R +--------------+ +--R +---+ | 2 +--R \|- a \|a x + b x + c +--R + +--R 4 3 2 6 3 2 3 5 +--R (64a c + 48a b )x + (336a b c + 60a b )x +--R + +--R 3 2 2 2 4 4 2 2 3 5 3 +--R (384a c + 408a b c - 6a b )x + (816a b c + 32a b c - 15b )x +--R + +--R 2 3 2 2 4 2 3 3 2 +--R (384a c + 240a b c - 60b c)x + (192a b c - 48b c )x +--R * +--R +---+ +-+ +--R \|- a \|c +--R / +--R 3 2 2 2 2 2 2 +---+ +-+ +--R ((192a c + 144a b )x + 768a b c x + 768a c )\|- a \|c +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 2 2 2 +--R (- 288a b c - 24a b )x + (- 576a c - 432a b c)x - 1152a b c x +--R + +--R 2 3 +--R - 768a c +--R * +--R +---+ +--R \|- a +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 54 +t1:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 55 +bb1:=(a*x^2+b*x+c)^(3/2)/(3*a)-(b*(2*a*x+b))/(8*a^2)*sqrt(a*x^2+b*x+c)-(b*(4*a*c-b^2))/(16*a^2)*t1.1 +--R +--R (3) +--R 3 +--R (- 12a b c + 3b ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R 2 2 2 +-+ | 2 +--R (16a x + 4a b x + 16a c - 6b )\|a \|a x + b x + c +--R / +--R 2 +-+ +--R 48a \|a +--R Type: Expression Integer +--E + +--S 56 +bb2:=(a*x^2+b*x+c)^(3/2)/(3*a)-(b*(2*a*x+b))/(8*a^2)*sqrt(a*x^2+b*x+c)-(b*(4*a*c-b^2))/(16*a^2)*t1.2 +--R +--R (4) +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R 3 \|- a \|a x + b x + c - \|- a \|c +--R (- 12a b c + 3b )atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R 2 2 2 +---+ | 2 +--R (8a x + 2a b x + 8a c - 3b )\|- a \|a x + b x + c +--R / +--R 2 +---+ +--R 24a \|- a +--R Type: Expression Integer +--E + +--S 57 +cc1:=aa.1-bb1 +--R +--R (5) +--R 2 2 3 5 2 2 2 4 3 +--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a b c +--R + +--R 3 2 +--R - 96b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 2 3 3 2 5 2 +--R (- 144a b c + 24a b c + 3b )x + (- 288a b c - 144a b c + 54b c)x +--R + +--R 2 3 4 2 4 3 3 +--R (- 576a b c + 144b c )x - 384a b c + 96b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 2 3 5 2 2 2 4 3 +--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a b c +--R + +--R 3 2 +--R - 96b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 2 3 3 2 5 2 +--R (- 144a b c + 24a b c + 3b )x + (- 288a b c - 144a b c + 54b c)x +--R + +--R 2 3 4 2 4 3 3 +--R (- 576a b c + 144b c )x - 384a b c + 96b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 3 2 2 4 2 3 3 2 4 +--R (128a c + 48a b c - 36b c)x + (512a b c - 192b c )x + 512a c +--R + +--R 2 3 +--R - 192b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|a \|a x + b x + c +--R + +--R 2 2 3 5 3 2 3 2 2 4 2 +--R (- 192a b c + 56a b c + 6b )x + (- 384a c - 144a b c + 108b c)x +--R + +--R 3 3 2 4 2 3 +--R (- 768a b c + 288b c )x - 512a c + 192b c +--R * +--R +-+ +-+ +--R \|a \|c +--R / +--R +--------------+ +--R 3 2 2 2 2 2 2 +-+ +-+ | 2 +--R ((384a c + 288a b )x + 1536a b c x + 1536a c )\|a \|c \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 2 2 2 +--R (- 576a b c - 48a b )x + (- 1152a c - 864a b c)x - 2304a b c x +--R + +--R 2 3 +--R - 1536a c +--R * +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 58 +cc2:=aa.2-bb1 +--R +--R (6) +--R 2 2 3 5 2 2 2 4 3 +--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a b c +--R + +--R 3 2 +--R - 96b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 +--R (- 144a b c + 24a b c + 3b )x +--R + +--R 2 3 3 2 5 2 2 3 4 2 +--R (- 288a b c - 144a b c + 54b c)x + (- 576a b c + 144b c )x +--R + +--R 4 3 3 +--R - 384a b c + 96b c +--R * +--R +---+ +--R \|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 2 3 5 2 2 2 4 +--R (- 192a b c - 96a b c + 36b )x + (- 768a b c + 192b c)x +--R + +--R 3 3 2 +--R - 768a b c + 192b c +--R * +--R +--------------+ +--R +-+ +-+ | 2 +--R \|a \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 +--R (288a b c - 48a b c - 6b )x +--R + +--R 2 3 3 2 5 2 2 3 4 2 +--R (576a b c + 288a b c - 108b c)x + (1152a b c - 288b c )x +--R + +--R 4 3 3 +--R 768a b c - 192b c +--R * +--R +-+ +--R \|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 2 3 2 2 4 2 3 3 2 4 +--R (128a c + 48a b c - 36b c)x + (512a b c - 192b c )x + 512a c +--R + +--R 2 3 +--R - 192b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|a \|a x + b x + c +--R + +--R 2 2 3 5 3 2 3 2 2 4 2 +--R (- 192a b c + 56a b c + 6b )x + (- 384a c - 144a b c + 108b c)x +--R + +--R 3 3 2 4 2 3 +--R (- 768a b c + 288b c )x - 512a c + 192b c +--R * +--R +---+ +-+ +-+ +--R \|- a \|a \|c +--R / +--R 3 2 2 2 2 2 2 +---+ +-+ +-+ +--R ((384a c + 288a b )x + 1536a b c x + 1536a c )\|- a \|a \|c +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 2 2 2 +--R (- 576a b c - 48a b )x + (- 1152a c - 864a b c)x - 2304a b c x +--R + +--R 2 3 +--R - 1536a c +--R * +--R +---+ +-+ +--R \|- a \|a +--R Type: Expression Integer +--E + +--S 59 +cc3:=aa.1-bb2 +--R +--R (7) +--R 2 2 3 5 2 2 2 4 3 +--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a b c +--R + +--R 3 2 +--R - 96b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 +--R (- 144a b c + 24a b c + 3b )x +--R + +--R 2 3 3 2 5 2 2 3 4 2 +--R (- 288a b c - 144a b c + 54b c)x + (- 576a b c + 144b c )x +--R + +--R 4 3 3 +--R - 384a b c + 96b c +--R * +--R +---+ +--R \|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 2 3 5 2 2 2 4 +--R (192a b c + 96a b c - 36b )x + (768a b c - 192b c)x +--R + +--R 3 3 2 +--R 768a b c - 192b c +--R * +--R +--------------+ +--R +-+ +-+ | 2 +--R \|a \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 +--R (- 288a b c + 48a b c + 6b )x +--R + +--R 2 3 3 2 5 2 2 3 4 2 +--R (- 576a b c - 288a b c + 108b c)x + (- 1152a b c + 288b c )x +--R + +--R 4 3 3 +--R - 768a b c + 192b c +--R * +--R +-+ +--R \|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 2 3 2 2 4 2 3 3 2 4 +--R (128a c + 48a b c - 36b c)x + (512a b c - 192b c )x + 512a c +--R + +--R 2 3 +--R - 192b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|a \|a x + b x + c +--R + +--R 2 2 3 5 3 2 3 2 2 4 2 +--R (- 192a b c + 56a b c + 6b )x + (- 384a c - 144a b c + 108b c)x +--R + +--R 3 3 2 4 2 3 +--R (- 768a b c + 288b c )x - 512a c + 192b c +--R * +--R +---+ +-+ +-+ +--R \|- a \|a \|c +--R / +--R 3 2 2 2 2 2 2 +---+ +-+ +-+ +--R ((384a c + 288a b )x + 1536a b c x + 1536a c )\|- a \|a \|c +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 2 2 2 +--R (- 576a b c - 48a b )x + (- 1152a c - 864a b c)x - 2304a b c x +--R + +--R 2 3 +--R - 1536a c +--R * +--R +---+ +-+ +--R \|- a \|a +--R Type: Expression Integer +--E + +--S 60 +cc4:=aa.2-bb2 +--R +--R (8) +--R 2 3 2 2 4 2 3 3 2 4 +--R (64a c + 24a b c - 18b c)x + (256a b c - 96b c )x + 256a c +--R + +--R 2 3 +--R - 96b c +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 2 3 5 3 2 3 2 2 4 2 +--R (- 96a b c + 28a b c + 3b )x + (- 192a c - 72a b c + 54b c)x +--R + +--R 3 3 2 4 2 3 +--R (- 384a b c + 144b c )x - 256a c + 96b c +--R * +--R +-+ +--R \|c +--R / +--R +--------------+ +--R 3 2 2 2 2 2 2 +-+ | 2 +--R ((192a c + 144a b )x + 768a b c x + 768a c )\|c \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 2 2 2 2 3 +--R (- 288a b c - 24a b )x + (- 576a c - 432a b c)x - 1152a b c x - 768a c +--R Type: Expression Integer +--E + +--S 61 14:286 Schaums and Axiom differ by a constant +dd4:=ratDenom cc4 +--R +--R 2 +-+ +--R (8a c - 3b )\|c +--R (9) ---------------- +--R 2 +--R 24a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.287~~~~~$\displaystyle +\int{x^2\sqrt{ax^2+bx+c}}~dx$} +$$\int{x^2\sqrt{ax^2+bx+c}}= +\frac{6ax-5b}{24a^2}(ax^2+bx+c)^{3/2}+ +\frac{5b^2-4ac}{16a^2}\int{\sqrt{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 62 +aa:=integrate(x^2*sqrt(a*x^2+b*x+c),x) +--R +--R +--R (1) +--R [ +--R 3 3 2 3 2 5 7 3 +--R (1536a b c - 1920a b c - 96a b c + 120b )x +--R + +--R 3 4 2 2 3 4 2 6 2 +--R (3072a c - 768a b c - 4800a b c + 1200b c)x +--R + +--R 2 4 3 3 5 2 2 5 2 4 +--R (9216a b c - 13824a b c + 2880b c )x + 6144a c - 9216a b c +--R + +--R 4 3 +--R 1920b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 4 4 2 4 2 6 8 4 +--R (- 768a c + 1440a b c - 288a b c - 15b )x +--R + +--R 3 4 2 3 3 5 2 7 3 +--R (- 6144a b c + 7680a b c + 384a b c - 480b c)x +--R + +--R 3 5 2 2 4 4 3 6 2 2 +--R (- 6144a c + 1536a b c + 9600a b c - 2400b c )x +--R + +--R 2 5 3 4 5 3 2 6 2 5 +--R (- 12288a b c + 18432a b c - 3840b c )x - 6144a c + 9216a b c +--R + +--R 4 4 +--R - 1920b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 5 2 4 2 3 4 7 +--R (- 1536a c - 2304a b c - 96a b )x +--R + +--R 4 2 3 3 2 5 6 +--R (- 12544a b c - 3456a b c - 16a b )x +--R + +--R 4 3 3 2 2 2 4 6 5 +--R (- 13056a c - 18240a b c - 80a b c + 20a b )x +--R + +--R 3 3 2 3 2 5 7 4 +--R (- 31104a b c + 480a b c + 24a b c - 30b )x +--R + +--R 3 4 2 2 3 4 2 6 3 +--R (- 18432a c + 768a b c + 2816a b c - 720b c)x +--R + +--R 2 4 3 3 5 2 2 +--R (- 7680a b c + 11520a b c - 2400b c )x +--R + +--R 2 5 2 4 4 3 +--R (- 6144a c + 9216a b c - 1920b c )x +--R * +--R +--------------+ +--R +-+ | 2 +--R \|a \|a x + b x + c +--R + +--R 5 4 3 8 5 2 4 2 3 4 7 +--R (3072a b c + 768a b )x + (6144a c + 11264a b c + 896a b )x +--R + +--R 4 2 3 3 2 5 6 +--R (30208a b c + 9984a b c - 32a b )x +--R + +--R 4 3 3 2 2 2 4 6 5 +--R (21504a c + 31488a b c - 320a b c + 80a b )x +--R + +--R 3 3 2 3 2 5 7 4 +--R (42624a b c - 4896a b c + 152a b c + 210b )x +--R + +--R 3 4 2 2 3 4 2 6 3 +--R (21504a c - 2304a b c - 6464a b c + 1680b c)x +--R + +--R 2 4 3 3 5 2 2 +--R (10752a b c - 16128a b c + 3360b c )x +--R + +--R 2 5 2 4 4 3 +--R (6144a c - 9216a b c + 1920b c )x +--R * +--R +-+ +-+ +--R \|a \|c +--R / +--R 4 3 3 3 4 2 3 2 2 +--R (12288a b c + 3072a b )x + (24576a c + 30720a b c)x +--R + +--R 3 2 3 3 +--R 73728a b c x + 49152a c +--R * +--R +--------------+ +--R +-+ +-+ | 2 +--R \|a \|c \|a x + b x + c +--R + +--R 5 2 4 2 3 4 4 +--R (- 6144a c - 9216a b c - 384a b )x +--R + +--R 4 2 3 3 3 4 3 3 2 2 2 +--R (- 49152a b c - 12288a b c)x + (- 49152a c - 61440a b c )x +--R + +--R 3 3 3 4 +--R - 98304a b c x - 49152a c +--R * +--R +-+ +--R \|a +--R , +--R +--R 3 3 2 3 2 5 7 3 +--R (- 1536a b c + 1920a b c + 96a b c - 120b )x +--R + +--R 3 4 2 2 3 4 2 6 2 +--R (- 3072a c + 768a b c + 4800a b c - 1200b c)x +--R + +--R 2 4 3 3 5 2 2 5 +--R (- 9216a b c + 13824a b c - 2880b c )x - 6144a c +--R + +--R 2 4 4 3 +--R 9216a b c - 1920b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 4 4 2 4 2 6 8 4 +--R (768a c - 1440a b c + 288a b c + 15b )x +--R + +--R 3 4 2 3 3 5 2 7 3 +--R (6144a b c - 7680a b c - 384a b c + 480b c)x +--R + +--R 3 5 2 2 4 4 3 6 2 2 +--R (6144a c - 1536a b c - 9600a b c + 2400b c )x +--R + +--R 2 5 3 4 5 3 2 6 2 5 +--R (12288a b c - 18432a b c + 3840b c )x + 6144a c - 9216a b c +--R + +--R 4 4 +--R 1920b c +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 5 2 4 2 3 4 7 +--R (- 768a c - 1152a b c - 48a b )x +--R + +--R 4 2 3 3 2 5 6 +--R (- 6272a b c - 1728a b c - 8a b )x +--R + +--R 4 3 3 2 2 2 4 6 5 +--R (- 6528a c - 9120a b c - 40a b c + 10a b )x +--R + +--R 3 3 2 3 2 5 7 4 +--R (- 15552a b c + 240a b c + 12a b c - 15b )x +--R + +--R 3 4 2 2 3 4 2 6 3 +--R (- 9216a c + 384a b c + 1408a b c - 360b c)x +--R + +--R 2 4 3 3 5 2 2 +--R (- 3840a b c + 5760a b c - 1200b c )x +--R + +--R 2 5 2 4 4 3 +--R (- 3072a c + 4608a b c - 960b c )x +--R * +--R +--------------+ +--R +---+ | 2 +--R \|- a \|a x + b x + c +--R + +--R 5 4 3 8 5 2 4 2 3 4 7 +--R (1536a b c + 384a b )x + (3072a c + 5632a b c + 448a b )x +--R + +--R 4 2 3 3 2 5 6 +--R (15104a b c + 4992a b c - 16a b )x +--R + +--R 4 3 3 2 2 2 4 6 5 +--R (10752a c + 15744a b c - 160a b c + 40a b )x +--R + +--R 3 3 2 3 2 5 7 4 +--R (21312a b c - 2448a b c + 76a b c + 105b )x +--R + +--R 3 4 2 2 3 4 2 6 3 +--R (10752a c - 1152a b c - 3232a b c + 840b c)x +--R + +--R 2 4 3 3 5 2 2 +--R (5376a b c - 8064a b c + 1680b c )x +--R + +--R 2 5 2 4 4 3 +--R (3072a c - 4608a b c + 960b c )x +--R * +--R +---+ +-+ +--R \|- a \|c +--R / +--R 4 3 3 3 4 2 3 2 2 +--R (6144a b c + 1536a b )x + (12288a c + 15360a b c)x +--R + +--R 3 2 3 3 +--R 36864a b c x + 24576a c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|c \|a x + b x + c +--R + +--R 5 2 4 2 3 4 4 +--R (- 3072a c - 4608a b c - 192a b )x +--R + +--R 4 2 3 3 3 4 3 3 2 2 2 +--R (- 24576a b c - 6144a b c)x + (- 24576a c - 30720a b c )x +--R + +--R 3 3 3 4 +--R - 49152a b c x - 24576a c +--R * +--R +---+ +--R \|- a +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 63 +t1:=integrate(sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((16a b c - 4b )x + 32a c - 8b c)\|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +--R (- 16a c + b )x + (- 32a b c + 8b c)x - 32a c + 8b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 2 3 3 2 2 2 +-+ +--R ((- 16a c - 4a b )x + (- 40a b c - 2b )x + (- 32a c - 8b c)x)\|a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 4 2 2 3 3 2 +--R 16a b x + (32a c + 24a b )x + (56a b c + 6b )x +--R + +--R 2 2 +--R (32a c + 8b c)x +--R * +--R +-+ +-+ +--R \|a \|c +--R / +--R +--------------+ +--R +-+ +-+ | 2 +--R (32a b x + 64a c)\|a \|c \|a x + b x + c +--R + +--R 2 2 2 2 +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|a +--R , +--R +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((16a b c - 4b )x + 32a c - 8b c)\|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +--R (- 16a c + b )x + (- 32a b c + 8b c)x - 32a c + 8b c +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 2 2 3 3 2 2 2 +---+ +--R ((- 8a c - 2a b )x + (- 20a b c - b )x + (- 16a c - 4b c)x)\|- a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 4 2 2 3 3 2 2 2 +--R (8a b x + (16a c + 12a b )x + (28a b c + 3b )x + (16a c + 4b c)x) +--R * +--R +---+ +-+ +--R \|- a \|c +--R / +--R +--------------+ +--R +---+ +-+ | 2 +--R (16a b x + 32a c)\|- a \|c \|a x + b x + c +--R + +--R 2 2 2 2 +---+ +--R ((- 16a c - 4a b )x - 32a b c x - 32a c )\|- a +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 64 +bb1:=(6*a*x-5*b)/(24*a^2)*(a*x^2+b*x+c)^(3/2)+(5*b^2-4*a*c)/(16*a^2)*t1.1 +--R +--R (3) +--R 2 2 3 5 2 3 2 2 4 +--R ((- 192a b c + 288a b c - 60b )x - 384a c + 576a b c - 120b c) +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 3 3 2 2 2 4 6 2 +--R (192a c - 240a b c - 12a b c + 15b )x +--R + +--R 2 3 3 2 5 2 4 2 3 4 2 +--R (384a b c - 576a b c + 120b c)x + 384a c - 576a b c + 120b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 4 3 2 5 3 2 3 4 +--R (- 384a c - 96a b )x + (- 832a b c - 16a b )x +--R + +--R 3 2 2 2 4 3 2 2 3 5 2 +--R (- 960a c - 96a b c + 20a b )x + (- 96a b c + 144a b c - 30b )x +--R + +--R 2 3 2 2 4 3 +--R (- 384a c + 896a b c - 120b c)x + 640a b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|a \|a x + b x + c +--R + +--R 4 6 4 3 2 5 3 2 3 4 +--R 384a b x + (768a c + 448a b )x + (1472a b c - 16a b )x +--R + +--R 3 2 2 2 4 3 2 2 3 5 2 +--R (1152a c - 192a b c + 40a b )x + (- 32a b c - 512a b c + 90b )x +--R + +--R 2 3 2 2 4 3 +--R (384a c - 1216a b c + 120b c)x - 640a b c +--R * +--R +-+ +-+ +--R \|a \|c +--R / +--R +--------------+ +--R 3 3 +-+ +-+ | 2 +--R (1536a b x + 3072a c)\|a \|c \|a x + b x + c +--R + +--R 4 3 2 2 3 3 2 +-+ +--R ((- 1536a c - 384a b )x - 3072a b c x - 3072a c )\|a +--R Type: Expression Integer +--E + +--S 65 +bb2:=(6*a*x-5*b)/(24*a^2)*(a*x^2+b*x+c)^(3/2)+(5*b^2-4*a*c)/(16*a^2)*t1.2 +--R +--R (4) +--R 2 2 3 5 2 3 2 2 4 +--R ((- 192a b c + 288a b c - 60b )x - 384a c + 576a b c - 120b c) +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 3 3 2 2 2 4 6 2 +--R (192a c - 240a b c - 12a b c + 15b )x +--R + +--R 2 3 3 2 5 2 4 2 3 4 2 +--R (384a b c - 576a b c + 120b c)x + 384a c - 576a b c + 120b c +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 4 3 2 5 3 2 3 4 +--R (- 192a c - 48a b )x + (- 416a b c - 8a b )x +--R + +--R 3 2 2 2 4 3 2 2 3 5 2 +--R (- 480a c - 48a b c + 10a b )x + (- 48a b c + 72a b c - 15b )x +--R + +--R 2 3 2 2 4 3 +--R (- 192a c + 448a b c - 60b c)x + 320a b c +--R * +--R +--------------+ +--R +---+ | 2 +--R \|- a \|a x + b x + c +--R + +--R 4 6 4 3 2 5 3 2 3 4 +--R 192a b x + (384a c + 224a b )x + (736a b c - 8a b )x +--R + +--R 3 2 2 2 4 3 2 2 3 5 2 +--R (576a c - 96a b c + 20a b )x + (- 16a b c - 256a b c + 45b )x +--R + +--R 2 3 2 2 4 3 +--R (192a c - 608a b c + 60b c)x - 320a b c +--R * +--R +---+ +-+ +--R \|- a \|c +--R / +--R +--------------+ +--R 3 3 +---+ +-+ | 2 +--R (768a b x + 1536a c)\|- a \|c \|a x + b x + c +--R + +--R 4 3 2 2 3 3 2 +---+ +--R ((- 768a c - 192a b )x - 1536a b c x - 1536a c )\|- a +--R Type: Expression Integer +--E + +--S 66 +cc1:=aa.1-bb1 +--R +--R (5) +--R 4 4 3 3 3 2 5 2 7 9 5 +--R (9216a b c - 6144a b c - 8064a b c + 1536a b c + 180b )x +--R + +--R 4 5 3 2 4 2 4 3 8 4 +--R (18432a c + 36864a b c - 77568a b c + 4200b c)x +--R + +--R 3 5 2 3 4 5 3 7 2 3 +--R (147456a b c - 135168a b c - 82944a b c + 26880b c )x +--R + +--R 3 6 2 2 5 4 4 6 3 2 +--R (98304a c + 73728a b c - 301056a b c + 69120b c )x +--R + +--R 2 6 3 5 5 4 2 7 +--R (245760a b c - 368640a b c + 76800b c )x + 98304a c +--R + +--R 2 6 4 5 +--R - 147456a b c + 30720b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 5 5 4 2 4 3 4 3 2 6 2 8 +--R - 3072a c - 6912a b c + 13440a b c + 672a b c - 828a b c +--R + +--R 10 +--R - 15b +--R * +--R 6 +--R x +--R + +--R 4 5 3 3 4 2 5 3 7 2 9 5 +--R (- 55296a b c + 36864a b c + 48384a b c - 9216a b c - 1080b c)x +--R + +--R 4 6 3 2 5 2 4 4 8 2 4 +--R (- 55296a c - 110592a b c + 232704a b c - 12600b c )x +--R + +--R 3 6 2 3 5 5 4 7 3 3 +--R (- 294912a b c + 270336a b c + 165888a b c - 53760b c )x +--R + +--R 3 7 2 2 6 4 5 6 4 2 +--R (- 147456a c - 110592a b c + 451584a b c - 103680b c )x +--R + +--R 2 7 3 6 5 5 2 8 +--R (- 294912a b c + 442368a b c - 92160b c )x - 98304a c +--R + +--R 2 7 4 6 +--R 147456a b c - 30720b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 4 4 3 3 3 2 5 2 7 9 5 +--R (9216a b c - 6144a b c - 8064a b c + 1536a b c + 180b )x +--R + +--R 4 5 3 2 4 2 4 3 8 4 +--R (18432a c + 36864a b c - 77568a b c + 4200b c)x +--R + +--R 3 5 2 3 4 5 3 7 2 3 +--R (147456a b c - 135168a b c - 82944a b c + 26880b c )x +--R + +--R 3 6 2 2 5 4 4 6 3 2 +--R (98304a c + 73728a b c - 301056a b c + 69120b c )x +--R + +--R 2 6 3 5 5 4 2 7 +--R (245760a b c - 368640a b c + 76800b c )x + 98304a c +--R + +--R 2 6 4 5 +--R - 147456a b c + 30720b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 5 5 4 2 4 3 4 3 2 6 2 8 +--R - 3072a c - 6912a b c + 13440a b c + 672a b c - 828a b c +--R + +--R 10 +--R - 15b +--R * +--R 6 +--R x +--R + +--R 4 5 3 3 4 2 5 3 7 2 9 5 +--R (- 55296a b c + 36864a b c + 48384a b c - 9216a b c - 1080b c)x +--R + +--R 4 6 3 2 5 2 4 4 8 2 4 +--R (- 55296a c - 110592a b c + 232704a b c - 12600b c )x +--R + +--R 3 6 2 3 5 5 4 7 3 3 +--R (- 294912a b c + 270336a b c + 165888a b c - 53760b c )x +--R + +--R 3 7 2 2 6 4 5 6 4 2 +--R (- 147456a c - 110592a b c + 451584a b c - 103680b c )x +--R + +--R 2 7 3 6 5 5 2 8 +--R (- 294912a b c + 442368a b c - 92160b c )x - 98304a c +--R + +--R 2 7 4 6 +--R 147456a b c - 30720b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 3 2 4 2 4 3 6 2 5 +--R (- 15360a b c - 12800a b c - 960a b c )x +--R + +--R 3 5 2 3 4 5 3 4 +--R (- 30720a b c - 107520a b c - 22400a b c )x +--R + +--R 2 2 5 4 4 3 2 6 3 5 2 +--R (- 245760a b c - 143360a b c )x + (- 163840a b c - 368640a b c )x +--R + +--R 2 6 7 +--R - 409600a b c x - 163840a b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|a \|a x + b x + c +--R + +--R 4 4 3 3 3 2 5 2 7 6 +--R (5120a b c + 19200a b c + 4800a b c + 80a b c)x +--R + +--R 3 2 4 2 4 3 6 2 5 +--R (92160a b c + 76800a b c + 5760a b c )x +--R + +--R 3 5 2 3 4 5 3 4 +--R (92160a b c + 322560a b c + 67200a b c )x +--R + +--R 2 2 5 4 4 3 2 6 3 5 2 +--R (491520a b c + 286720a b c )x + (245760a b c + 552960a b c )x +--R + +--R 2 6 7 +--R 491520a b c x + 163840a b c +--R * +--R +-+ +-+ +--R \|a \|c +--R / +--R 5 2 4 3 3 5 5 +--R (73728a b c + 61440a b c + 4608a b )x +--R + +--R 5 3 4 2 2 3 4 4 +--R (147456a c + 516096a b c + 107520a b c)x +--R + +--R 4 3 3 3 2 3 4 4 3 2 3 2 +--R (1179648a b c + 688128a b c )x + (786432a c + 1769472a b c )x +--R + +--R 3 4 3 5 +--R 1966080a b c x + 786432a c +--R * +--R +--------------+ +--R +-+ +-+ | 2 +--R \|a \|c \|a x + b x + c +--R + +--R 6 3 5 2 2 4 4 3 6 6 +--R (- 24576a c - 92160a b c - 23040a b c - 384a b )x +--R + +--R 5 3 4 3 2 3 5 5 +--R (- 442368a b c - 368640a b c - 27648a b c)x +--R + +--R 5 4 4 2 3 3 4 2 4 +--R (- 442368a c - 1548288a b c - 322560a b c )x +--R + +--R 4 4 3 3 3 3 +--R (- 2359296a b c - 1376256a b c )x +--R + +--R 4 5 3 2 4 2 3 5 3 6 +--R (- 1179648a c - 2654208a b c )x - 2359296a b c x - 786432a c +--R * +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 67 +cc2:=aa.2-bb1 +--R +--R (6) +--R 4 4 3 3 3 2 5 2 7 9 5 +--R (9216a b c - 6144a b c - 8064a b c + 1536a b c + 180b )x +--R + +--R 4 5 3 2 4 2 4 3 8 4 +--R (18432a c + 36864a b c - 77568a b c + 4200b c)x +--R + +--R 3 5 2 3 4 5 3 7 2 3 +--R (147456a b c - 135168a b c - 82944a b c + 26880b c )x +--R + +--R 3 6 2 2 5 4 4 6 3 2 +--R (98304a c + 73728a b c - 301056a b c + 69120b c )x +--R + +--R 2 6 3 5 5 4 2 7 +--R (245760a b c - 368640a b c + 76800b c )x + 98304a c +--R + +--R 2 6 4 5 +--R - 147456a b c + 30720b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|c \|a x + b x + c +--R + +--R 5 5 4 2 4 3 4 3 2 6 2 8 +--R - 3072a c - 6912a b c + 13440a b c + 672a b c - 828a b c +--R + +--R 10 +--R - 15b +--R * +--R 6 +--R x +--R + +--R 4 5 3 3 4 2 5 3 7 2 +--R - 55296a b c + 36864a b c + 48384a b c - 9216a b c +--R + +--R 9 +--R - 1080b c +--R * +--R 5 +--R x +--R + +--R 4 6 3 2 5 2 4 4 8 2 4 +--R (- 55296a c - 110592a b c + 232704a b c - 12600b c )x +--R + +--R 3 6 2 3 5 5 4 7 3 3 +--R (- 294912a b c + 270336a b c + 165888a b c - 53760b c )x +--R + +--R 3 7 2 2 6 4 5 6 4 2 +--R (- 147456a c - 110592a b c + 451584a b c - 103680b c )x +--R + +--R 2 7 3 6 5 5 2 8 +--R (- 294912a b c + 442368a b c - 92160b c )x - 98304a c +--R + +--R 2 7 4 6 +--R 147456a b c - 30720b c +--R * +--R +---+ +--R \|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 4 4 3 3 3 2 5 2 7 9 5 +--R (- 18432a b c + 12288a b c + 16128a b c - 3072a b c - 360b )x +--R + +--R 4 5 3 2 4 2 4 3 8 4 +--R (- 36864a c - 73728a b c + 155136a b c - 8400b c)x +--R + +--R 3 5 2 3 4 5 3 7 2 3 +--R (- 294912a b c + 270336a b c + 165888a b c - 53760b c )x +--R + +--R 3 6 2 2 5 4 4 6 3 2 +--R (- 196608a c - 147456a b c + 602112a b c - 138240b c )x +--R + +--R 2 6 3 5 5 4 2 7 +--R (- 491520a b c + 737280a b c - 153600b c )x - 196608a c +--R + +--R 2 6 4 5 +--R 294912a b c - 61440b c +--R * +--R +--------------+ +--R +-+ +-+ | 2 +--R \|a \|c \|a x + b x + c +--R + +--R 5 5 4 2 4 3 4 3 2 6 2 8 +--R 6144a c + 13824a b c - 26880a b c - 1344a b c + 1656a b c +--R + +--R 10 +--R 30b +--R * +--R 6 +--R x +--R + +--R 4 5 3 3 4 2 5 3 7 2 +--R 110592a b c - 73728a b c - 96768a b c + 18432a b c +--R + +--R 9 +--R 2160b c +--R * +--R 5 +--R x +--R + +--R 4 6 3 2 5 2 4 4 8 2 4 +--R (110592a c + 221184a b c - 465408a b c + 25200b c )x +--R + +--R 3 6 2 3 5 5 4 7 3 3 +--R (589824a b c - 540672a b c - 331776a b c + 107520b c )x +--R + +--R 3 7 2 2 6 4 5 6 4 2 +--R (294912a c + 221184a b c - 903168a b c + 207360b c )x +--R + +--R 2 7 3 6 5 5 2 8 +--R (589824a b c - 884736a b c + 184320b c )x + 196608a c +--R + +--R 2 7 4 6 +--R - 294912a b c + 61440b c +--R * +--R +-+ +--R \|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 3 2 4 2 4 3 6 2 5 +--R (- 15360a b c - 12800a b c - 960a b c )x +--R + +--R 3 5 2 3 4 5 3 4 +--R (- 30720a b c - 107520a b c - 22400a b c )x +--R + +--R 2 2 5 4 4 3 2 6 3 5 2 +--R (- 245760a b c - 143360a b c )x + (- 163840a b c - 368640a b c )x +--R + +--R 2 6 7 +--R - 409600a b c x - 163840a b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|a \|a x + b x + c +--R + +--R 4 4 3 3 3 2 5 2 7 6 +--R (5120a b c + 19200a b c + 4800a b c + 80a b c)x +--R + +--R 3 2 4 2 4 3 6 2 5 +--R (92160a b c + 76800a b c + 5760a b c )x +--R + +--R 3 5 2 3 4 5 3 4 +--R (92160a b c + 322560a b c + 67200a b c )x +--R + +--R 2 2 5 4 4 3 2 6 3 5 2 +--R (491520a b c + 286720a b c )x + (245760a b c + 552960a b c )x +--R + +--R 2 6 7 +--R 491520a b c x + 163840a b c +--R * +--R +---+ +-+ +-+ +--R \|- a \|a \|c +--R / +--R 5 2 4 3 3 5 5 +--R (73728a b c + 61440a b c + 4608a b )x +--R + +--R 5 3 4 2 2 3 4 4 +--R (147456a c + 516096a b c + 107520a b c)x +--R + +--R 4 3 3 3 2 3 4 4 3 2 3 2 +--R (1179648a b c + 688128a b c )x + (786432a c + 1769472a b c )x +--R + +--R 3 4 3 5 +--R 1966080a b c x + 786432a c +--R * +--R +--------------+ +--R +---+ +-+ +-+ | 2 +--R \|- a \|a \|c \|a x + b x + c +--R + +--R 6 3 5 2 2 4 4 3 6 6 +--R (- 24576a c - 92160a b c - 23040a b c - 384a b )x +--R + +--R 5 3 4 3 2 3 5 5 +--R (- 442368a b c - 368640a b c - 27648a b c)x +--R + +--R 5 4 4 2 3 3 4 2 4 +--R (- 442368a c - 1548288a b c - 322560a b c )x +--R + +--R 4 4 3 3 3 3 +--R (- 2359296a b c - 1376256a b c )x +--R + +--R 4 5 3 2 4 2 3 5 3 6 +--R (- 1179648a c - 2654208a b c )x - 2359296a b c x - 786432a c +--R * +--R +---+ +-+ +--R \|- a \|a +--R Type: Expression Integer +--E + +--S 68 +cc3:=aa.1-bb2 +--R +--R (7) +--R 4 4 3 3 3 2 5 2 7 9 5 +--R (9216a b c - 6144a b c - 8064a b c + 1536a b c + 180b )x +--R + +--R 4 5 3 2 4 2 4 3 8 4 +--R (18432a c + 36864a b c - 77568a b c + 4200b c)x +--R + +--R 3 5 2 3 4 5 3 7 2 3 +--R (147456a b c - 135168a b c - 82944a b c + 26880b c )x +--R + +--R 3 6 2 2 5 4 4 6 3 2 +--R (98304a c + 73728a b c - 301056a b c + 69120b c )x +--R + +--R 2 6 3 5 5 4 2 7 +--R (245760a b c - 368640a b c + 76800b c )x + 98304a c +--R + +--R 2 6 4 5 +--R - 147456a b c + 30720b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|c \|a x + b x + c +--R + +--R 5 5 4 2 4 3 4 3 2 6 2 8 +--R - 3072a c - 6912a b c + 13440a b c + 672a b c - 828a b c +--R + +--R 10 +--R - 15b +--R * +--R 6 +--R x +--R + +--R 4 5 3 3 4 2 5 3 7 2 +--R - 55296a b c + 36864a b c + 48384a b c - 9216a b c +--R + +--R 9 +--R - 1080b c +--R * +--R 5 +--R x +--R + +--R 4 6 3 2 5 2 4 4 8 2 4 +--R (- 55296a c - 110592a b c + 232704a b c - 12600b c )x +--R + +--R 3 6 2 3 5 5 4 7 3 3 +--R (- 294912a b c + 270336a b c + 165888a b c - 53760b c )x +--R + +--R 3 7 2 2 6 4 5 6 4 2 +--R (- 147456a c - 110592a b c + 451584a b c - 103680b c )x +--R + +--R 2 7 3 6 5 5 2 8 +--R (- 294912a b c + 442368a b c - 92160b c )x - 98304a c +--R + +--R 2 7 4 6 +--R 147456a b c - 30720b c +--R * +--R +---+ +--R \|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 4 4 3 3 3 2 5 2 7 9 5 +--R (18432a b c - 12288a b c - 16128a b c + 3072a b c + 360b )x +--R + +--R 4 5 3 2 4 2 4 3 8 4 +--R (36864a c + 73728a b c - 155136a b c + 8400b c)x +--R + +--R 3 5 2 3 4 5 3 7 2 3 +--R (294912a b c - 270336a b c - 165888a b c + 53760b c )x +--R + +--R 3 6 2 2 5 4 4 6 3 2 +--R (196608a c + 147456a b c - 602112a b c + 138240b c )x +--R + +--R 2 6 3 5 5 4 2 7 +--R (491520a b c - 737280a b c + 153600b c )x + 196608a c +--R + +--R 2 6 4 5 +--R - 294912a b c + 61440b c +--R * +--R +--------------+ +--R +-+ +-+ | 2 +--R \|a \|c \|a x + b x + c +--R + +--R 5 5 4 2 4 3 4 3 2 6 2 +--R - 6144a c - 13824a b c + 26880a b c + 1344a b c +--R + +--R 8 10 +--R - 1656a b c - 30b +--R * +--R 6 +--R x +--R + +--R 4 5 3 3 4 2 5 3 7 2 +--R - 110592a b c + 73728a b c + 96768a b c - 18432a b c +--R + +--R 9 +--R - 2160b c +--R * +--R 5 +--R x +--R + +--R 4 6 3 2 5 2 4 4 8 2 4 +--R (- 110592a c - 221184a b c + 465408a b c - 25200b c )x +--R + +--R 3 6 2 3 5 5 4 7 3 3 +--R (- 589824a b c + 540672a b c + 331776a b c - 107520b c )x +--R + +--R 3 7 2 2 6 4 5 6 4 2 +--R (- 294912a c - 221184a b c + 903168a b c - 207360b c )x +--R + +--R 2 7 3 6 5 5 2 8 +--R (- 589824a b c + 884736a b c - 184320b c )x - 196608a c +--R + +--R 2 7 4 6 +--R 294912a b c - 61440b c +--R * +--R +-+ +--R \|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 3 2 4 2 4 3 6 2 5 +--R (- 15360a b c - 12800a b c - 960a b c )x +--R + +--R 3 5 2 3 4 5 3 4 +--R (- 30720a b c - 107520a b c - 22400a b c )x +--R + +--R 2 2 5 4 4 3 2 6 3 5 2 +--R (- 245760a b c - 143360a b c )x + (- 163840a b c - 368640a b c )x +--R + +--R 2 6 7 +--R - 409600a b c x - 163840a b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|a \|a x + b x + c +--R + +--R 4 4 3 3 3 2 5 2 7 6 +--R (5120a b c + 19200a b c + 4800a b c + 80a b c)x +--R + +--R 3 2 4 2 4 3 6 2 5 +--R (92160a b c + 76800a b c + 5760a b c )x +--R + +--R 3 5 2 3 4 5 3 4 +--R (92160a b c + 322560a b c + 67200a b c )x +--R + +--R 2 2 5 4 4 3 2 6 3 5 2 +--R (491520a b c + 286720a b c )x + (245760a b c + 552960a b c )x +--R + +--R 2 6 7 +--R 491520a b c x + 163840a b c +--R * +--R +---+ +-+ +-+ +--R \|- a \|a \|c +--R / +--R 5 2 4 3 3 5 5 +--R (73728a b c + 61440a b c + 4608a b )x +--R + +--R 5 3 4 2 2 3 4 4 +--R (147456a c + 516096a b c + 107520a b c)x +--R + +--R 4 3 3 3 2 3 4 4 3 2 3 2 +--R (1179648a b c + 688128a b c )x + (786432a c + 1769472a b c )x +--R + +--R 3 4 3 5 +--R 1966080a b c x + 786432a c +--R * +--R +--------------+ +--R +---+ +-+ +-+ | 2 +--R \|- a \|a \|c \|a x + b x + c +--R + +--R 6 3 5 2 2 4 4 3 6 6 +--R (- 24576a c - 92160a b c - 23040a b c - 384a b )x +--R + +--R 5 3 4 3 2 3 5 5 +--R (- 442368a b c - 368640a b c - 27648a b c)x +--R + +--R 5 4 4 2 3 3 4 2 4 +--R (- 442368a c - 1548288a b c - 322560a b c )x +--R + +--R 4 4 3 3 3 3 +--R (- 2359296a b c - 1376256a b c )x +--R + +--R 4 5 3 2 4 2 3 5 3 6 +--R (- 1179648a c - 2654208a b c )x - 2359296a b c x - 786432a c +--R * +--R +---+ +-+ +--R \|- a \|a +--R Type: Expression Integer +--E + +--S 69 +cc4:=aa.2-bb2 +--R +--R (8) +--R 2 2 4 4 3 6 2 5 +--R (- 960a b c - 800a b c - 60b c )x +--R + +--R 2 5 3 4 5 3 4 +--R (- 1920a b c - 6720a b c - 1400b c )x +--R + +--R 2 5 4 4 3 6 3 5 2 +--R (- 15360a b c - 8960b c )x + (- 10240a b c - 23040b c )x +--R + +--R 2 6 7 +--R - 25600b c x - 10240b c +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 3 4 2 3 3 5 2 7 6 +--R (320a b c + 1200a b c + 300a b c + 5b c)x +--R + +--R 2 2 4 4 3 6 2 5 +--R (5760a b c + 4800a b c + 360b c )x +--R + +--R 2 5 3 4 5 3 4 2 5 4 4 3 +--R (5760a b c + 20160a b c + 4200b c )x + (30720a b c + 17920b c )x +--R + +--R 6 3 5 2 2 6 7 +--R (15360a b c + 34560b c )x + 30720b c x + 10240b c +--R * +--R +-+ +--R \|c +--R / +--R 4 2 3 3 2 5 5 +--R (4608a b c + 3840a b c + 288a b )x +--R + +--R 4 3 3 2 2 2 4 4 +--R (9216a c + 32256a b c + 6720a b c)x +--R + +--R 3 3 2 3 2 3 3 4 2 2 3 2 +--R (73728a b c + 43008a b c )x + (49152a c + 110592a b c )x +--R + +--R 2 4 2 5 +--R 122880a b c x + 49152a c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 5 3 4 2 2 3 4 2 6 6 +--R (- 1536a c - 5760a b c - 1440a b c - 24a b )x +--R + +--R 4 3 3 3 2 2 5 5 +--R (- 27648a b c - 23040a b c - 1728a b c)x +--R + +--R 4 4 3 2 3 2 4 2 4 +--R (- 27648a c - 96768a b c - 20160a b c )x +--R + +--R 3 4 2 3 3 3 3 5 2 2 4 2 +--R (- 147456a b c - 86016a b c )x + (- 73728a c - 165888a b c )x +--R + +--R 2 5 2 6 +--R - 147456a b c x - 49152a c +--R Type: Expression Integer +--E + +--S 70 14:287 Schaums and Axiom differ by a constant +dd4:=ratDenom cc4 +--R +--R +-+ +--R 5b c\|c +--R (9) - -------- +--R 2 +--R 24a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.288~~~~~$\displaystyle +\int{\frac{\sqrt{ax^2+bx+c}}{x}}~dx$} +$$\int{\frac{\sqrt{ax^2+bx+c}}{x}}= +\sqrt{ax^2+bx+c}+\frac{b}{2}\int{\frac{1}{\sqrt{ax^2+bx+c}}} ++c\int{\frac{1}{x\sqrt{ax^2+bx+c}}} +$$ +<<*>>= +)clear all + +--S 71 +aa:=integrate(sqrt(a*x^2+b*x+c)/x,x) +--R +--R +--R (1) +--R [ +--R +--------------+ +--R +-+ | 2 +-+ +-+ +--R (4c\|a \|a x + b x + c + (- 2b x - 4c)\|a \|c ) +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R +-+ +--R 2x\|c +--R + +--R +--------------+ +--R +-+ | 2 2 +--R (2b\|c \|a x + b x + c - b x - 2b c) +--R * +--R log +--R 2 +-+ 2 2 +-+ +--R ((- 2a b x - 8a c x)\|c + (4a c x + 4b c x + 8c )\|a ) +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 3 2 2 2 +-+ +-+ 2 3 +--R (- 2a b x + (- 8a c - b )x - 8b c x - 8c )\|a \|c + 4a c x +--R + +--R 2 2 +--R 6a b c x + 8a c x +--R / +--R +--------------+ +--R 2 | 2 +--R (4b c x + 8c )\|a x + b x + c +--R + +--R 2 2 2 +-+ +--R ((- 4a c - b )x - 8b c x - 8c )\|c +--R + +--R +--------------+ +--R +-+ | 2 2 +-+ +-+ +--R - 2b x\|a \|a x + b x + c + (4a x + 2b x)\|a \|c +--R / +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R 4\|a \|c \|a x + b x + c + (- 2b x - 4c)\|a +--R , +--R +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R (2c\|- a \|a x + b x + c + (- b x - 2c)\|- a \|c ) +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R +-+ +--R 2x\|c +--R + +--R +--------------+ +--R +-+ | 2 2 +--R (2b\|c \|a x + b x + c - b x - 2b c) +--R * +--R +--------------+ +--R +---+ +-+ | 2 +---+ +--R \|- a \|c \|a x + b x + c - c\|- a +--R atan(-------------------------------------) +--R +-+ +--R a x\|c +--R + +--R +--------------+ +--R +---+ | 2 2 +---+ +-+ +--R - b x\|- a \|a x + b x + c + (2a x + b x)\|- a \|c +--R / +--R +--------------+ +--R +---+ +-+ | 2 +---+ +--R 2\|- a \|c \|a x + b x + c + (- b x - 2c)\|- a +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 72 +t1:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 73 +t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) +--R +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R (3) -------------------------------------- +--R +-+ +--R \|c +--R Type: Union(Expression Integer,...) +--E + +--S 74 +bb1:=sqrt(a*x^2+b*x+c)+b/2*t1.1+c*t2 +--R +--R (4) +--R +--------------+ +--R +-+ | 2 +--R +-+ 2\|c \|a x + b x + c - b x - 2c +--R 2c\|a log(---------------------------------) +--R x +--R + +--R +-+ +--R b\|c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ +-+ | 2 +--R 2\|a \|c \|a x + b x + c +--R / +--R +-+ +-+ +--R 2\|a \|c +--R Type: Expression Integer +--E + +--S 75 +bb2:=sqrt(a*x^2+b*x+c)+b/2*t1.2+c*t2 +--R +--R (5) +--R +--------------+ +--R +-+ | 2 +--R +---+ 2\|c \|a x + b x + c - b x - 2c +--R c\|- a log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R +-+ \|- a \|a x + b x + c - \|- a \|c +--R b\|c atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|c \|a x + b x + c +--R / +--R +---+ +-+ +--R \|- a \|c +--R Type: Expression Integer +--E + +--S 76 +cc1:=aa.1-bb1 +--R +--R (6) +--R +--------------+ +--R +-+ | 2 +--R +-+ 2\|c \|a x + b x + c - b x - 2c +--R - 2c\|a log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +-+ | 2 +--R +-+ 2\|c \|a x + b x + c - b x - 2c +--R 2c\|a log(---------------------------------) +--R +-+ +--R 2x\|c +--R + +--R - +--R +-+ +--R b\|c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +-+ +--R b\|c +--R * +--R log +--R 2 +-+ 2 2 +-+ +--R ((- 2a b x - 8a c x)\|c + (4a c x + 4b c x + 8c )\|a ) +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 3 2 2 2 +-+ +-+ 2 3 +--R (- 2a b x + (- 8a c - b )x - 8b c x - 8c )\|a \|c + 4a c x +--R + +--R 2 2 +--R 6a b c x + 8a c x +--R / +--R +--------------+ +--R 2 | 2 +--R (4b c x + 8c )\|a x + b x + c +--R + +--R 2 2 2 +-+ +--R ((- 4a c - b )x - 8b c x - 8c )\|c +--R + +--R +-+ +--R 2c\|a +--R / +--R +-+ +-+ +--R 2\|a \|c +--R Type: Expression Integer +--E + +--S 77 +cc2:=aa.2-bb1 +--R +--R (7) +--R +--------------+ +--R +-+ | 2 +--R +---+ +-+ 2\|c \|a x + b x + c - b x - 2c +--R - 2c\|- a \|a log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +-+ | 2 +--R +---+ +-+ 2\|c \|a x + b x + c - b x - 2c +--R 2c\|- a \|a log(---------------------------------) +--R +-+ +--R 2x\|c +--R + +--R - +--R +---+ +-+ +--R b\|- a \|c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +---+ +-+ | 2 +---+ +--R +-+ +-+ \|- a \|c \|a x + b x + c - c\|- a +---+ +-+ +--R 2b\|a \|c atan(-------------------------------------) + 2c\|- a \|a +--R +-+ +--R a x\|c +--R / +--R +---+ +-+ +-+ +--R 2\|- a \|a \|c +--R Type: Expression Integer +--E + +--S 78 +cc3:=aa.1-bb2 +--R +--R (8) +--R +--------------+ +--R +-+ | 2 +--R +---+ +-+ 2\|c \|a x + b x + c - b x - 2c +--R - 2c\|- a \|a log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +-+ | 2 +--R +---+ +-+ 2\|c \|a x + b x + c - b x - 2c +--R 2c\|- a \|a log(---------------------------------) +--R +-+ +--R 2x\|c +--R + +--R +---+ +-+ +--R b\|- a \|c +--R * +--R log +--R 2 +-+ 2 2 +-+ +--R ((- 2a b x - 8a c x)\|c + (4a c x + 4b c x + 8c )\|a ) +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 3 2 2 2 +-+ +-+ 2 3 +--R (- 2a b x + (- 8a c - b )x - 8b c x - 8c )\|a \|c + 4a c x +--R + +--R 2 2 +--R 6a b c x + 8a c x +--R / +--R +--------------+ +--R 2 | 2 +--R (4b c x + 8c )\|a x + b x + c +--R + +--R 2 2 2 +-+ +--R ((- 4a c - b )x - 8b c x - 8c )\|c +--R + +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R +-+ +-+ \|- a \|a x + b x + c - \|- a \|c +---+ +-+ +--R - 2b\|a \|c atan(------------------------------------) + 2c\|- a \|a +--R a x +--R / +--R +---+ +-+ +-+ +--R 2\|- a \|a \|c +--R Type: Expression Integer +--E + +--S 79 +cc4:=aa.2-bb2 +--R +--R (9) +--R +--------------+ +--R +-+ | 2 +--R +---+ 2\|c \|a x + b x + c - b x - 2c +--R - c\|- a log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +-+ | 2 +--R +---+ 2\|c \|a x + b x + c - b x - 2c +--R c\|- a log(---------------------------------) +--R +-+ +--R 2x\|c +--R + +--R +--------------+ +--R +---+ +-+ | 2 +---+ +--R +-+ \|- a \|c \|a x + b x + c - c\|- a +--R b\|c atan(-------------------------------------) +--R +-+ +--R a x\|c +--R + +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R +-+ \|- a \|a x + b x + c - \|- a \|c +---+ +--R - b\|c atan(------------------------------------) + c\|- a +--R a x +--R / +--R +---+ +-+ +--R \|- a \|c +--R Type: Expression Integer +--E + +--S 80 +dd4:=ratDenom cc4 +--R +--R (10) +--R +--------------+ +--R +-+ | 2 +--R +-+ 2\|c \|a x + b x + c - b x - 2c +--R - \|c log(---------------------------------) +--R x +--R + +--R +--------------+ +--R | 2 +-+ +--R +-+ 2c\|a x + b x + c + (- b x - 2c)\|c +-+ +--R \|c log(--------------------------------------) + \|c +--R 2c x +--R Type: Expression Integer +--E + +--S 81 +ee4:=expandLog dd4 +--R +--R (11) +--R +--------------+ +--R +-+ +-+ | 2 +--R - \|c log(2\|c \|a x + b x + c - b x - 2c) +--R + +--R +--------------+ +--R +-+ | 2 +-+ +-+ +--R \|c log(2c\|a x + b x + c + (- b x - 2c)\|c ) + (- log(c) - log(2) + 1)\|c +--R Type: Expression Integer +--E + +--S 82 14:288 Schaums and Axiom differ by a constant +ff4:=complexNormalize ee4 +--R +--R +-+ +--R (- log(c) - 2log(2) + 2)\|c +--R (12) ---------------------------- +--R 2 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.289~~~~~$\displaystyle +\int{\frac{\sqrt{ax^2+bx+c}}{x^2}}~dx$} +$$\int{\frac{\sqrt{ax^2+bx+c}}{x^2}}= +-\frac{\sqrt{ax^2+bx+c}}{x^2} ++a\int{\frac{1}{\sqrt{ax^2+bx+c}}} ++\frac{b}{2}\int{\frac{1}{x\sqrt{ax^2+bx+c}}} +$$ +<<*>>= +)clear all + +--S 83 +aa:=integrate(sqrt(a*x^2+b*x+c)/x^2,x) +--R +--R +--R (1) +--R [ +--R +--------------+ +--R +-+ | 2 2 2 +--R (4b x\|c \|a x + b x + c - 2b x - 4b c x) +--R * +--R +--------------+ +--R | 2 +-+ +--R 2c\|a x + b x + c + (- b x - 2c)\|c +--R log(--------------------------------------) +--R 2c x +--R + +--R +--------------+ +--R +-+ | 2 2 +-+ +-+ +--R (8c x\|a \|a x + b x + c + (- 4b x - 8c x)\|a \|c ) +--R * +--R +--------------+ +--R +-+ +-+ | 2 +-+ +-+ 2 +--R (2\|c - 2x\|a )\|a x + b x + c + 2x\|a \|c - 2a x - b x - 2c +--R log(-----------------------------------------------------------------) +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 2 2 2 +--R (2b x + 8c)\|c \|a x + b x + c + (- 8a c + b )x - 6b c x - 8c +--R / +--R +--------------+ +--R | 2 2 +-+ +--R 8c x\|a x + b x + c + (- 4b x - 8c x)\|c +--R , +--R +--R +--------------+ +--R +-+ | 2 2 2 +--R (4b x\|c \|a x + b x + c - 2b x - 4b c x) +--R * +--R +--------------+ +--R | 2 +-+ +--R 2c\|a x + b x + c + (- b x - 2c)\|c +--R log(--------------------------------------) +--R 2c x +--R + +--R +--------------+ +--R +---+ | 2 2 +---+ +-+ +--R (16c x\|- a \|a x + b x + c + (- 8b x - 16c x)\|- a \|c ) +--R * +--R +--------------+ +--R | 2 +-+ +--R \|a x + b x + c - \|c +--R atan(------------------------) +--R +---+ +--R x\|- a +--R + +--R +--------------+ +--R +-+ | 2 2 2 2 +--R (2b x + 8c)\|c \|a x + b x + c + (- 8a c + b )x - 6b c x - 8c +--R / +--R +--------------+ +--R | 2 2 +-+ +--R 8c x\|a x + b x + c + (- 4b x - 8c x)\|c +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 84 +t1:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 85 +t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) +--R +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R (3) -------------------------------------- +--R +-+ +--R \|c +--R Type: Union(Expression Integer,...) +--E + +--S 86 +bb1:=-sqrt(a*x^2+b*x+c)/x+a*t1.1+b/2*t2 +--R +--R (4) +--R +--------------+ +--R +-+ | 2 +--R +-+ 2\|c \|a x + b x + c - b x - 2c +--R b x\|a log(---------------------------------) +--R x +--R + +--R +-+ +--R 2a x\|c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ +-+ | 2 +--R - 2\|a \|c \|a x + b x + c +--R / +--R +-+ +-+ +--R 2x\|a \|c +--R Type: Expression Integer +--E + +--S 87 +bb2:=-sqrt(a*x^2+b*x+c)/x+a*t1.2+b/2*t2 +--R +--R (5) +--R +--------------+ +--R +-+ | 2 +--R +---+ 2\|c \|a x + b x + c - b x - 2c +--R b x\|- a log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R +-+ \|- a \|a x + b x + c - \|- a \|c +--R 4a x\|c atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ +-+ | 2 +--R - 2\|- a \|c \|a x + b x + c +--R / +--R +---+ +-+ +--R 2x\|- a \|c +--R Type: Expression Integer +--E + +--S 88 +cc1:=aa.1-bb1 +--R +--R (6) +--R +--------------+ +--R +-+ | 2 2 +-+ +-+ +--R (- 4b c\|a \|a x + b x + c + (2b x + 4b c)\|a \|c ) +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +-+ | 2 2 +-+ +-+ +--R (4b c\|a \|a x + b x + c + (- 2b x - 4b c)\|a \|c ) +--R * +--R +--------------+ +--R | 2 +-+ +--R 2c\|a x + b x + c + (- b x - 2c)\|c +--R log(--------------------------------------) +--R 2c x +--R + +--R +--------------+ +--R +-+ | 2 2 +--R (- 8a c\|c \|a x + b x + c + 4a b c x + 8a c ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 2 +--R (8a c\|c \|a x + b x + c - 4a b c x - 8a c ) +--R * +--R +--------------+ +--R +-+ +-+ | 2 +-+ +-+ 2 +--R (2\|c - 2x\|a )\|a x + b x + c + 2x\|a \|c - 2a x - b x - 2c +--R log(-----------------------------------------------------------------) +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 2 +-+ +-+ +--R - 2b c\|a \|a x + b x + c + (b x + 2b c)\|a \|c +--R / +--R +--------------+ +--R +-+ +-+ | 2 2 +-+ +--R 8c\|a \|c \|a x + b x + c + (- 4b c x - 8c )\|a +--R Type: Expression Integer +--E + +--S 89 +cc2:=aa.2-bb1 +--R +--R (7) +--R +--------------+ +--R +-+ | 2 2 +-+ +-+ +--R (- 4b c\|a \|a x + b x + c + (2b x + 4b c)\|a \|c ) +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +-+ | 2 2 +-+ +-+ +--R (4b c\|a \|a x + b x + c + (- 2b x - 4b c)\|a \|c ) +--R * +--R +--------------+ +--R | 2 +-+ +--R 2c\|a x + b x + c + (- b x - 2c)\|c +--R log(--------------------------------------) +--R 2c x +--R + +--R +--------------+ +--R +-+ | 2 2 +--R (- 8a c\|c \|a x + b x + c + 4a b c x + 8a c ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +---+ +-+ +-+ | 2 2 +---+ +-+ +--R (16c\|- a \|a \|c \|a x + b x + c + (- 8b c x - 16c )\|- a \|a ) +--R * +--R +--------------+ +--R | 2 +-+ +--R \|a x + b x + c - \|c +--R atan(------------------------) +--R +---+ +--R x\|- a +--R + +--R +--------------+ +--R +-+ | 2 2 +-+ +-+ +--R - 2b c\|a \|a x + b x + c + (b x + 2b c)\|a \|c +--R / +--R +--------------+ +--R +-+ +-+ | 2 2 +-+ +--R 8c\|a \|c \|a x + b x + c + (- 4b c x - 8c )\|a +--R Type: Expression Integer +--E + +--S 90 +cc3:=aa.1-bb2 +--R +--R (8) +--R +--------------+ +--R +---+ | 2 2 +---+ +-+ +--R (- 4b c\|- a \|a x + b x + c + (2b x + 4b c)\|- a \|c ) +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +---+ | 2 2 +---+ +-+ +--R (4b c\|- a \|a x + b x + c + (- 2b x - 4b c)\|- a \|c ) +--R * +--R +--------------+ +--R | 2 +-+ +--R 2c\|a x + b x + c + (- b x - 2c)\|c +--R log(--------------------------------------) +--R 2c x +--R + +--R +--------------+ +--R +---+ +-+ +-+ | 2 2 +---+ +-+ +--R (8c\|- a \|a \|c \|a x + b x + c + (- 4b c x - 8c )\|- a \|a ) +--R * +--R +--------------+ +--R +-+ +-+ | 2 +-+ +-+ 2 +--R (2\|c - 2x\|a )\|a x + b x + c + 2x\|a \|c - 2a x - b x - 2c +--R log(-----------------------------------------------------------------) +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 2 +--R (- 16a c\|c \|a x + b x + c + 8a b c x + 16a c ) +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ | 2 2 +---+ +-+ +--R - 2b c\|- a \|a x + b x + c + (b x + 2b c)\|- a \|c +--R / +--R +--------------+ +--R +---+ +-+ | 2 2 +---+ +--R 8c\|- a \|c \|a x + b x + c + (- 4b c x - 8c )\|- a +--R Type: Expression Integer +--E + +--S 91 +cc4:=aa.2-bb2 +--R +--R (9) +--R +--------------+ +--R +---+ | 2 2 +---+ +-+ +--R (- 4b c\|- a \|a x + b x + c + (2b x + 4b c)\|- a \|c ) +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +---+ | 2 2 +---+ +-+ +--R (4b c\|- a \|a x + b x + c + (- 2b x - 4b c)\|- a \|c ) +--R * +--R +--------------+ +--R | 2 +-+ +--R 2c\|a x + b x + c + (- b x - 2c)\|c +--R log(--------------------------------------) +--R 2c x +--R + +--R +--------------+ +--R +-+ | 2 2 +--R (- 16a c\|c \|a x + b x + c + 8a b c x + 16a c ) +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +-+ | 2 2 +--R (- 16a c\|c \|a x + b x + c + 8a b c x + 16a c ) +--R * +--R +--------------+ +--R | 2 +-+ +--R \|a x + b x + c - \|c +--R atan(------------------------) +--R +---+ +--R x\|- a +--R + +--R +--------------+ +--R +---+ | 2 2 +---+ +-+ +--R - 2b c\|- a \|a x + b x + c + (b x + 2b c)\|- a \|c +--R / +--R +--------------+ +--R +---+ +-+ | 2 2 +---+ +--R 8c\|- a \|c \|a x + b x + c + (- 4b c x - 8c )\|- a +--R Type: Expression Integer +--E + +--S 92 +dd4:=ratDenom cc4 +--R +--R (10) +--R +--------------+ +--R +-+ | 2 +--R +-+ 2\|c \|a x + b x + c - b x - 2c +--R - 2b\|c log(---------------------------------) +--R x +--R + +--R +--------------+ +--R | 2 +-+ +--R +-+ 2c\|a x + b x + c + (- b x - 2c)\|c +-+ +--R 2b\|c log(--------------------------------------) - b\|c +--R 2c x +--R / +--R 4c +--R Type: Expression Integer +--E + +--S 93 +ee4:=expandLog dd4 +--R +--R (11) +--R +--------------+ +--R +-+ +-+ | 2 +--R - 2b\|c log(2\|c \|a x + b x + c - b x - 2c) +--R + +--R +--------------+ +--R +-+ | 2 +-+ +--R 2b\|c log(2c\|a x + b x + c + (- b x - 2c)\|c ) +--R + +--R +-+ +--R (- 2b log(c) - 2b log(2) - b)\|c +--R / +--R 4c +--R Type: Expression Integer +--E + +--S 94 14:289 Schaums and Axiom differ by a constant +ff4:=complexNormalize ee4 +--R +--R +-+ +--R (- b log(c) - 2b log(2) - b)\|c +--R (12) -------------------------------- +--R 4c +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.290~~~~~$\displaystyle +\int{\frac{dx}{(ax^2+bx+c)^{3/2}}}$} +$$ +\int{\frac{1}{(ax^2+bx+c)^{3/2}}}= +\frac{2(2ax+b)}{(4ac-b^2)\sqrt{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 95 +aa:=integrate(1/(a*x^2+b*x+c)^(3/2),x) +--R +--R +--R +--------------+ +--R | 2 +-+ +--R - 2x\|a x + b x + c + 2x\|c +--R (1) -------------------------------------------------------- +--R +--------------+ +--R +-+ | 2 2 2 +--R (b x + 2c)\|c \|a x + b x + c - 2a c x - 2b c x - 2c +--R Type: Union(Expression Integer,...) +--E + +--S 96 +bb:=(2*(2*a*x+b))/((4*a*c-b^2)*sqrt(a*x^2+b*x+c)) +--R +--R 4a x + 2b +--R (2) ---------------------------- +--R +--------------+ +--R 2 | 2 +--R (4a c - b )\|a x + b x + c +--R Type: Expression Integer +--E + +--S 97 +cc:=aa-bb +--R +--R (3) +--R +--------------+ +--R +-+ | 2 2 +--R 4b\|c \|a x + b x + c - 2b x - 4b c +--R ----------------------------------------------------------------------- +--R +--------------+ +--R 2 2 | 2 3 2 2 +-+ +--R (8a c - 2b c)\|a x + b x + c + ((- 4a b c + b )x - 8a c + 2b c)\|c +--R Type: Expression Integer +--E + +--S 98 14:290 Schaums and Axiom differ by a constant +dd:=ratDenom cc +--R +--R +-+ +--R 2b\|c +--R (4) ----------- +--R 2 2 +--R 4a c - b c +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.291~~~~~$\displaystyle +\int{\frac{x~dx}{(ax^2+bx+c)^{3/2}}}$} +$$\int{\frac{x}{(ax^2+bx+c)^{3/2}}}= +\frac{2(bx+2c)}{(b^2-4ac)\sqrt{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 99 +aa:=integrate(x/(a*x^2+b*x+c)^(3/2),x) +--R +--R +--R 2 +-+ +--R 2x \|c +--R (1) -------------------------------------------------------- +--R +--------------+ +--R +-+ | 2 2 2 +--R (b x + 2c)\|c \|a x + b x + c - 2a c x - 2b c x - 2c +--R Type: Union(Expression Integer,...) +--E + +--S 100 +bb:=(2*(b*x+2*c))/((b^2-4*a*c)*sqrt(a*x^2+b*x+c)) +--R +--R - 2b x - 4c +--R (2) ---------------------------- +--R +--------------+ +--R 2 | 2 +--R (4a c - b )\|a x + b x + c +--R Type: Expression Integer +--E + +--S 101 +cc:=aa-bb +--R +--R (3) +--R +--------------+ +--R +-+ | 2 2 +--R - 8c\|c \|a x + b x + c + 4b c x + 8c +--R ----------------------------------------------------------------------- +--R +--------------+ +--R 2 2 | 2 3 2 2 +-+ +--R (8a c - 2b c)\|a x + b x + c + ((- 4a b c + b )x - 8a c + 2b c)\|c +--R Type: Expression Integer +--E + +--S 102 14:291 Schaums and Axiom differ by a constant +dd:=ratDenom cc +--R +--R +-+ +--R 4\|c +--R (4) - --------- +--R 2 +--R 4a c - b +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.292~~~~~$\displaystyle +\int{\frac{x^2~dx}{(ax^2+bx+c)^{3/2}}}$} +$$\int{\frac{x^2}{(ax^2+bx+c)^{3/2}}}= +\frac{(2b^2-4ac)x+2bc}{a(4ac-b^2)\sqrt{ax^2+bx+c}} ++\frac{1}{a}\int{\frac{1}{\sqrt{ax^2+bx+c}}} +$$ +<<*>>= +)clear all + +--S 103 +aa:=integrate(x^2/(a*x^2+b*x+c)^(3/2),x) +--R +--R +--R (1) +--R [ +--R +--------------+ +--R +-+ | 2 2 2 +--R ((b x + 2c)\|c \|a x + b x + c - 2a c x - 2b c x - 2c ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 2 +-+ +-+ +--R 2c x\|a \|a x + b x + c + (- 2b x - 2c x)\|a \|c +--R / +--R +--------------+ +--R +-+ +-+ | 2 +--R (a b x + 2a c)\|a \|c \|a x + b x + c +--R + +--R 2 2 2 +-+ +--R (- 2a c x - 2a b c x - 2a c )\|a +--R , +--R +--R +--------------+ +--R +-+ | 2 2 2 +--R ((2b x + 4c)\|c \|a x + b x + c - 4a c x - 4b c x - 4c ) +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ | 2 2 +---+ +-+ +--R 2c x\|- a \|a x + b x + c + (- 2b x - 2c x)\|- a \|c +--R / +--R +--------------+ +--R +---+ +-+ | 2 +--R (a b x + 2a c)\|- a \|c \|a x + b x + c +--R + +--R 2 2 2 +---+ +--R (- 2a c x - 2a b c x - 2a c )\|- a +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 104 +t1:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 105 +bb1:=((2*b^2-4*a*c)*x+2*b*c)/(a*(4*a*c-b^2)*sqrt(a*x^2+b*x+c))+1/a*t1.1 +--R +--R (3) +--R +--------------+ +--R 2 | 2 +--R (4a c - b )\|a x + b x + c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 +-+ +--R ((- 4a c + 2b )x + 2b c)\|a +--R / +--R +--------------+ +--R 2 2 +-+ | 2 +--R (4a c - a b )\|a \|a x + b x + c +--R Type: Expression Integer +--E + +--S 106 +bb2:=((2*b^2-4*a*c)*x+2*b*c)/(a*(4*a*c-b^2)*sqrt(a*x^2+b*x+c))+1/a*t1.2 +--R +--R (4) +--R +--------------+ +--R +--------------+ +---+ | 2 +---+ +-+ +--R 2 | 2 \|- a \|a x + b x + c - \|- a \|c +--R (8a c - 2b )\|a x + b x + c atan(------------------------------------) +--R a x +--R + +--R 2 +---+ +--R ((- 4a c + 2b )x + 2b c)\|- a +--R / +--R +--------------+ +--R 2 2 +---+ | 2 +--R (4a c - a b )\|- a \|a x + b x + c +--R Type: Expression Integer +--E + +--S 107 +cc1:=aa.1-bb1 +--R +--R (5) +--R +--------------+ +--R +-+ | 2 2 2 +--R 4b c\|c \|a x + b x + c - 2b c x - 4b c +--R ----------------------------------------------------------------------------- +--R +--------------+ +--R 2 2 2 | 2 2 3 2 2 2 +-+ +--R (8a c - 2a b c)\|a x + b x + c + ((- 4a b c + a b )x - 8a c + 2a b c)\|c +--R Type: Expression Integer +--E + +--S 108 +cc2:=aa.2-bb1 +--R +--R (6) +--R +--------------+ +--R 2 2 +---+ | 2 +--R (- 8a c + 2b c)\|- a \|a x + b x + c +--R + +--R 3 2 2 +---+ +-+ +--R ((4a b c - b )x + 8a c - 2b c)\|- a \|c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R 2 2 +-+ | 2 +--R (16a c - 4b c)\|a \|a x + b x + c +--R + +--R 3 2 2 +-+ +-+ +--R ((- 8a b c + 2b )x - 16a c + 4b c)\|a \|c +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ +-+ +-+ | 2 2 2 +---+ +-+ +--R 4b c\|- a \|a \|c \|a x + b x + c + (- 2b c x - 4b c )\|- a \|a +--R / +--R +--------------+ +--R 2 2 2 +---+ +-+ | 2 +--R (8a c - 2a b c)\|- a \|a \|a x + b x + c +--R + +--R 2 3 2 2 2 +---+ +-+ +-+ +--R ((- 4a b c + a b )x - 8a c + 2a b c)\|- a \|a \|c +--R Type: Expression Integer +--E + +--S 109 +cc3:=aa.1-bb2 +--R +--R (7) +--R +--------------+ +--R 2 2 +---+ | 2 +--R (8a c - 2b c)\|- a \|a x + b x + c +--R + +--R 3 2 2 +---+ +-+ +--R ((- 4a b c + b )x - 8a c + 2b c)\|- a \|c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R 2 2 +-+ | 2 +--R (- 16a c + 4b c)\|a \|a x + b x + c +--R + +--R 3 2 2 +-+ +-+ +--R ((8a b c - 2b )x + 16a c - 4b c)\|a \|c +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ +-+ +-+ | 2 2 2 +---+ +-+ +--R 4b c\|- a \|a \|c \|a x + b x + c + (- 2b c x - 4b c )\|- a \|a +--R / +--R +--------------+ +--R 2 2 2 +---+ +-+ | 2 +--R (8a c - 2a b c)\|- a \|a \|a x + b x + c +--R + +--R 2 3 2 2 2 +---+ +-+ +-+ +--R ((- 4a b c + a b )x - 8a c + 2a b c)\|- a \|a \|c +--R Type: Expression Integer +--E + +--S 110 +cc4:=aa.2-bb2 +--R +--R (8) +--R +--------------+ +--R +-+ | 2 2 2 +--R 4b c\|c \|a x + b x + c - 2b c x - 4b c +--R ----------------------------------------------------------------------------- +--R +--------------+ +--R 2 2 2 | 2 2 3 2 2 2 +-+ +--R (8a c - 2a b c)\|a x + b x + c + ((- 4a b c + a b )x - 8a c + 2a b c)\|c +--R Type: Expression Integer +--E + +--S 111 14:292 Schaums and Axiom differ by a constant +dd4:=ratDenom cc4 +--R +--R +-+ +--R 2b\|c +--R (9) ----------- +--R 2 2 +--R 4a c - a b +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.293~~~~~$\displaystyle +\int{\frac{dx}{x(ax^2+bx+c)^{3/2}}}$} +$$\int{\frac{1}{x(ax^2+bx+c)^{3/2}}}= +\frac{1}{c\sqrt{ax^2+bx+c}} ++\frac{1}{c}\int{\frac{1}{x\sqrt{ax^2+bx+c}}} +-\frac{b}{2c}\int{\frac{1}{(ax^2+bx+c)^{3/2}}} +$$ +<<*>>= +)clear all + +--S 112 +aa:=integrate(1/(x*(a*x^2+b*x+c)^(3/2)),x) +--R +--R +--R (1) +--R +--------------+ +--R | 2 2 +-+ +--R ((b x + 2c)\|a x + b x + c + (- 2a x - 2b x - 2c)\|c ) +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R +-+ +--R 2x\|c +--R + +--R +--------------+ +--R | 2 2 +-+ +--R 2b x\|a x + b x + c + (- 2a x - 2b x)\|c +--R / +--R +--------------+ +--R 2 +-+ | 2 2 2 2 3 +--R (b c x + 2c )\|c \|a x + b x + c - 2a c x - 2b c x - 2c +--R Type: Union(Expression Integer,...) +--E + +--S 113 +t1:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) +--R +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R (2) -------------------------------------- +--R +-+ +--R \|c +--R Type: Union(Expression Integer,...) +--E + +--S 114 +t2:=integrate(1/(a*x^2+b*x+c)^(3/2),x) +--R +--R +--------------+ +--R | 2 +-+ +--R - 2x\|a x + b x + c + 2x\|c +--R (3) -------------------------------------------------------- +--R +--------------+ +--R +-+ | 2 2 2 +--R (b x + 2c)\|c \|a x + b x + c - 2a c x - 2b c x - 2c +--R Type: Union(Expression Integer,...) +--E + +--S 115 +bb:=1/(c*sqrt(a*x^2+b*x+c))+1/c*t1-b/(2*c)*t2 +--R +--R (4) +--R +--------------+ +--R 2 2 | 2 +--R (2a c x + 2b c x + 2c )\|a x + b x + c +--R + +--R 3 2 2 2 +-+ +--R (- a b x + (- 2a c - b )x - 3b c x - 2c )\|c +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R 2 | 2 3 2 2 2 +-+ +--R - 2c \|a x + b x + c + (- a b x + (2a c - b )x + b c x + 2c )\|c +--R / +--R +--------------+ +--R 2 2 2 3 +-+ | 2 2 3 +--R (2a c x + 2b c x + 2c )\|c \|a x + b x + c - a b c x +--R + +--R 3 2 2 2 3 4 +--R (- 2a c - b c )x - 3b c x - 2c +--R Type: Expression Integer +--E + +--S 116 +cc:=aa-bb +--R +--R (5) +--R +--------------+ +--R 2 2 2 | 2 +--R ((- 4a c - b )x - 8b c x - 8c )\|a x + b x + c +--R + +--R 3 2 2 2 +-+ +--R (4a b x + (8a c + 4b )x + 12b c x + 8c )\|c +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R 2 2 2 | 2 +--R ((4a c + b )x + 8b c x + 8c )\|a x + b x + c +--R + +--R 3 2 2 2 +-+ +--R (- 4a b x + (- 8a c - 4b )x - 12b c x - 8c )\|c +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R +-+ +--R 2x\|c +--R + +--R +--------------+ +--R 2 2 2 | 2 +--R ((4a c + b )x + 8b c x + 8c )\|a x + b x + c +--R + +--R 3 2 2 2 +-+ +--R (- 4a b x + (- 8a c - 4b )x - 12b c x - 8c )\|c +--R / +--R +--------------+ +--R 2 2 2 2 3 +-+ | 2 2 3 +--R ((4a c + b c)x + 8b c x + 8c )\|c \|a x + b x + c - 4a b c x +--R + +--R 3 2 2 2 3 4 +--R (- 8a c - 4b c )x - 12b c x - 8c +--R Type: Expression Integer +--E + +--S 117 +dd:=ratDenom cc +--R +--R (6) +--R +--------------+ +--R +-+ | 2 +--R +-+ 2\|c \|a x + b x + c - b x - 2c +--R - \|c log(---------------------------------) +--R x +--R + +--R +--------------+ +--R | 2 +-+ +--R +-+ 2c\|a x + b x + c + (- b x - 2c)\|c +-+ +--R \|c log(--------------------------------------) + \|c +--R 2c x +--R / +--R 2 +--R c +--R Type: Expression Integer +--E + +--S 118 +ee:=expandLog dd +--R +--R (7) +--R +--------------+ +--R +-+ +-+ | 2 +--R - \|c log(2\|c \|a x + b x + c - b x - 2c) +--R + +--R +--------------+ +--R +-+ | 2 +-+ +--R \|c log(2c\|a x + b x + c + (- b x - 2c)\|c ) +--R + +--R +-+ +--R (- log(c) - log(2) + 1)\|c +--R / +--R 2 +--R c +--R Type: Expression Integer +--E + +--S 119 14:293 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R +-+ +--R (- log(c) - 2log(2) + 2)\|c +--R (8) ---------------------------- +--R 2 +--R 2c +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.294~~~~~$\displaystyle +\int{\frac{dx}{x^2(ax^2+bx+c)^{3/2}}}$} +$$\begin{array}{rl} +\displaystyle\int{\frac{1}{x^2(ax^2+bx+c)^{3/2}}}= +&\displaystyle-\frac{ax^2+bx+c}{c^2x\sqrt{ax^2+bx+c}}\\ +&\\ +&\displaystyle+\frac{b^2-2ac}{2c^2}\int{\frac{1}{(ax^2+bx+c)^{3/2}}}\\ +&\\ +&\displaystyle-\frac{3b}{2c^2}\int{\frac{1}{x\sqrt{ax^2+bx+c}}} +\end{array} +$$ +<<*>>= +)clear all + +--S 120 +aa:=integrate(1/(x^2*(a*x^2+b*x+c)^(3/2)),x) +--R +--R +--R (1) +--R +--------------+ +--R 3 3 2 2 2 +-+ | 2 +--R ((- 24a b c - 6b )x - 48b c x - 48b c x)\|c \|a x + b x + c +--R + +--R 2 4 2 3 3 2 2 2 3 +--R 24a b c x + (48a b c + 24b c)x + 72b c x + 48b c x +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R +-+ +--R 2x\|c +--R + +--R 3 3 2 2 2 2 3 +-+ +--R ((4a b c - 9b )x + (64a c - 24b c)x + 40b c x + 32c )\|c +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 2 2 4 2 3 3 3 2 2 2 +--R (- 32a c + 24a b c)x + (- 48a b c + 24b c)x + (- 80a c + 8b c )x +--R + +--R 3 4 +--R - 56b c x - 32c +--R / +--R +--------------+ +--R 4 2 3 3 4 2 5 | 2 +--R ((16a c + 4b c )x + 32b c x + 32c x)\|a x + b x + c +--R + +--R 3 4 4 2 3 3 4 2 5 +-+ +--R (- 16a b c x + (- 32a c - 16b c )x - 48b c x - 32c x)\|c +--R Type: Union(Expression Integer,...) +--E + +--S 121 +t1:=integrate(1/(a*x^2+b*x+c)^(3/2),x) +--R +--R +--------------+ +--R | 2 +-+ +--R - 2x\|a x + b x + c + 2x\|c +--R (2) -------------------------------------------------------- +--R +--------------+ +--R +-+ | 2 2 2 +--R (b x + 2c)\|c \|a x + b x + c - 2a c x - 2b c x - 2c +--R Type: Union(Expression Integer,...) +--E + +--S 122 +t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) +--R +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R (3) -------------------------------------- +--R +-+ +--R \|c +--R Type: Union(Expression Integer,...) +--E + +--S 123 +bb:=-(a*x^2+2*b*x+c)/(c^2*x*sqrt(a*x^2+b*x+c))+(b^2-2*a*c)/(2*c^2)*t1-(3*b)/(2*c^2)*t2 +--R +--R (4) +--R +--------------+ +--R 3 2 2 2 | 2 +--R (- 6a b c x - 6b c x - 6b c x)\|a x + b x + c +--R + +--R 2 4 3 3 2 2 2 +-+ +--R (3a b x + (6a b c + 3b )x + 9b c x + 6b c x)\|c +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R 3 2 2 2 2 3 | 2 +--R (2a b c x + (8a c + 2b c)x + 10b c x + 4c )\|a x + b x + c +--R + +--R 2 2 4 3 3 2 2 2 +--R (- 8a c + 2a b )x + (- 16a b c + 2b )x + (- 12a c - 6b c)x +--R + +--R 2 3 +--R - 12b c x - 4c +--R * +--R +-+ +--R \|c +--R / +--R +--------------+ +--R 3 3 3 2 4 +-+ | 2 3 4 +--R (4a c x + 4b c x + 4c x)\|c \|a x + b x + c - 2a b c x +--R + +--R 4 2 3 3 4 2 5 +--R (- 4a c - 2b c )x - 6b c x - 4c x +--R Type: Expression Integer +--E + +--S 124 +cc:=aa-bb +--R +--R (5) +--R 2 4 3 2 3 2 2 2 3 +--R ((72a b c + 6b )x + (144a b c + 108b c)x + 288b c x + 192b c ) +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 2 2 3 4 2 2 4 3 +--R (- 48a b c - 36a b c)x + (- 240a b c - 36b c)x +--R + +--R 3 3 2 2 2 3 4 +--R (- 240a b c - 228b c )x - 384b c x - 192b c +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R + +--R 2 4 3 2 3 2 2 2 +--R (- 72a b c - 6b )x + (- 144a b c - 108b c)x - 288b c x +--R + +--R 3 +--R - 192b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 2 2 3 4 2 2 4 3 +--R (48a b c + 36a b c)x + (240a b c + 36b c)x +--R + +--R 3 3 2 2 2 3 4 +--R (240a b c + 228b c )x + 384b c x + 192b c +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R +-+ +--R 2x\|c +--R + +--R 2 4 3 2 3 2 2 2 3 +--R ((- 60a b c - 5b )x + (- 120a b c - 90b c)x - 240b c x - 160b c ) +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 2 2 3 4 2 2 4 3 3 3 2 2 +--R (40a b c + 30a b c)x + (200a b c + 30b c)x + (200a b c + 190b c )x +--R + +--R 2 3 4 +--R 320b c x + 160b c +--R / +--R 4 3 3 3 5 2 4 2 5 6 +--R ((48a b c + 4b c )x + (96a c + 72b c )x + 192b c x + 128c ) +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 4 2 3 4 4 3 3 3 +--R (- 32a c - 24a b c )x + (- 160a b c - 24b c )x +--R + +--R 5 2 4 2 5 6 +--R (- 160a c - 152b c )x - 256b c x - 128c +--R * +--R +-+ +--R \|c +--R Type: Expression Integer +--E + +--S 125 +dd:=ratDenom cc +--R +--R (6) +--R +--------------+ +--R +-+ | 2 +--R +-+ 2\|c \|a x + b x + c - b x - 2c +--R 6b\|c log(---------------------------------) +--R x +--R + +--R +--------------+ +--R | 2 +-+ +--R +-+ 2c\|a x + b x + c + (- b x - 2c)\|c +-+ +--R - 6b\|c log(--------------------------------------) - 5b\|c +--R 2c x +--R / +--R 3 +--R 4c +--R Type: Expression Integer +--E + +--S 126 +ee:=expandLog dd +--R +--R (7) +--R +--------------+ +--R +-+ +-+ | 2 +--R 6b\|c log(2\|c \|a x + b x + c - b x - 2c) +--R + +--R +--------------+ +--R +-+ | 2 +-+ +--R - 6b\|c log(2c\|a x + b x + c + (- b x - 2c)\|c ) +--R + +--R +-+ +--R (6b log(c) + 6b log(2) - 5b)\|c +--R / +--R 3 +--R 4c +--R Type: Expression Integer +--E + +--S 127 14:294 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R +-+ +--R (3b log(c) + 6b log(2) - 5b)\|c +--R (8) -------------------------------- +--R 3 +--R 4c +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.295~~~~~$\displaystyle +\int{(ax^2+bx+c)^{n+1/2}}~dx$} +$$\begin{array}{rl} +\displaystyle\int{(ax^2+bx+c)^{n+1/2}}= +&\displaystyle\frac{(2ax+b)(ax^2+bx+c)^{n+1/2}}{4a(n+1)}\\ +&\\ +&\displaystyle+\frac{(2n+1)(4ac-b^2)}{8a(n+1)} +\int{(ax^2+bx+c)^{n-1/2}} +\end{array} +$$ +<<*>>= +)clear all + +--S 128 14:295 Axiom cannot compute this integral +aa:=integrate((a*x^2+b*x+c)^(n+1/2),x) +--R +--R +--R 2n + 1 +--R x ------ +--R ++ 2 2 +--I (1) | (c + %N b + %N a) d%N +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.296~~~~~$\displaystyle +\int{x(ax^2+bx+c)^{n+1/2}}~dx$} +$$\int{x(ax^2+bx+c)^{n+1/2}}= +\frac{(ax^2+bx+c)^{n+3/2}}{a(2n+3)}- +\frac{b}{2a}\int{(ax^2+bx+c)^{n+1/2}} +$$ +<<*>>= +)clear all + +--S 129 14:296 Axiom cannot compute this integral +aa:=integrate(x*(a*x^2+b*x+c)^(n+1/2),x) +--R +--R +--R 2n + 1 +--R x ------ +--R ++ 2 2 +--I (1) | %N (c + %N b + %N a) d%N +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.297~~~~~$\displaystyle +\int{\frac{dx}{(ax^2+bx+c)^{n+1/2}}}$} +$$\begin{array}{rl} +\displaystyle\int{\frac{1}{(ax^2+bx+c)^{n+1/2}}}= +&\displaystyle\frac{2(2ax+b)}{(2n-1)(4ac-b^2)(ax^2+bx+c)^{n-1/2}}\\ +&\\ +&\displaystyle ++\frac{8a(n-1)}{(2n-1)(4ac-b^2)}\int{\frac{1}{(ax^2+bx+c)^{n-1/2}}} +\end{array} +$$ +<<*>>= +)clear all + +--S 130 14:297 Axiom cannot compute this integral +aa:=integrate(1/(a*x^2+b*x+c)^(n+1/2),x) +--R +--R +--R x +--R ++ 1 +--I (1) | ----------------------- d%N +--R ++ 2n + 1 +--R ------ +--R 2 2 +--I (c + %N b + %N a) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.298~~~~~$\displaystyle +\int{\frac{dx}{x(ax^2+bx+c)^{n+1/2}}}$} +$$\begin{array}{rl} +\displaystyle\int{\frac{1}{x(ax^2+bx+c)^{n+1/2}}}= +&\displaystyle\frac{1}{(2n-1)c(ax^2+bx+c)^{n-1/2}}\\ +&\\ +&\displaystyle ++\frac{1}{c}\int{\frac{1}{x(ax^2+bx+c)^{n-1/2}}}\\ +&\\ +&\displaystyle +-\frac{b}{2c}\int{\frac{1}{(ax^2+bx+c)^{n+1/2}}} +\end{array} +$$ +<<*>>= +)clear all + +--S 131 14:298 Axiom cannot compute this integral +aa:=integrate(1/(x*(a*x^2+b*x+c)^(n+1/2)),x) +--R +--R +--R x +--R ++ 1 +--I (1) | -------------------------- d%N +--R ++ 2n + 1 +--R ------ +--R 2 2 +--I %N (c + %N b + %N a) +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp72-73 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum13.input.pdf b/src/axiom-website/CATS/schaum13.input.pdf new file mode 100644 index 0000000..91d46bd --- /dev/null +++ b/src/axiom-website/CATS/schaum13.input.pdf @@ -0,0 +1,6474 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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+\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.299~~~~~$\displaystyle +\int{\frac{dx}{x^3+a^3}}$} +$$\int{\frac{1}{x^3+a^3}}= +\frac{1}{6a^2}\ln\frac{(x+a)^2}{x^2-ax+a^2} ++\frac{1}{a^2\sqrt{3}}\tan^{-1}\frac{2x-a}{a\sqrt{3}} +$$ +<<*>>= +)spool schaum14.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(1/(x^3+a^3),x) +--R +--R +--R +-+ +--R +-+ 2 2 +-+ (2x - a)\|3 +--R - \|3 log(x - a x + a ) + 2\|3 log(x + a) + 6atan(------------) +--R 3a +--R (1) ---------------------------------------------------------------- +--R 2 +-+ +--R 6a \|3 +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=1/(6*a^2)*log((x+a)^2/(x^2-a*x+a^2))+1/(a^2*sqrt(3))*atan((2*x-a)/(a*sqrt(3))) +--R +--R 2 2 +-+ +--R x + 2a x + a +-+ (2x - a)\|3 +--R log(--------------) + 2\|3 atan(------------) +--R 2 2 3a +--R x - a x + a +--R (2) --------------------------------------------- +--R 2 +--R 6a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R 2 2 +--R 2 2 x + 2a x + a +--R - log(x - a x + a ) + 2log(x + a) - log(--------------) +--R 2 2 +--R x - a x + a +--R (3) -------------------------------------------------------- +--R 2 +--R 6a +--R Type: Expression Integer +--E + +--S 4 14:299 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.300~~~~~$\displaystyle +\int{\frac{x~dx}{x^3+a^3}}$} +$$\int{\frac{x}{x^3+a^3}}= +\frac{1}{6a}\ln\frac{x^2-ax+a^2}{(x+a)^2} ++\frac{1}{a\sqrt{3}}\tan^{-1}\frac{2x-a}{a\sqrt{3}} +$$ +<<*>>= +)clear all + +--S 5 +aa:=integrate(x/(x^3+a^3),x) +--R +--R +--R +-+ +--R +-+ 2 2 +-+ (2x - a)\|3 +--R \|3 log(x - a x + a ) - 2\|3 log(x + a) + 6atan(------------) +--R 3a +--R (1) -------------------------------------------------------------- +--R +-+ +--R 6a\|3 +--R Type: Union(Expression Integer,...) +--E + +--S 6 +bb:=1/(6*a)*log((x^2-a*x+a^2)/(x+a)^2)+1/(a*sqrt(3))*atan((2*x-a)/(a*sqrt(3))) +--R +--R 2 2 +-+ +--R x - a x + a +-+ (2x - a)\|3 +--R log(--------------) + 2\|3 atan(------------) +--R 2 2 3a +--R x + 2a x + a +--R (2) --------------------------------------------- +--R 6a +--R Type: Expression Integer +--E + +--S 7 +cc:=aa-bb +--R +--R 2 2 +--R 2 2 x - a x + a +--R log(x - a x + a ) - 2log(x + a) - log(--------------) +--R 2 2 +--R x + 2a x + a +--R (3) ------------------------------------------------------ +--R 6a +--R Type: Expression Integer +--E + +--S 8 14:300 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.301~~~~~$\displaystyle +\int{\frac{x^2~dx}{x^3+a^3}}$} +$$\int{\frac{x^2}{x^3+a^3}}= +\frac{1}{3}\ln(x^3+a^3) +$$ +<<*>>= +)clear all + +--S 9 +aa:=integrate(x^2/(x^3+a^3),x) +--R +--R +--R 3 3 +--R log(x + a ) +--R (1) ------------ +--R 3 +--R Type: Union(Expression Integer,...) +--E + +--S 10 +bb:=1/3*log(x^3+a^3) +--R +--R 3 3 +--R log(x + a ) +--R (2) ------------ +--R 3 +--R Type: Expression Integer +--E + +--S 11 14:301 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.302~~~~~$\displaystyle +\int{\frac{dx}{x(x^3+a^3)}}$} +$$\int{\frac{1}{x(x^3+a^3)}}= +\frac{1}{3a^3}\ln\left(\frac{x^3}{x^3+a^3}\right) +$$ +<<*>>= +)clear all + +--S 12 +aa:=integrate(1/(x*(x^3+a^3)),x) +--R +--R +--R 3 3 +--R - log(x + a ) + 3log(x) +--R (1) ------------------------ +--R 3 +--R 3a +--R Type: Union(Expression Integer,...) +--E + +--S 13 +bb:=1/(3*a^3)*log(x^3/(x^3+a^3)) +--R +--R 3 +--R x +--R log(-------) +--R 3 3 +--R x + a +--R (2) ------------ +--R 3 +--R 3a +--R Type: Expression Integer +--E + +--S 14 +cc:=aa-bb +--R +--R 3 +--R 3 3 x +--R - log(x + a ) + 3log(x) - log(-------) +--R 3 3 +--R x + a +--R (3) --------------------------------------- +--R 3 +--R 3a +--R Type: Expression Integer +--E + +--S 15 14:302 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.303~~~~~$\displaystyle +\int{\frac{dx}{x^2(x^3+a^3)}}~dx$} +$$\int{\frac{1}{x^2(x^3+a^3)}}= +-\frac{1}{a^3x}-\frac{1}{6a^4}\ln\frac{x^2-ax+a^2}{(x+a)^2} +-\frac{1}{a^4\sqrt{3}}\tan^{-1}\frac{2x-a}{a\sqrt{3}} +$$ +<<*>>= +)clear all + +--S 15 +aa:=integrate(1/(x^2*(x^3+a^3)),x) +--R +--R +--R (1) +--R +-+ +--R +-+ 2 2 +-+ (2x - a)\|3 +-+ +--R - x\|3 log(x - a x + a ) + 2x\|3 log(x + a) - 6x atan(------------) - 6a\|3 +--R 3a +--R ----------------------------------------------------------------------------- +--R 4 +-+ +--R 6a x\|3 +--R Type: Union(Expression Integer,...) +--E + +--S 16 +bb:=-1/(a^3*x)-1/(6*a^4)*log((x^2-a*x+a^2)/(x+a)^2)-1/(a^4*sqrt(3))*atan((2*x-a)/(a*sqrt(3))) +--R +--R 2 2 +-+ +--R x - a x + a +-+ (2x - a)\|3 +--R - x log(--------------) - 2x\|3 atan(------------) - 6a +--R 2 2 3a +--R x + 2a x + a +--R (2) ------------------------------------------------------- +--R 4 +--R 6a x +--R Type: Expression Integer +--E + +--S 17 +cc:=aa-bb +--R +--R 2 2 +--R 2 2 x - a x + a +--R - log(x - a x + a ) + 2log(x + a) + log(--------------) +--R 2 2 +--R x + 2a x + a +--R (3) -------------------------------------------------------- +--R 4 +--R 6a +--R Type: Expression Integer +--E + +--S 18 14:303 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.304~~~~~$\displaystyle +\int{\frac{dx}{(x^3+a^3)^2}}$} +$$\int{\frac{1}{(x^3+a^3)^2}}= +\frac{x}{3a^3(x^3+a^3)} ++\frac{1}{9a^5}\ln\frac{(x+a)^2}{x^2-ax+a^2} ++\frac{2}{3a^5\sqrt{3}}\tan^{-1}\frac{2x-a}{a\sqrt{3}} +$$ +<<*>>= +)clear all + +--S 19 +aa:=integrate(1/(x^3+a^3)^2,x) +--R +--R +--R (1) +--R 3 3 +-+ 2 2 3 3 +-+ +--R (- x - a )\|3 log(x - a x + a ) + (2x + 2a )\|3 log(x + a) +--R + +--R +-+ +--R 3 3 (2x - a)\|3 2 +-+ +--R (6x + 6a )atan(------------) + 3a x\|3 +--R 3a +--R / +--R 5 3 8 +-+ +--R (9a x + 9a )\|3 +--R Type: Union(Expression Integer,...) +--E + +--S 20 +bb:=x/(3*a^3*(x^3+a^3))+1/(9*a^5)*log((x+a)^2/(x^2-a*x+a^2))+2/(3*a^5*sqrt(3))*atan((2*x-a)/(a*sqrt(3))) +--R +--R (2) +--R 2 2 +-+ +--R 3 3 x + 2a x + a 3 3 +-+ (2x - a)\|3 2 +--R (x + a )log(--------------) + (2x + 2a )\|3 atan(------------) + 3a x +--R 2 2 3a +--R x - a x + a +--R ----------------------------------------------------------------------- +--R 5 3 8 +--R 9a x + 9a +--R Type: Expression Integer +--E + +--S 21 +cc:=aa-bb +--R +--R 2 2 +--R 2 2 x + 2a x + a +--R - log(x - a x + a ) + 2log(x + a) - log(--------------) +--R 2 2 +--R x - a x + a +--R (3) -------------------------------------------------------- +--R 5 +--R 9a +--R Type: Expression Integer +--E + +--S 22 14:304 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.305~~~~~$\displaystyle +\int{\frac{x~dx}{(x^3+a^3)^2}}$} +$$\int{\frac{x}{(x^3+a^3)^2}}= +\frac{x^2}{3a^3(x^3+a^3)} ++\frac{1}{18a^4}\ln\frac{x^2-ax+a^2}{(x+a)^2} ++\frac{1}{3a^4\sqrt{3}}\tan^{-1}\frac{2x-a}{a\sqrt{3}} +$$ +<<*>>= +)clear all + +--S 23 +aa:=integrate(x/(x^3+a^3)^2,x) +--R +--R +--R (1) +--R 3 3 +-+ 2 2 3 3 +-+ +--R (x + a )\|3 log(x - a x + a ) + (- 2x - 2a )\|3 log(x + a) +--R + +--R +-+ +--R 3 3 (2x - a)\|3 2 +-+ +--R (6x + 6a )atan(------------) + 6a x \|3 +--R 3a +--R / +--R 4 3 7 +-+ +--R (18a x + 18a )\|3 +--R Type: Union(Expression Integer,...) +--E + +--S 24 +bb:=x^2/(3*a^3*(x^3+a^3))+1/(18*a^4)*log((x^2-a*x+a^2)/(x+a)^2)+1/(3*a^4*sqrt(3))*atan((2*x-a)/(a*sqrt(3))) +--R +--R (2) +--R 2 2 +-+ +--R 3 3 x - a x + a 3 3 +-+ (2x - a)\|3 2 +--R (x + a )log(--------------) + (2x + 2a )\|3 atan(------------) + 6a x +--R 2 2 3a +--R x + 2a x + a +--R ------------------------------------------------------------------------ +--R 4 3 7 +--R 18a x + 18a +--R Type: Expression Integer +--E + +--S 25 +cc:=aa-bb +--R +--R 2 2 +--R 2 2 x - a x + a +--R log(x - a x + a ) - 2log(x + a) - log(--------------) +--R 2 2 +--R x + 2a x + a +--R (3) ------------------------------------------------------ +--R 4 +--R 18a +--R Type: Expression Integer +--E + +--S 26 14:305 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.306~~~~~$\displaystyle +\int{\frac{x^2~dx}{(x^3+a^3)^2}}$} +$$\int{\frac{x^2}{(x^3+a^3)^2}}= +-\frac{1}{3(x^3+a^3)} +$$ +<<*>>= +)clear all + +--S 27 +aa:=integrate(x^2/(x^3+a^3)^2,x) +--R +--R +--R 1 +--R (1) - --------- +--R 3 3 +--R 3x + 3a +--R Type: Union(Expression Integer,...) +--E + +--S 28 +bb:=-1/(3*(x^3+a^3)) +--R +--R 1 +--R (2) - --------- +--R 3 3 +--R 3x + 3a +--R Type: Fraction Polynomial Integer +--E + +--S 29 14:306 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.307~~~~~$\displaystyle +\int{\frac{dx}{x(x^3+a^3)^2}}$} +$$\int{\frac{1}{x(x^3+a^3)^2}}= +\frac{1}{3a^3(x^3+a^3)}+\frac{1}{3a^6}\ln\left(\frac{x^3}{x^3+a^3}\right) +$$ +<<*>>= +)clear all + +--S 30 +aa:=integrate(1/(x*(x^3+a^3)^2),x) +--R +--R +--R 3 3 3 3 3 3 3 +--R (- x - a )log(x + a ) + (3x + 3a )log(x) + a +--R (1) ------------------------------------------------ +--R 6 3 9 +--R 3a x + 3a +--R Type: Union(Expression Integer,...) +--E + +--S 31 +bb:=1/(3*a^3*(x^3+a^3))+1/(3*a^6)*log(x^3/(x^3+a^3)) +--R +--R 3 +--R 3 3 x 3 +--R (x + a )log(-------) + a +--R 3 3 +--R x + a +--R (2) -------------------------- +--R 6 3 9 +--R 3a x + 3a +--R Type: Expression Integer +--E + +--S 32 +cc:=aa-bb +--R +--R 3 +--R 3 3 x +--R - log(x + a ) + 3log(x) - log(-------) +--R 3 3 +--R x + a +--R (3) --------------------------------------- +--R 6 +--R 3a +--R Type: Expression Integer +--E + +--S 33 14:307 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.308~~~~~$\displaystyle +\int{\frac{dx}{x^2(x^3+a^3)^2}}~dx$} +$$\int{\frac{1}{x^2(x^3+a^3)^2}}= +-\frac{1}{a^6x}-\frac{x^2}{3a^6(x^3+a^3)} +-\frac{4}{3a^6}\int{\frac{x}{x^3+a^3}} +$$ +<<*>>= +)clear all + +--S 34 +aa:=integrate(1/(x^2*(x^3+a^3)^2),x) +--R +--R +--R (1) +--R 4 3 +-+ 2 2 4 3 +-+ +--R (- 2x - 2a x)\|3 log(x - a x + a ) + (4x + 4a x)\|3 log(x + a) +--R + +--R +-+ +--R 4 3 (2x - a)\|3 3 4 +-+ +--R (- 12x - 12a x)atan(------------) + (- 12a x - 9a )\|3 +--R 3a +--R / +--R 7 4 10 +-+ +--R (9a x + 9a x)\|3 +--R Type: Union(Expression Integer,...) +--E + +--S 35 +t1:=integrate(x/(x^3+a^3),x) +--R +--R +-+ +--R +-+ 2 2 +-+ (2x - a)\|3 +--R \|3 log(x - a x + a ) - 2\|3 log(x + a) + 6atan(------------) +--R 3a +--R (2) -------------------------------------------------------------- +--R +-+ +--R 6a\|3 +--R Type: Union(Expression Integer,...) +--E + +--S 36 +bb:=-1/(a^6*x)-x^2/(3*a^6*(x^3+a^3))-4/(3*a^6)*t1 +--R +--R (3) +--R 4 3 +-+ 2 2 4 3 +-+ +--R (- 2x - 2a x)\|3 log(x - a x + a ) + (4x + 4a x)\|3 log(x + a) +--R + +--R +-+ +--R 4 3 (2x - a)\|3 3 4 +-+ +--R (- 12x - 12a x)atan(------------) + (- 12a x - 9a )\|3 +--R 3a +--R / +--R 7 4 10 +-+ +--R (9a x + 9a x)\|3 +--R Type: Expression Integer +--E + +--S 37 14:308 Schaums and Axiom agree +cc:=aa-bb +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.309~~~~~$\displaystyle +\int{\frac{x^m~dx}{x^3+a^3}}$} +$$\int{\frac{x^m}{x^3+a^3}}= +\frac{x^{m-2}}{m-2}-a^3\int{\frac{x^{m-3}}{x^3+a^3}} +$$ +<<*>>= +)clear all + +--S 38 14:309 Axiom cannot compute this integral +aa:=integrate(x^m/(x^3+a^3),x) +--R +--R +--R x m +--I ++ %L +--I (1) | -------- d%L +--R ++ 3 3 +--I a + %L +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.310~~~~~$\displaystyle +\int{\frac{dx}{x^n(x^3+a^3)}}$} +$$\int{\frac{1}{x^n(x^3+a^3)}}= +\frac{-1}{a^3(n-1)x^{n-1}}-\frac{1}{a^3}\int{\frac{1}{x^{n-3}(x^3+a^3)}} +$$ +<<*>>= +)clear all + +--S 39 14:310 Axiom cannot compute this integral +aa:=integrate(1/(x^n*(x^3+a^3)),x) +--R +--R +--R x +--R ++ 1 +--I (1) | ------------- d%L +--R ++ 3 3 n +--I (a + %L )%L +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p73 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum14.input.pdf b/src/axiom-website/CATS/schaum14.input.pdf new file mode 100644 index 0000000..107608d --- /dev/null +++ b/src/axiom-website/CATS/schaum14.input.pdf @@ -0,0 +1,2081 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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a/src/axiom-website/CATS/schaum15.input.pamphlet b/src/axiom-website/CATS/schaum15.input.pamphlet new file mode 100644 index 0000000..0c01fbb --- /dev/null +++ b/src/axiom-website/CATS/schaum15.input.pamphlet @@ -0,0 +1,1332 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum15.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.311~~~~~$\displaystyle +\int{\frac{dx}{x^4+a^4}}$} +$$\int{\frac{1}{x^4+a^4}}= +\frac{1}{4a^3\sqrt{2}} +\ln\left(\frac{x^2+ax\sqrt{2}+a^2}{x^2-ax\sqrt{2}+a^2}\right) +-\frac{1}{2a^3\sqrt{2}}\tan^{-1}\frac{ax\sqrt{2}}{x^2-a^2} +$$ +<<*>>= +)spool schaum15.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(1/(x^4+a^4),x) +--R +--R +--R (1) +--R +------+ +------+2 +------+ +--R | 1 8 | 1 4 +-+ | 1 2 +--R |------ log(16a |------ + 4a x\|2 |------ + x ) +--R 4| 12 4| 12 4| 12 +--R \|256a \|256a \|256a +--R + +--R +------+ +------+2 +------+ +--R | 1 8 | 1 4 +-+ | 1 2 +--R - |------ log(16a |------ - 4a x\|2 |------ + x ) +--R 4| 12 4| 12 4| 12 +--R \|256a \|256a \|256a +--R + +--R +------+ +------+ +--R 4 | 1 4 | 1 +--R 4a |------ 4a |------ +--R +------+ 4| 12 +------+ 4| 12 +--R | 1 \|256a | 1 \|256a +--R 2 |------ atan(-------------------- - 2 |------ atan(--------------------) +--R 4| 12 +------+ 4| 12 +------+ +--R \|256a 4 | 1 +-+ \|256a 4 | 1 +-+ +--R 4a |------ - x\|2 4a |------ + x\|2 +--R 4| 12 4| 12 +--R \|256a \|256a +--R / +--R +-+ +--R \|2 +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=1/(4*a^3*sqrt(2))*log((x^2+a*x*sqrt(2)+a^2)/(x^2-a*x*sqrt(2)+a^2))-1/(2*a^3*sqrt(2))*atan((a*x*sqrt(2))/(x^2-a^2)) +--R +--R +-+ 2 2 +-+ +--R +-+ - a x\|2 - x - a +-+ a x\|2 +--R \|2 log(-------------------) - 2\|2 atan(-------) +--R +-+ 2 2 2 2 +--R a x\|2 - x - a x - a +--R (2) ------------------------------------------------- +--R 3 +--R 8a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R (3) +--R +------+ +------+2 +------+ +--R 3 | 1 8 | 1 4 +-+ | 1 2 +--R 4a |------ log(16a |------ + 4a x\|2 |------ + x ) +--R 4| 12 4| 12 4| 12 +--R \|256a \|256a \|256a +--R + +--R +------+ +------+2 +------+ +--R 3 | 1 8 | 1 4 +-+ | 1 2 +--R - 4a |------ log(16a |------ - 4a x\|2 |------ + x ) +--R 4| 12 4| 12 4| 12 +--R \|256a \|256a \|256a +--R + +--R +------+ +--R 4 | 1 +--R 4a |------ +--R +------+ 4| 12 +--R 3 | 1 \|256a +--R 8a |------ atan(--------------------) +--R 4| 12 +------+ +--R \|256a 4 | 1 +-+ +--R 4a |------ - x\|2 +--R 4| 12 +--R \|256a +--R + +--R +------+ +--R 4 | 1 +--R 4a |------ +--R +------+ 4| 12 +-+ 2 2 +--R 3 | 1 \|256a - a x\|2 - x - a +--R - 8a |------ atan(-------------------- - log(-------------------) +--R 4| 12 +------+ +-+ 2 2 +--R \|256a 4 | 1 +-+ a x\|2 - x - a +--R 4a |------ + x\|2 +--R 4| 12 +--R \|256a +--R + +--R +-+ +--R a x\|2 +--R 2atan(-------) +--R 2 2 +--R x - a +--R / +--R 3 +-+ +--R 4a \|2 +--R Type: Expression Integer +--E + +--S 4 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 5 +dd:=atanrule cc +--R +--R (5) +--R +------+ +------+2 +------+ +--R 3 | 1 8 | 1 4 +-+ | 1 2 +--R 4a |------ log(16a |------ + 4a x\|2 |------ + x ) +--R 4| 12 4| 12 4| 12 +--R \|256a \|256a \|256a +--R + +--R +------+ +------+2 +------+ +--R 3 | 1 8 | 1 4 +-+ | 1 2 +--R - 4a |------ log(16a |------ - 4a x\|2 |------ + x ) +--R 4| 12 4| 12 4| 12 +--R \|256a \|256a \|256a +--R + +--R +------+ +--R 4 | 1 +-+ +--R (- 4 + 4%i)a |------ + %i x\|2 +--R +------+ 4| 12 +--R 3 | 1 \|256a +--R 4%i a |------ log(---------------------------------) +--R 4| 12 +------+ +--R \|256a 4 | 1 +-+ +--R (4 + 4%i)a |------ + %i x\|2 +--R 4| 12 +--R \|256a +--R + +--R +------+ +--R 4 | 1 +-+ +--R (- 4 + 4%i)a |------ - %i x\|2 +--R +------+ 4| 12 +--R 3 | 1 \|256a +--R - 4%i a |------ log(---------------------------------) +--R 4| 12 +------+ +--R \|256a 4 | 1 +-+ +--R (4 + 4%i)a |------ - %i x\|2 +--R 4| 12 +--R \|256a +--R + +--R +-+ 2 2 +-+ 2 2 +--R - a x\|2 + %i x - %i a - a x\|2 - x - a +--R - %i log(-------------------------) - log(-------------------) +--R +-+ 2 2 +-+ 2 2 +--R a x\|2 + %i x - %i a a x\|2 - x - a +--R / +--R 3 +-+ +--R 4a \|2 +--R Type: Expression Complex Integer +--E + +--S 6 +ee:=rootSimp dd +--R +--R (6) +--R +-+ +--R +-+ 2 2 x\|2 + (1 + %i)a +--R log(a x\|2 + x + a ) + %i log(-----------------) +--R +-+ +--R x\|2 + (1 - %i)a +--R + +--R +-+ +-+ 2 2 +--R x\|2 + (- 1 - %i)a - a x\|2 + %i x - %i a +--R - %i log(-------------------) - %i log(-------------------------) +--R +-+ +-+ 2 2 +--R x\|2 + (- 1 + %i)a a x\|2 + %i x - %i a +--R + +--R +-+ 2 2 +--R - a x\|2 - x - a +-+ 2 2 +--R - log(-------------------) - log(- a x\|2 + x + a ) +--R +-+ 2 2 +--R a x\|2 - x - a +--R / +--R 3 +-+ +--R 4a \|2 +--R Type: Expression Complex Integer +--E + +--S 7 +ff:=expandLog ee +--R +--R (7) +--R +-+ 2 2 +-+ 2 2 +--R %i log(a x\|2 + %i x - %i a ) - %i log(a x\|2 - %i x + %i a ) +--R + +--R +-+ +-+ +--R %i log(x\|2 + (1 + %i)a) - %i log(x\|2 + (1 - %i)a) +--R + +--R +-+ +-+ +--R %i log(x\|2 + (- 1 + %i)a) - %i log(x\|2 + (- 1 - %i)a) +--R + +--R (- 2 - %i)log(- 1) +--R / +--R 3 +-+ +--R 4a \|2 +--R Type: Expression Complex Integer +--E + +--S 8 +gg:=complexNormalize ff +--R +--R %i %i +--R %i log(--) - %i log(- --) + (- 2 - %i)log(- 1) +--R 2 2 +--R (8) ---------------------------------------------- +--R 3 +-+ +--R 4a \|2 +--R Type: Expression Complex Integer +--E + +--S 9 14:311 Schaums and Axiom differ by a constant +hh:=expandLog gg +--R +--R %i log(%i) - %i log(- %i) + (- 2 - %i)log(- 1) +--R (9) ---------------------------------------------- +--R 3 +-+ +--R 4a \|2 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.312~~~~~$\displaystyle +\int{\frac{x~dx}{x^4+a^4}}$} +$$\int{\frac{x}{x^4+a^4}}= +\frac{1}{2a^2}\tan^{-1}\frac{x^2}{a^2} +$$ +<<*>>= +)clear all + +--S 10 +aa:=integrate(x/(x^4+a^4),x) +--R +--R +--R 2 +--R x +--R atan(--) +--R 2 +--R a +--R (1) -------- +--R 2 +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 11 +bb:=1/(2*a^2)*atan(x^2/a^2) +--R +--R 2 +--R x +--R atan(--) +--R 2 +--R a +--R (2) -------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 12 14:312 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.313~~~~~$\displaystyle +\int{\frac{x^2~dx}{x^4+a^4}}$} +$$\int{\frac{x^2}{x^4+a^4}}= +\frac{1}{4a\sqrt{2}} +\ln\left(\frac{x^2-ax\sqrt{2}+a^2}{x^2+ax\sqrt{2}+a^2}\right) +-\frac{1}{2a\sqrt{2}}\tan^{-1}\frac{ax\sqrt{2}}{x^2-a^2} +$$ +<<*>>= +)clear all + +--S 13 +aa:=integrate(x^2/(x^4+a^4),x) +--R +--R +--R (1) +--R +-----+ +-----+3 +-----+2 +--R | 1 4 +-+ | 1 4 | 1 2 +--R - |----- log(64a x\|2 |----- + 16a |----- + x ) +--R 4| 4 4| 4 4| 4 +--R \|256a \|256a \|256a +--R + +--R +-----+ +-----+3 +-----+2 +--R | 1 4 +-+ | 1 4 | 1 2 +--R |----- log(- 64a x\|2 |----- + 16a |----- + x ) +--R 4| 4 4| 4 4| 4 +--R \|256a \|256a \|256a +--R + +--R +-----+3 +-----+3 +--R 4 | 1 4 | 1 +--R 64a |----- 64a |----- +--R +-----+ 4| 4 +-----+ 4| 4 +--R | 1 \|256a | 1 \|256a +--R 2 |----- atan(--------------------- - 2 |----- atan(---------------------) +--R 4| 4 +-----+3 4| 4 +-----+3 +--R \|256a 4 | 1 +-+ \|256a 4 | 1 +-+ +--R 64a |----- - x\|2 64a |----- + x\|2 +--R 4| 4 4| 4 +--R \|256a \|256a +--R / +--R +-+ +--R \|2 +--R Type: Union(Expression Integer,...) +--E + +--S 14 +bb:=1/(4*a*sqrt(2))*log((x^2-a*x*sqrt(2)+a^2)/(x^2+a*x*sqrt(2)+a^2))-1/(2*a*sqrt(2))*atan((a*x*sqrt(2))/(x^2-a^2)) +--R +--R +-+ 2 2 +-+ +--R +-+ - a x\|2 + x + a +-+ a x\|2 +--R \|2 log(-------------------) - 2\|2 atan(-------) +--R +-+ 2 2 2 2 +--R a x\|2 + x + a x - a +--R (2) ------------------------------------------------- +--R 8a +--R Type: Expression Integer +--E + +--S 15 +cc:=aa-bb +--R +--R (3) +--R +-----+ +-----+3 +-----+2 +--R | 1 4 +-+ | 1 4 | 1 2 +--R - 4a |----- log(64a x\|2 |----- + 16a |----- + x ) +--R 4| 4 4| 4 4| 4 +--R \|256a \|256a \|256a +--R + +--R +-----+ +-----+3 +-----+2 +--R | 1 4 +-+ | 1 4 | 1 2 +--R 4a |----- log(- 64a x\|2 |----- + 16a |----- + x ) +--R 4| 4 4| 4 4| 4 +--R \|256a \|256a \|256a +--R + +--R +-----+3 +--R 4 | 1 +--R 64a |----- +--R +-----+ 4| 4 +--R | 1 \|256a +--R 8a |----- atan(---------------------) +--R 4| 4 +-----+3 +--R \|256a 4 | 1 +-+ +--R 64a |----- - x\|2 +--R 4| 4 +--R \|256a +--R + +--R +-----+3 +--R 4 | 1 +--R 64a |----- +--R +-----+ 4| 4 +-+ 2 2 +--R | 1 \|256a - a x\|2 + x + a +--R - 8a |----- atan(--------------------- - log(-------------------) +--R 4| 4 +-----+3 +-+ 2 2 +--R \|256a 4 | 1 +-+ a x\|2 + x + a +--R 64a |----- + x\|2 +--R 4| 4 +--R \|256a +--R + +--R +-+ +--R a x\|2 +--R 2atan(-------) +--R 2 2 +--R x - a +--R / +--R +-+ +--R 4a\|2 +--R Type: Expression Integer +--E + +--S 16 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 17 +dd:=atanrule cc +--R +--R (5) +--R +-----+ +-----+3 +-----+2 +--R | 1 4 +-+ | 1 4 | 1 2 +--R - 4a |----- log(64a x\|2 |----- + 16a |----- + x ) +--R 4| 4 4| 4 4| 4 +--R \|256a \|256a \|256a +--R + +--R +-----+3 +--R 4 | 1 +-+ +--R (- 64 + 64%i)a |----- + %i x\|2 +--R +-----+ 4| 4 +--R | 1 \|256a +--R 4%i a |----- log(-----------------------------------) +--R 4| 4 +-----+3 +--R \|256a 4 | 1 +-+ +--R (64 + 64%i)a |----- + %i x\|2 +--R 4| 4 +--R \|256a +--R + +--R +-----+3 +--R 4 | 1 +-+ +--R (- 64 + 64%i)a |----- - %i x\|2 +--R +-----+ 4| 4 +--R | 1 \|256a +--R - 4%i a |----- log(-----------------------------------) +--R 4| 4 +-----+3 +--R \|256a 4 | 1 +-+ +--R (64 + 64%i)a |----- - %i x\|2 +--R 4| 4 +--R \|256a +--R + +--R +-----+ +-----+3 +-----+2 +--R | 1 4 +-+ | 1 4 | 1 2 +--R 4a |----- log(- 64a x\|2 |----- + 16a |----- + x ) +--R 4| 4 4| 4 4| 4 +--R \|256a \|256a \|256a +--R + +--R +-+ 2 2 +-+ 2 2 +--R - a x\|2 + x + a - a x\|2 + %i x - %i a +--R - log(-------------------) - %i log(-------------------------) +--R +-+ 2 2 +-+ 2 2 +--R a x\|2 + x + a a x\|2 + %i x - %i a +--R / +--R +-+ +--R 4a\|2 +--R Type: Expression Complex Integer +--E + +--S 18 +ee:=expandLog dd +--R +--R (6) +--R +-----+ +-----+3 +-----+2 +--R | 1 4 +-+ | 1 4 | 1 2 +--R - 4a |----- log(64a x\|2 |----- + 16a |----- + x ) +--R 4| 4 4| 4 4| 4 +--R \|256a \|256a \|256a +--R + +--R +-----+ +-----+3 +-----+2 +--R | 1 4 +-+ | 1 4 | 1 2 +--R 4a |----- log(64a x\|2 |----- - 16a |----- - x ) +--R 4| 4 4| 4 4| 4 +--R \|256a \|256a \|256a +--R + +--R +-----+ +-----+3 +--R | 1 4 | 1 +-+ +--R 4%i a |----- log((64 + 64%i)a |----- + x\|2 ) +--R 4| 4 4| 4 +--R \|256a \|256a +--R + +--R +-----+ +-----+3 +--R | 1 4 | 1 +-+ +--R - 4%i a |----- log((64 + 64%i)a |----- + %i x\|2 ) +--R 4| 4 4| 4 +--R \|256a \|256a +--R + +--R +-----+ +-----+3 +--R | 1 4 | 1 +-+ +--R 4%i a |----- log((64 + 64%i)a |----- - %i x\|2 ) +--R 4| 4 4| 4 +--R \|256a \|256a +--R + +--R +-----+ +-----+3 +-----+ +--R | 1 4 | 1 +-+ | 1 +--R - 4%i a |----- log((64 + 64%i)a |----- - x\|2 + 4a log(- 1) |----- +--R 4| 4 4| 4 4| 4 +--R \|256a \|256a \|256a +--R + +--R +-+ 2 2 +-+ 2 2 +--R log(a x\|2 + x + a ) + %i log(a x\|2 + %i x - %i a ) +--R + +--R +-+ 2 2 +-+ 2 2 +--R - %i log(a x\|2 - %i x + %i a ) - log(a x\|2 - x - a ) +--R + +--R (- 1 - %i)log(- 1) +--R / +--R +-+ +--R 4a\|2 +--R Type: Expression Complex Integer +--E + +--S 19 +ff:=rootSimp ee +--R +--R (7) +--R +-+ 2 2 +-+ 2 2 +--R %i log(a x\|2 + %i x - %i a ) - %i log(a x\|2 - %i x + %i a ) +--R + +--R +-+ +-+ +--R %i log(x\|2 + (1 + %i)a) - %i log(%i x\|2 + (1 + %i)a) +--R + +--R +-+ +-+ +--R %i log(- %i x\|2 + (1 + %i)a) - %i log(- x\|2 + (1 + %i)a) - %i log(- 1) +--R / +--R +-+ +--R 4a\|2 +--R Type: Expression Complex Integer +--E + +--S 20 14:313 Schaums and Axiom differ by a constant +gg:=complexNormalize ff +--R +--R %i log(2) - %i log(- 1) - %i log(- 2) +--R (8) ------------------------------------- +--R +-+ +--R 4a\|2 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.314~~~~~$\displaystyle +\int{\frac{x^3~dx}{x^4+a^4}}$} +$$\int{\frac{x^3}{x^4+a^4}}= +\frac{1}{4}\ln(x^4+a^4) +$$ +<<*>>= +)clear all + +--S 21 +aa:=integrate(x^3/(x^4+a^4),x) +--R +--R +--R 4 4 +--R log(x + a ) +--R (1) ------------ +--R 4 +--R Type: Union(Expression Integer,...) +--E + +--S 22 +bb:=1/4*log(x^4+a^4) +--R +--R 4 4 +--R log(x + a ) +--R (2) ------------ +--R 4 +--R Type: Expression Integer +--E + +--S 23 14:314 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.315~~~~~$\displaystyle +\int{\frac{dx}{x(x^4+a^4)}}~dx$} +$$\int{\frac{1}{x(x^4+a^4)}}= +\frac{1}{4a^4}\ln\left(\frac{x^4}{x^4+a^4}\right) +$$ +<<*>>= +)clear all + +--S 24 +aa:=integrate(1/(x*(x^4+a^4)),x) +--R +--R +--R 4 4 +--R - log(x + a ) + 4log(x) +--R (1) ------------------------ +--R 4 +--R 4a +--R Type: Union(Expression Integer,...) +--E + +--S 25 +bb:=1/(4*a^4)*log(x^4/(x^4+a^4)) +--R +--R 4 +--R x +--R log(-------) +--R 4 4 +--R x + a +--R (2) ------------ +--R 4 +--R 4a +--R Type: Expression Integer +--E + +--S 26 +cc:=aa-bb +--R +--R 4 +--R 4 4 x +--R - log(x + a ) + 4log(x) - log(-------) +--R 4 4 +--R x + a +--R (3) --------------------------------------- +--R 4 +--R 4a +--R Type: Expression Integer +--E + +--S 27 14:315 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.316~~~~~$\displaystyle +\int{\frac{dx}{x^2(x^4+a^4)}}$} +$$\int{\frac{1}{x^2(x^4+a^4)}}= +-\frac{1}{a^4x}-\frac{1}{4a^5\sqrt{2}} +\ln\left(\frac{x^2-ax\sqrt{2}+a^2}{x^2+ax\sqrt{2}+a^2}\right) ++\frac{1}{2a^5\sqrt{2}}\tan^{-1}\frac{ax\sqrt{2}}{x^2-a^2} +$$ +<<*>>= +)clear all + +--S 28 +aa:=integrate(1/(x^2*(x^4+a^4)),x) +--R +--R +--R (1) +--R +------+ +------+3 +------+2 +--R 4 | 1 16 +-+ | 1 12 | 1 2 +--R a x |------ log(64a x\|2 |------ + 16a |------ + x ) +--R 4| 20 4| 20 4| 20 +--R \|256a \|256a \|256a +--R + +--R +------+ +------+3 +------+2 +--R 4 | 1 16 +-+ | 1 12 | 1 2 +--R - a x |------ log(- 64a x\|2 |------ + 16a |------ + x ) +--R 4| 20 4| 20 4| 20 +--R \|256a \|256a \|256a +--R + +--R +------+3 +--R 16 | 1 +--R 64a |------ +--R +------+ 4| 20 +--R 4 | 1 \|256a +--R - 2a x |------ atan(-----------------------) +--R 4| 20 +------+3 +--R \|256a 16 | 1 +-+ +--R 64a |------ - x\|2 +--R 4| 20 +--R \|256a +--R + +--R +------+3 +--R 16 | 1 +--R 64a |------ +--R +------+ 4| 20 +--R 4 | 1 \|256a +-+ +--R 2a x |------ atan(----------------------- - \|2 +--R 4| 20 +------+3 +--R \|256a 16 | 1 +-+ +--R 64a |------ + x\|2 +--R 4| 20 +--R \|256a +--R / +--R 4 +-+ +--R a x\|2 +--R Type: Union(Expression Integer,...) +--E + +--S 29 +bb:=-1/(a^4*x)-1/(4*a^5*sqrt(2))*log((x^2-a*x*sqrt(2)+a^2)/(x^2+a*x*sqrt(2)+a^2))+1/(2*a^5*sqrt(2))*atan((a*x*sqrt(2))/(x^2-a^2)) +--R +--R +-+ 2 2 +-+ +--R +-+ - a x\|2 + x + a +-+ a x\|2 +--R - x\|2 log(-------------------) + 2x\|2 atan(-------) - 8a +--R +-+ 2 2 2 2 +--R a x\|2 + x + a x - a +--R (2) ---------------------------------------------------------- +--R 5 +--R 8a x +--R Type: Expression Integer +--E + +--S 30 +cc:=aa-bb +--R +--R (3) +--R +------+ +------+3 +------+2 +--R 5 | 1 16 +-+ | 1 12 | 1 2 +--R 4a |------ log(64a x\|2 |------ + 16a |------ + x ) +--R 4| 20 4| 20 4| 20 +--R \|256a \|256a \|256a +--R + +--R +------+ +------+3 +------+2 +--R 5 | 1 16 +-+ | 1 12 | 1 2 +--R - 4a |------ log(- 64a x\|2 |------ + 16a |------ + x ) +--R 4| 20 4| 20 4| 20 +--R \|256a \|256a \|256a +--R + +--R +------+3 +--R 16 | 1 +--R 64a |------ +--R +------+ 4| 20 +--R 5 | 1 \|256a +--R - 8a |------ atan(-----------------------) +--R 4| 20 +------+3 +--R \|256a 16 | 1 +-+ +--R 64a |------ - x\|2 +--R 4| 20 +--R \|256a +--R + +--R +------+3 +--R 16 | 1 +--R 64a |------ +--R +------+ 4| 20 +-+ 2 2 +--R 5 | 1 \|256a - a x\|2 + x + a +--R 8a |------ atan(----------------------- + log(-------------------) +--R 4| 20 +------+3 +-+ 2 2 +--R \|256a 16 | 1 +-+ a x\|2 + x + a +--R 64a |------ + x\|2 +--R 4| 20 +--R \|256a +--R + +--R +-+ +--R a x\|2 +--R - 2atan(-------) +--R 2 2 +--R x - a +--R / +--R 5 +-+ +--R 4a \|2 +--R Type: Expression Integer +--E + +--S 31 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 32 +dd:=atanrule cc +--R +--R (5) +--R +------+ +------+3 +------+2 +--R 5 | 1 16 +-+ | 1 12 | 1 2 +--R 4a |------ log(64a x\|2 |------ + 16a |------ + x ) +--R 4| 20 4| 20 4| 20 +--R \|256a \|256a \|256a +--R + +--R +------+3 +--R 16 | 1 +-+ +--R (- 64 + 64%i)a |------ + %i x\|2 +--R +------+ 4| 20 +--R 5 | 1 \|256a +--R - 4%i a |------ log(-------------------------------------) +--R 4| 20 +------+3 +--R \|256a 16 | 1 +-+ +--R (64 + 64%i)a |------ + %i x\|2 +--R 4| 20 +--R \|256a +--R + +--R +------+3 +--R 16 | 1 +-+ +--R (- 64 + 64%i)a |------ - %i x\|2 +--R +------+ 4| 20 +--R 5 | 1 \|256a +--R 4%i a |------ log(-------------------------------------) +--R 4| 20 +------+3 +--R \|256a 16 | 1 +-+ +--R (64 + 64%i)a |------ - %i x\|2 +--R 4| 20 +--R \|256a +--R + +--R +------+ +------+3 +------+2 +--R 5 | 1 16 +-+ | 1 12 | 1 2 +--R - 4a |------ log(- 64a x\|2 |------ + 16a |------ + x ) +--R 4| 20 4| 20 4| 20 +--R \|256a \|256a \|256a +--R + +--R +-+ 2 2 +-+ 2 2 +--R - a x\|2 + x + a - a x\|2 + %i x - %i a +--R log(-------------------) + %i log(-------------------------) +--R +-+ 2 2 +-+ 2 2 +--R a x\|2 + x + a a x\|2 + %i x - %i a +--R / +--R 5 +-+ +--R 4a \|2 +--R Type: Expression Complex Integer +--E + +--S 33 +ee:=expandLog dd +--R +--R (6) +--R +------+ +------+3 +------+2 +--R 5 | 1 16 +-+ | 1 12 | 1 2 +--R 4a |------ log(64a x\|2 |------ + 16a |------ + x ) +--R 4| 20 4| 20 4| 20 +--R \|256a \|256a \|256a +--R + +--R +------+ +------+3 +------+2 +--R 5 | 1 16 +-+ | 1 12 | 1 2 +--R - 4a |------ log(64a x\|2 |------ - 16a |------ - x ) +--R 4| 20 4| 20 4| 20 +--R \|256a \|256a \|256a +--R + +--R +------+ +------+3 +--R 5 | 1 16 | 1 +-+ +--R - 4%i a |------ log((64 + 64%i)a |------ + x\|2 ) +--R 4| 20 4| 20 +--R \|256a \|256a +--R + +--R +------+ +------+3 +--R 5 | 1 16 | 1 +-+ +--R 4%i a |------ log((64 + 64%i)a |------ + %i x\|2 ) +--R 4| 20 4| 20 +--R \|256a \|256a +--R + +--R +------+ +------+3 +--R 5 | 1 16 | 1 +-+ +--R - 4%i a |------ log((64 + 64%i)a |------ - %i x\|2 ) +--R 4| 20 4| 20 +--R \|256a \|256a +--R + +--R +------+ +------+3 +--R 5 | 1 16 | 1 +-+ +--R 4%i a |------ log((64 + 64%i)a |------ - x\|2 ) +--R 4| 20 4| 20 +--R \|256a \|256a +--R + +--R +------+ +--R 5 | 1 +-+ 2 2 +--R - 4a log(- 1) |------ - log(a x\|2 + x + a ) +--R 4| 20 +--R \|256a +--R + +--R +-+ 2 2 +-+ 2 2 +--R - %i log(a x\|2 + %i x - %i a ) + %i log(a x\|2 - %i x + %i a ) +--R + +--R +-+ 2 2 +--R log(a x\|2 - x - a ) + (1 + %i)log(- 1) +--R / +--R 5 +-+ +--R 4a \|2 +--R Type: Expression Complex Integer +--E + +--S 34 +ff:=rootSimp ee +--R +--R (7) +--R +-+ 2 2 +-+ 2 2 +--R - %i log(a x\|2 + %i x - %i a ) + %i log(a x\|2 - %i x + %i a ) +--R + +--R +-+ +-+ +--R - %i log(x\|2 + (1 + %i)a) + %i log(%i x\|2 + (1 + %i)a) +--R + +--R +-+ +-+ +--R - %i log(- %i x\|2 + (1 + %i)a) + %i log(- x\|2 + (1 + %i)a) +--R + +--R %i log(- 1) +--R / +--R 5 +-+ +--R 4a \|2 +--R Type: Expression Complex Integer +--E + +--S 35 14:316 Schaums and Axiom differ by a constant +gg:=complexNormalize ff +--R +--R - %i log(2) + %i log(- 1) + %i log(- 2) +--R (8) --------------------------------------- +--R 5 +-+ +--R 4a \|2 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.317~~~~~$\displaystyle +\int{\frac{dx}{x^3(x^4+a^4)}}$} +$$\int{\frac{1}{x^3(x^4+a^4)}}= +-\frac{1}{2a^4x^2}-\frac{1}{2a^6}\tan^{-1}\frac{x^2}{a^2} +$$ +<<*>>= +)clear all + +--S 36 +aa:=integrate(1/(x^3*(x^4+a^4)),x) +--R +--R +--R 2 +--R 2 x 2 +--R - x atan(--) - a +--R 2 +--R a +--R (1) ----------------- +--R 6 2 +--R 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 37 +bb:=-1/(2*a^4*x^2)-1/(2*a^6)*atan(x^2/a^2) +--R +--R 2 +--R 2 x 2 +--R - x atan(--) - a +--R 2 +--R a +--R (2) ----------------- +--R 6 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 38 14:317 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.318~~~~~$\displaystyle +\int{\frac{dx}{(x^4-a^4)}}$} +$$\int{\frac{1}{(x^4-a^4)}}= +\frac{1}{4a^3}\ln\left(\frac{x-a}{x+a}\right) +-\frac{1}{2a^3}\tan^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 39 +aa:=integrate(1/(x^4-a^4),x) +--R +--R +--R x +--R - log(x + a) + log(x - a) - 2atan(-) +--R a +--R (1) ------------------------------------ +--R 3 +--R 4a +--R Type: Union(Expression Integer,...) +--E + +--S 40 +bb:=1/(4*a^3)*log((x-a)/(x+a))-1/(2*a^3)*atan(x/a) +--R +--R x - a x +--R log(-----) - 2atan(-) +--R x + a a +--R (2) --------------------- +--R 3 +--R 4a +--R Type: Expression Integer +--E + +--S 41 +cc:=aa-bb +--R +--R x - a +--R - log(x + a) + log(x - a) - log(-----) +--R x + a +--R (3) -------------------------------------- +--R 3 +--R 4a +--R Type: Expression Integer +--E + +--S 42 14:318 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.319~~~~~$\displaystyle +\int{\frac{x~dx}{(x^4-a^4)}}$} +$$\int{\frac{x}{(x^4-a^4)}}= +\frac{1}{4a^2}\ln\left(\frac{x^2-a^2}{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 43 +aa:=integrate(x/(x^4-a^4),x) +--R +--R +--R 2 2 2 2 +--R - log(x + a ) + log(x - a ) +--R (1) ----------------------------- +--R 2 +--R 4a +--R Type: Union(Expression Integer,...) +--E + +--S 44 +bb:=1/(4*a^2)*log((x^2-a^2)/(x^2+a^2)) +--R +--R 2 2 +--R x - a +--R log(-------) +--R 2 2 +--R x + a +--R (2) ------------ +--R 2 +--R 4a +--R Type: Expression Integer +--E + +--S 45 +cc:=aa-bb +--R +--R 2 2 +--R 2 2 2 2 x - a +--R - log(x + a ) + log(x - a ) - log(-------) +--R 2 2 +--R x + a +--R (3) -------------------------------------------- +--R 2 +--R 4a +--R Type: Expression Integer +--E + +--S 46 14:319 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.320~~~~~$\displaystyle +\int{\frac{x^2~dx}{x^4-a^4}}$} +$$\int{\frac{x^2}{x^4-a^4}}= +\frac{1}{4a}\ln\left(\frac{x-a}{x+a}\right) ++\frac{1}{2a}\tan^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 47 +aa:=integrate(x^2/(x^4-a^4),x) +--R +--R +--R x +--R - log(x + a) + log(x - a) + 2atan(-) +--R a +--R (1) ------------------------------------ +--R 4a +--R Type: Union(Expression Integer,...) +--E + +--S 48 +bb:=1/(4*a)*log((x-a)/(x+a))+1/(2*a)*atan(x/a) +--R +--R x - a x +--R log(-----) + 2atan(-) +--R x + a a +--R (2) --------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 49 +cc:=aa-bb +--R +--R x - a +--R - log(x + a) + log(x - a) - log(-----) +--R x + a +--R (3) -------------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 50 14:320 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.321~~~~~$\displaystyle +\int{\frac{x^3~dx}{x^4-a^4}}$} +$$\int{\frac{x^3}{x^4-a^4}}= +\frac{1}{4}\ln(x^4-a^4) +$$ +<<*>>= +)clear all + +--S 51 +aa:=integrate(x^3/(x^4-a^4),x) +--R +--R +--R 4 4 +--R log(x - a ) +--R (1) ------------ +--R 4 +--R Type: Union(Expression Integer,...) +--E + +--S 52 +bb:=1/4*log(x^4-a^4) +--R +--R 4 4 +--R log(x - a ) +--R (2) ------------ +--R 4 +--R Type: Expression Integer +--E + +--S 53 14:321 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.322~~~~~$\displaystyle +\int{\frac{dx}{x(x^4-a^4)}}$} +$$\int{\frac{1}{x(x^4-a^4)}}= +\frac{1}{4a^4}\ln\left(\frac{x^4-a^4}{x^4}\right) +$$ +<<*>>= +)clear all + +--S 54 +aa:=integrate(1/(x*(x^4-a^4)),x) +--R +--R +--R 4 4 +--R log(x - a ) - 4log(x) +--R (1) ---------------------- +--R 4 +--R 4a +--R Type: Union(Expression Integer,...) +--E + +--S 55 +bb:=1/(4*a^4)*log((x^4-a^4)/x^4) +--R +--R 4 4 +--R x - a +--R log(-------) +--R 4 +--R x +--R (2) ------------ +--R 4 +--R 4a +--R Type: Expression Integer +--E + +--S 56 +cc:=aa-bb +--R +--R 4 4 +--R 4 4 x - a +--R log(x - a ) - 4log(x) - log(-------) +--R 4 +--R x +--R (3) ------------------------------------- +--R 4 +--R 4a +--R Type: Expression Integer +--E + +--S 57 14:322 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.323~~~~~$\displaystyle +\int{\frac{dx}{x^2(x^4-a^4)}}$} +$$\int{\frac{1}{x^2(x^4-a^4)}}= +\frac{1}{a^4x}+\frac{1}{4a^5}\ln\left(\frac{x-a}{x+a}\right) ++\frac{1}{2a^5}\tan^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 58 +aa:=integrate(1/(x^2*(x^4-a^4)),x) +--R +--R +--R x +--R - x log(x + a) + x log(x - a) + 2x atan(-) + 4a +--R a +--R (1) ----------------------------------------------- +--R 5 +--R 4a x +--R Type: Union(Expression Integer,...) +--E + +--S 59 +bb:=1/(a^4*x)+1/(4*a^5)*log((x-a)/(x+a))+1/(2*a^5)*atan(x/a) +--R +--R x - a x +--R x log(-----) + 2x atan(-) + 4a +--R x + a a +--R (2) ------------------------------ +--R 5 +--R 4a x +--R Type: Expression Integer +--E + +--S 60 +cc:=aa-bb +--R +--R x - a +--R - log(x + a) + log(x - a) - log(-----) +--R x + a +--R (3) -------------------------------------- +--R 5 +--R 4a +--R Type: Expression Integer +--E + +--S 61 14:323 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.324~~~~~$\displaystyle +\int{\frac{dx}{x^3(x^4-a^4)}}$} +$$\int{\frac{1}{x^3(x^4-a^4)}}= +\frac{1}{2a^4x^2}+\frac{1}{4a^6}\ln\left(\frac{x^2-a^2}{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 62 +aa:=integrate(1/(x^3*(x^4-a^4)),x) +--R +--R +--R 2 2 2 2 2 2 2 +--R - x log(x + a ) + x log(x - a ) + 2a +--R (1) --------------------------------------- +--R 6 2 +--R 4a x +--R Type: Union(Expression Integer,...) +--E + +--S 63 +bb:=1/(2*a^4*x^2)+1/(4*a^6)*log((x^2-a^2)/(x^2+a^2)) +--R +--R 2 2 +--R 2 x - a 2 +--R x log(-------) + 2a +--R 2 2 +--R x + a +--R (2) -------------------- +--R 6 2 +--R 4a x +--R Type: Expression Integer +--E + +--S 64 +cc:=aa-bb +--R +--R 2 2 +--R 2 2 2 2 x - a +--R - log(x + a ) + log(x - a ) - log(-------) +--R 2 2 +--R x + a +--R (3) -------------------------------------------- +--R 6 +--R 4a +--R Type: Expression Integer +--E + +--S 65 14:324 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp73-74 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b/src/axiom-website/CATS/schaum16.input.pamphlet @@ -0,0 +1,748 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum16.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.325~~~~~$\displaystyle +\int{\frac{dx}{x(x^n+a^n)}}$} +$$\int{\frac{1}{x(x^n+a^n)}}= +\frac{1}{na^n}\ln\frac{x^n}{x^n+a^n} +$$ +<<*>>= +)spool schaum16.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(1/(x*(x^n+a^n)),x) +--R +--R n log(x) n +--R - log(%e + a ) + n log(x) +--R (1) --------------------------------- +--R n +--R n a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=1/(n*a^n)*log(x^n/(x^n+a^n)) +--R +--R n +--R x +--R log(-------) +--R n n +--R x + a +--R (2) ------------ +--R n +--R n a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R n +--R n log(x) n x +--R - log(%e + a ) - log(-------) + n log(x) +--R n n +--R x + a +--R (3) ------------------------------------------------ +--R n +--R n a +--R Type: Expression Integer +--E + +--S 4 +dd:=expandLog cc +--R +--R n log(x) n n n n +--R - log(%e + a ) + log(x + a ) - log(x ) + n log(x) +--R (4) ---------------------------------------------------------- +--R n +--R n a +--R Type: Expression Integer +--E + +--S 5 14:325 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.326~~~~~$\displaystyle +\int{\frac{x^{n-1}~dx}{x^n+a^n}}$} +$$\int{\frac{x^{n-1}}{x^n+a^n}}= +\frac{1}{n}\ln(x^n+a^n) +$$ +<<*>>= +)clear all + +--S 8 +aa:=integrate(x^(n-1)/(x^n+a^n),x) +--R +--R +--R n log(x) n +--R log(%e + a ) +--R (1) -------------------- +--R n +--R Type: Union(Expression Integer,...) +--E + +--S 9 +bb:=1/n*log(x^n+a^n) +--R +--R n n +--R log(x + a ) +--R (2) ------------ +--R n +--R Type: Expression Integer +--E + +--S 10 +cc:=aa-bb +--R +--R n log(x) n n n +--R log(%e + a ) - log(x + a ) +--R (3) ----------------------------------- +--R n +--R Type: Expression Integer +--E + +--S 11 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 12 14:326 Schaums and Axiom agree +dd:=explog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.327~~~~~$\displaystyle +\int{\frac{x^m~dx}{(x^n+a^n)^r}}$} +$$\int{\frac{x^m}{(x^n+a^n)^r}}= +\int{\frac{x^{m-n}}{(x^n+a^n)^{r-1}}} +-a^n\int{\frac{x^{m-n}}{(x^n+a^n)^r}} +$$ +<<*>>= +)clear all + +--S 13 14:327 Axiom cannot compute this integral +aa:=integrate(x^m/(x^n+a^n)^r,x) +--R +--R +--R x m +--I ++ %J +--I (1) | ----------- d%J +--R ++ n n r +--I (a + %J ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.328~~~~~$\displaystyle +\int{\frac{dx}{x^m(x^n+a^n)^r}}$} +$$\int{\frac{1}{x^m(x^n+a^n)^r}}= +\frac{1}{a^n}\int{\frac{1}{x^m(x^n+a^n)^{r-1}}} +-\frac{1}{a^n}\int{\frac{1}{x^{m-n}(x^n+a^n)^r}} +$$ +<<*>>= +)clear all + +--S 14 14:328 Axiom cannot compute this integral +aa:=integrate(1/(x^m*(x^n+a^n)^r),x) +--R +--R +--R x +--R ++ 1 +--I (1) | -------------- d%J +--R ++ m n n r +--I %J (a + %J ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.329~~~~~$\displaystyle +\int{\frac{dx}{x\sqrt{x^n+a^n}}}$} +$$\int{\frac{1}{x\sqrt{x^n+a^n}}}= +\frac{1}{n\sqrt{a^n}}\ln\left(\frac{\sqrt{x^n+a^n}-\sqrt{a^n}} +{\sqrt{x^n+a^n}+\sqrt{a^n}}\right) +$$ +<<*>>= +)clear all + +--S 15 +aa:=integrate(1/(x*sqrt(x^n+a^n)),x) +--R +--R +--R (1) +--R +---------------+ +--+ +--R n | n log(x) n n log(x) n | n +--R - 2a \|%e + a + (%e + 2a )\|a +--R log(-------------------------------------------------) +--R n log(x) +--R %e +--R [------------------------------------------------------, +--R +--+ +--R | n +--R n\|a +--R +----+ +---------------+ +--R | n | n log(x) n +--R \|- a \|%e + a +--R 2atan(-------------------------) +--R n +--R a +--R - --------------------------------] +--R +----+ +--R | n +--R n\|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 16 +bb:=1/(n*sqrt(a^n))*log((sqrt(x^n+a^n)-sqrt(a^n))/(sqrt(x^n+a^n)+sqrt(a^n))) +--R +--R +-------+ +--+ +--R | n n | n +--R \|x + a - \|a +--R log(------------------) +--R +-------+ +--+ +--R | n n | n +--R \|x + a + \|a +--R (2) ----------------------- +--R +--+ +--R | n +--R n\|a +--R Type: Expression Integer +--E + +--S 17 +cc1:=aa.1-bb +--R +--R (3) +--R +---------------+ +--+ +--R n | n log(x) n n log(x) n | n +--R - 2a \|%e + a + (%e + 2a )\|a +--R log(-------------------------------------------------) +--R n log(x) +--R %e +--R + +--R +-------+ +--+ +--R | n n | n +--R \|x + a - \|a +--R - log(------------------) +--R +-------+ +--+ +--R | n n | n +--R \|x + a + \|a +--R / +--R +--+ +--R | n +--R n\|a +--R Type: Expression Integer +--E + +--S 18 +dd1:=expandLog cc1 +--R +--R (4) +--R +---------------+ +--+ +--R n | n log(x) n n log(x) n | n +--R log(2a \|%e + a + (- %e - 2a )\|a ) +--R + +--R +-------+ +--+ +-------+ +--+ +--R | n n | n | n n | n +--R log(\|x + a + \|a ) - log(\|x + a - \|a ) - n log(x) + log(- 1) +--R / +--R +--+ +--R | n +--R n\|a +--R Type: Expression Integer +--E + +--S 19 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (5) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 20 +ee1:=explog dd1 +--R +--R (6) +--R +-------+ +--+ +-------+ +--+ +--R n | n n n n | n | n n | n +--R log(2a \|x + a + (- x - 2a )\|a ) + log(\|x + a + \|a ) +--R + +--R +-------+ +--+ +--R | n n | n +--R - log(\|x + a - \|a ) - n log(x) + log(- 1) +--R / +--R +--+ +--R | n +--R n\|a +--R Type: Expression Integer +--E + +--S 21 +ff1:=complexNormalize ee1 +--R +--R n log(a) + 4log(- 1) +--R (7) -------------------- +--R +----------+ +--R | n log(a) +--R 2n\|%e +--R Type: Expression Integer +--E + +--S 22 14:329 Schaums and Axiom differ by a constant +gg1:=explog ff1 +--R +--R n log(a) + 4log(- 1) +--R (8) -------------------- +--R +--+ +--R | n +--R 2n\|a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.330~~~~~$\displaystyle +\int{\frac{dx}{x(x^n-a^n)}}$} +$$\int{\frac{1}{x(x^n-a^n)}}= +\frac{1}{na^n}\ln\left(\frac{x^n-a^n}{x^n}\right) +$$ +<<*>>= +)clear all + +--S 23 +aa:=integrate(1/(x*(x^n-a^n)),x) +--R +--R +--R n log(x) n +--R log(%e - a ) - n log(x) +--R (1) ------------------------------- +--R n +--R n a +--R Type: Union(Expression Integer,...) +--E + +--S 24 +bb:=1/(n*a^n)*log((x^n-a^n)/x^n) +--R +--R n n +--R x - a +--R log(-------) +--R n +--R x +--R (2) ------------ +--R n +--R n a +--R Type: Expression Integer +--E + +--S 25 +cc:=aa-bb +--R +--R n n +--R n log(x) n x - a +--R log(%e - a ) - log(-------) - n log(x) +--R n +--R x +--R (3) ---------------------------------------------- +--R n +--R n a +--R Type: Expression Integer +--E + +--S 26 +dd:=expandLog cc +--R +--R n log(x) n n n n +--R log(%e - a ) + log(x ) - log(x - a ) - n log(x) +--R (4) -------------------------------------------------------- +--R n +--R n a +--R Type: Expression Integer +--E + +--S 27 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (5) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 28 +ee:=explog dd +--R +--R n +--R log(x ) - n log(x) +--R (6) ------------------ +--R n +--R n a +--R Type: Expression Integer +--E + +--S 29 +logpow:=rule(log(a^n) == n*log(a)) +--R +--R n +--R (7) log(a ) == n log(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 30 14:330 Schaums and Axiom agree +ff:=logpow ee +--R +--R (8) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.331~~~~~$\displaystyle +\int{\frac{x^{n-1}dx}{x^n-a^n}}$} +$$\int{\frac{x^{n-1}}{x^n-a^n}}= +\frac{1}{n}\ln(x^n-a^n) +$$ +<<*>>= +)clear all + +--S 31 +aa:=integrate(x^(n-1)/(x^n-a^n),x) +--R +--R +--R n log(x) n +--R log(%e - a ) +--R (1) -------------------- +--R n +--R Type: Union(Expression Integer,...) +--E + +--S 32 +bb:=1/n*log(x^n-a^n) +--R +--R n n +--R log(x - a ) +--R (2) ------------ +--R n +--R Type: Expression Integer +--E + +--S 33 +cc:=aa-bb +--R +--R n log(x) n n n +--R log(%e - a ) - log(x - a ) +--R (3) ----------------------------------- +--R n +--R Type: Expression Integer +--E + +--S 34 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 35 14:331 Schaums and Axiom agree +dd:=explog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.332~~~~~$\displaystyle +\int{\frac{x^m~dx}{(x^n-a^n)^r}}$} +$$\int{\frac{x^m}{(x^n-a^n)^r}}= +a^n\int{\frac{x^{m-n}}{(x^n-a^n)^r}} ++\int{\frac{x^{m-n}}{(x^n-a^n)^{r-1}}} +$$ +<<*>>= +)clear all + +--S 36 14:332 Axiom cannot compute this integral +aa:=integrate(x^m/(x^n-a^n)^r,x) +--R +--R +--R x m +--I ++ %J +--I (1) | ------------- d%J +--R ++ n n r +--I (- a + %J ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.333~~~~~$\displaystyle +\int{\frac{dx}{x^m(x^n-a^n)^r}}$} +$$\int{\frac{1}{x^m(x^n-a^n)^r}}= +\frac{1}{a^n}\int{\frac{1}{x^{m-n}(x^n-a^n)^r}} +-\frac{1}{a^n}\int{\frac{1}{x^m(x^n-a^n)^{r-1}}} +$$ +<<*>>= +)clear all + +--S 37 14:333 Axiom cannot compute this integral +aa:=integrate(1/(x^m*(x^n-a^n)^r),x) +--R +--R +--R x +--R ++ 1 +--I (1) | ---------------- d%J +--R ++ m n n r +--I %J (- a + %J ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.334~~~~~$\displaystyle +\int{\frac{dx}{x\sqrt{x^n-a^n}}}$} +$$\int{\frac{1}{x\sqrt{x^n-a^n}}}= +\frac{2}{n\sqrt{a^n}}\cos^{-1}\sqrt{\frac{a^n}{x^n}} +$$ +<<*>>= +)clear all + +--S 38 +aa:=integrate(1/(x*sqrt(x^n-a^n)),x) +--R +--R +--R (1) +--R +---------------+ +----+ +--R n | n log(x) n n log(x) n | n +--R 2a \|%e - a + (%e - 2a )\|- a +--R log(-------------------------------------------------) +--R n log(x) +--R %e +--R [------------------------------------------------------, +--R +----+ +--R | n +--R n\|- a +--R +--+ +---------------+ +--R | n | n log(x) n +--R \|a \|%e - a +--R 2atan(-----------------------) +--R n +--R a +--R ------------------------------] +--R +--+ +--R | n +--R n\|a +--R Type: Union(List Expression Integer,...) +--E + +--S 39 +bb:=2/(n*sqrt(a^n))*acos(sqrt(a^n/x^n)) +--R +--R +--+ +--R | n +--R |a +--R 2acos( |-- ) +--R | n +--R \|x +--R (2) ------------ +--R +--+ +--R | n +--R n\|a +--R Type: Expression Integer +--E + +--S 40 +cc1:=aa.1-bb +--R +--R (3) +--R +---------------+ +----+ +--R +--+ n | n log(x) n n log(x) n | n +--R | n 2a \|%e - a + (%e - 2a )\|- a +--R \|a log(-------------------------------------------------) +--R n log(x) +--R %e +--R + +--R +--+ +--R +----+ | n +--R | n |a +--R - 2\|- a acos( |-- ) +--R | n +--R \|x +--R / +--R +----+ +--+ +--R | n | n +--R n\|- a \|a +--R Type: Expression Integer +--E + +--S 41 14:334 Axiom cannot simplify this expression +cc2:=aa.2-bb +--R +--R +--+ +---------------+ +--+ +--R | n | n log(x) n | n +--R \|a \|%e - a |a +--R 2atan(-----------------------) - 2acos( |-- ) +--R n | n +--R a \|x +--R (4) --------------------------------------------- +--R +--+ +--R | n +--R n\|a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.335~~~~~$\displaystyle +\int{\frac{x^{p-1}~dx}{x^{2m}+a^{2m}}}$ provided $0>= +)clear all + +--S 42 14:335 Axiom cannot compute this integral +aa:=integrate(x^(p-1)/(x^(2*m)+a^(2*m)),x) +--R +--R +--R x p - 1 +--I ++ %J +--I (1) | ---------- d%J +--R ++ 2m 2m +--I a + %J +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.336~~~~~$\displaystyle +\int{\frac{x^{p-1}dx}{x^{2m}-a^{2m}}}$ provided $0>= +)clear all + +--S 43 14:336 Axiom cannot compute this integral +aa:=integrate(x^(p-1)/(x^(2*m)-a^(2*m)),x) +--R +--R +--R x p - 1 +--I ++ %J +--I (1) | - ---------- d%J +--R ++ 2m 2m +--I a - %J +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.337~~~~~$\displaystyle +\int{\frac{x^{p-1}~dx}{x^{2m+1}+a^{2m+1}}}$ provided $0>= +)clear all + +--S 44 14:337 Axiom cannot compute this integral +aa:=integrate(x^(p-1)/(x^(2*m+1)+a^(2*m+1)),x) +--R +--R +--R x p - 1 +--I ++ %J +--I (1) | ------------------ d%J +--R ++ 2m + 1 2m + 1 +--I a + %J +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.338~~~~~$\displaystyle +\int{\frac{x^{p-1}~dx}{x^{2m+1}-a^{2m+1}}}$ provided $0>= +)clear all + +--S 45 14:338 Axiom cannot compute this integral +aa:=integrate(x^(p-1)/(x^(2*m+1)-a^(2*m+1)),x) +--R +--R +--R x p - 1 +--I ++ %J +--I (1) | - ------------------ d%J +--R ++ 2m + 1 2m + 1 +--I a - %J +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp74-75 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum16.input.pdf b/src/axiom-website/CATS/schaum16.input.pdf new file mode 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message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(sin(a*x),x) +--R +--R +--R cos(a x) +--R (1) - -------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=-cos(a*x)/a +--R +--R cos(a x) +--R (2) - -------- +--R a +--R Type: Expression Integer +--E + +--S 3 14:339 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.340~~~~~$\displaystyle +\int{x\sin{ax}~dx}$} +$$\int{x\sin{ax}}= +\frac{\sin{ax}}{a^2}-\frac{x\cos{ax}}{a} +$$ +<<*>>= +)clear all + +--S 4 +aa:=integrate(x*sin(a*x),x) +--R +--R +--R sin(a x) - a x cos(a x) +--R (1) ----------------------- +--R 2 +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 5 +bb:=sin(a*x)/a^2-(x*cos(a*x))/a +--R +--R sin(a x) - a x cos(a x) +--R (2) ----------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 6 14:340 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.341~~~~~$\displaystyle +\int{x^2\sin{ax}~dx}$} +$$\int{x^2\sin{ax}}= +\frac{2x}{a^2}\sin{ax}+\left(\frac{2}{a^3}-\frac{x^2}{a}\right)\cos{ax} +$$ +<<*>>= +)clear all + +--S 7 +aa:=integrate(x^2*sin(a*x),x) +--R +--R +--R 2 2 +--R 2a x sin(a x) + (- a x + 2)cos(a x) +--R (1) ------------------------------------ +--R 3 +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 8 +bb:=(2*x)/a^2*sin(a*x)+(2/a^3-x^2/a)*cos(a*x) +--R +--R 2 2 +--R 2a x sin(a x) + (- a x + 2)cos(a x) +--R (2) ------------------------------------ +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 9 14:341 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.342~~~~~$\displaystyle +\int{x^3\sin{ax}~dx}$} +$$\int{x^3\sin{ax}}= +\left(\frac{3x^2}{a^2}-\frac{6}{a^4}\right)\sin{ax} ++\left(\frac{6x}{a^3}-\frac{x^3}{a}\right)\cos{ax} +$$ +<<*>>= +)clear all + +--S 10 +aa:=integrate(x^3*sin(a*x),x) +--R +--R +--R 2 2 3 3 +--R (3a x - 6)sin(a x) + (- a x + 6a x)cos(a x) +--R (1) --------------------------------------------- +--R 4 +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 11 +bb:=((3*x^2)/a^2-6/a^4)*sin(a*x)+(6*x/a^3-x^3/a)*cos(a*x) +--R +--R 2 2 3 3 +--R (3a x - 6)sin(a x) + (- a x + 6a x)cos(a x) +--R (2) --------------------------------------------- +--R 4 +--R a +--R Type: Expression Integer +--E + +--S 12 14:342 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.343~~~~~$\displaystyle +\int{\frac{\sin{ax}}{x}}~dx$} +$$\int{\frac{\sin{ax}}{x}}= +ax-\frac{(ax)^3}{3\cdot 3!}+\frac{(ax)^5}{5\cdot 5!}-\cdots +$$ +<<*>>= +)clear all + +--S 13 14:343 Schaums and Axiom agree by definition +aa:=integrate(sin(x)/x,x) +--R +--R +--R (1) Si(x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.344~~~~~$\displaystyle +\int{\frac{\sin{ax}}{x^2}}~dx$} +$$\int{\frac{\sin{ax}}{x^2}}= +-\frac{\sin{ax}}{x}+a\int{\frac{\cos{ax}}{x}} +$$ +<<*>>= +)clear all + +--S 14 14:344 Axiom cannot compute this integral +aa:=integrate(sin(a*x)/x^2,x) +--R +--R +--R x +--I ++ sin(%I a) +--I (1) | --------- d%I +--R ++ 2 +--I %I +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.345~~~~~$\displaystyle +\int{\frac{dx}{\sin{ax}}}$} +$$\int{\frac{1}{\sin{ax}}}= +\frac{1}{a}\ln(\csc{ax}-\cot{ax})= +\frac{1}{a}\ln\tan\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 15 +aa:=integrate(1/sin(a*x),x) +--R +--R +--R sin(a x) +--R log(------------) +--R cos(a x) + 1 +--R (1) ----------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 16 +bb:=1/a*log(tan((a*x)/2)) +--R +--R a x +--R log(tan(---)) +--R 2 +--R (2) ------------- +--R a +--R Type: Expression Integer +--E + +--S 17 +cc:=aa-bb +--R +--R a x sin(a x) +--R - log(tan(---)) + log(------------) +--R 2 cos(a x) + 1 +--R (3) ----------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 18 14:345 Schaums and Axiom agree +dd:=complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.346~~~~~$\displaystyle +\int{\frac{x~dx}{\sin{ax}}}$} +$$\int{\frac{x}{\sin{ax}}}= +\frac{1}{a^2}\left\{ +ax+\frac{(ax)^3}{18}+\frac{7(ax)^5}{1800}+\cdots+ +\frac{2(2^{2n-1}-1)B_n(ax)^{2n+1}}{(2n+1)!}+\cdots\right\} +$$ +<<*>>= +)clear all + +--S 19 14:346 Axiom cannot compute this integral +aa:=integrate(x/sin(a*x),x) +--R +--R +--R x +--I ++ %I +--I (1) | --------- d%I +--I ++ sin(%I a) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.347~~~~~$\displaystyle +\int{\sin^2{ax}}~dx$} +$$\int{\sin^2{ax}}= +\frac{x}{2}-\frac{\sin{2ax}}{4a} +$$ +<<*>>= +)clear all + +--S 20 +aa:=integrate(sin(a*x)^2,x) +--R +--R +--R - cos(a x)sin(a x) + a x +--R (1) ------------------------ +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 21 +bb:=x/2-sin(2*a*x)/(4*a) +--R +--R - sin(2a x) + 2a x +--R (2) ------------------ +--R 4a +--R Type: Expression Integer +--E + +--S 22 +cc:=aa-bb +--R +--R sin(2a x) - 2cos(a x)sin(a x) +--R (3) ----------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 23 14:347 Schaums and Axiom agreee +dd:=complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.348~~~~~$\displaystyle +\int{x\sin^2{ax}}~dx$} +$$\int{x\sin^2{ax}}= +\frac{x^2}{4}-\frac{x\sin{2ax}}{4a}-\frac{\cos{2ax}}{8a^2} +$$ +<<*>>= +)clear all + +--S 24 +aa:=integrate(x*sin(a*x)^2,x) +--R +--R +--R 2 2 2 +--R - 2a x cos(a x)sin(a x) - cos(a x) + a x +--R (1) ------------------------------------------ +--R 2 +--R 4a +--R Type: Union(Expression Integer,...) +--E + +--S 25 +bb:=x^2/4-(x*sin(2*a*x))/(4*a)-cos(2*a*x)/(8*a^2) +--R +--R 2 2 +--R - 2a x sin(2a x) - cos(2a x) + 2a x +--R (2) ------------------------------------ +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 26 +cc:=aa-bb +--R +--R 2 +--R 2a x sin(2a x) - 4a x cos(a x)sin(a x) + cos(2a x) - 2cos(a x) +--R (3) --------------------------------------------------------------- +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 27 14:348 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 1 +--R (4) - --- +--R 2 +--R 8a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.349~~~~~$\displaystyle +\int{\sin^3{ax}}~dx$} +$$\int{\sin^3{ax}}= +-\frac{\cos{ax}}{a}+\frac{\cos^3{ax}}{3a} +$$ +<<*>>= +)clear all + +--S 28 +aa:=integrate(sin(a*x)^3,x) +--R +--R +--R 3 +--R cos(a x) - 3cos(a x) +--R (1) --------------------- +--R 3a +--R Type: Union(Expression Integer,...) +--E + +--S 29 +bb:=-cos(a*x)/a+cos(a*x)^3/(3*a) +--R +--R 3 +--R cos(a x) - 3cos(a x) +--R (2) --------------------- +--R 3a +--R Type: Expression Integer +--E + +--S 30 14:349 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.350~~~~~$\displaystyle +\int{\sin^4{ax}}~dx$} +$$\int{\sin^4{ax}}= +\frac{3x}{8}-\frac{\sin{2ax}}{4a}+\frac{\sin{4ax}}{32a} +$$ +<<*>>= +)clear all + +--S 31 +aa:=integrate(sin(a*x)^4,x) +--R +--R +--R 3 +--R (2cos(a x) - 5cos(a x))sin(a x) + 3a x +--R (1) --------------------------------------- +--R 8a +--R Type: Union(Expression Integer,...) +--E + +--S 32 +bb:=(3*x)/8-sin(2*a*x)/(4*a)+sin(4*a*x)/(32*a) +--R +--R sin(4a x) - 8sin(2a x) + 12a x +--R (2) ------------------------------ +--R 32a +--R Type: Expression Integer +--E + +--S 33 +cc:=aa-bb +--R +--R 3 +--R - sin(4a x) + 8sin(2a x) + (8cos(a x) - 20cos(a x))sin(a x) +--R (3) ------------------------------------------------------------ +--R 32a +--R Type: Expression Integer +--E + +--S 34 14:350 Schaums and Axiom agree +dd:=complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.351~~~~~$\displaystyle +\int{\frac{dx}{\sin^2{ax}}}$} +$$\int{\frac{1}{\sin^2{ax}}}= +-\frac{1}{a}\cot{ax} +$$ +<<*>>= +)clear all + +--S 35 +aa:=integrate(1/sin(a*x)^2,x) +--R +--R +--R cos(a x) +--R (1) - ---------- +--R a sin(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 36 +bb:=-1/a*cot(a*x) +--R +--R cot(a x) +--R (2) - -------- +--R a +--R Type: Expression Integer +--E + +--S 37 +cc:=aa-bb +--R +--R cot(a x)sin(a x) - cos(a x) +--R (3) --------------------------- +--R a sin(a x) +--R Type: Expression Integer +--E + +--S 38 14:351 Schaums and Axiom agree +dd:=complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.352~~~~~$\displaystyle +\int{\frac{dx}{\sin^3{ax}}}$} +$$\int{\frac{1}{\sin^3{ax}}}= +-\frac{\cos{ax}}{2a\sin^2{ax}}+\frac{1}{2a}\ln\tan\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 39 +aa:=integrate(1/sin(a*x)^3,x) +--R +--R +--R 2 sin(a x) +--R (cos(a x) - 1)log(------------) + cos(a x) +--R cos(a x) + 1 +--R (1) ------------------------------------------- +--R 2 +--R 2a cos(a x) - 2a +--R Type: Union(Expression Integer,...) +--E + +--S 40 +bb:=-cos(a*x)/(2*a*sin(a*x)^2)+1/(2*a)*log(tan((a*x)/2)) +--R +--R 2 a x +--R sin(a x) log(tan(---)) - cos(a x) +--R 2 +--R (2) --------------------------------- +--R 2 +--R 2a sin(a x) +--R Type: Expression Integer +--E + +--S 41 +cc:=aa-bb +--R +--R (3) +--R 2 2 a x +--R (- cos(a x) + 1)sin(a x) log(tan(---)) +--R 2 +--R + +--R 2 2 sin(a x) 2 3 +--R (cos(a x) - 1)sin(a x) log(------------) + cos(a x)sin(a x) + cos(a x) +--R cos(a x) + 1 +--R + +--R - cos(a x) +--R / +--R 2 2 +--R (2a cos(a x) - 2a)sin(a x) +--R Type: Expression Integer +--E + +--S 42 +dd:=expandLog cc +--R +--R (4) +--R 2 2 a x +--R (- cos(a x) + 1)sin(a x) log(tan(---)) +--R 2 +--R + +--R 2 2 +--R (cos(a x) - 1)sin(a x) log(sin(a x)) +--R + +--R 2 2 2 +--R (- cos(a x) + 1)sin(a x) log(cos(a x) + 1) + cos(a x)sin(a x) +--R + +--R 3 +--R cos(a x) - cos(a x) +--R / +--R 2 2 +--R (2a cos(a x) - 2a)sin(a x) +--R Type: Expression Integer +--E + +--S 43 14:352 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.353~~~~~$\displaystyle +\int{\sin{px}\sin{qx}}~dx$} +$$\int{\sin{px}\sin{qx}}= +\frac{\sin{(p-q)x}}{2(p-q)}-\frac{\sin{(p+q)x}}{2(p+q)} +$$ +<<*>>= +)clear all + +--S 44 +aa:=integrate(sin(p*x)*sin(q*x),x) +--R +--R +--R p cos(p x)sin(q x) - q cos(q x)sin(p x) +--R (1) --------------------------------------- +--R 2 2 +--R q - p +--R Type: Union(Expression Integer,...) +--E + +--S 45 +bb:=sin((p-q)*x)/(2*(p-q))-sin((p+q)*x)/(2*(p+q)) +--R +--R (- q + p)sin((q + p)x) + (q + p)sin((q - p)x) +--R (2) --------------------------------------------- +--R 2 2 +--R 2q - 2p +--R Type: Expression Integer +--E + +--S 46 +cc:=aa-bb +--R +--R (3) +--R (q - p)sin((q + p)x) + 2p cos(p x)sin(q x) + (- q - p)sin((q - p)x) +--R + +--R - 2q cos(q x)sin(p x) +--R / +--R 2 2 +--R 2q - 2p +--R Type: Expression Integer +--E + +--S 47 14:353 Schams and Axiom agree +dd:=complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.354~~~~~$\displaystyle +\int{\frac{dx}{1-\sin{ax}}}$} +$$\int{\frac{1}{1-\sin{ax}}}= +\frac{1}{a}\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right) +$$ +<<*>>= +)clear all + +--S 48 +aa:=integrate(1/(1-sin(a*x)),x) +--R +--R +--R - 2cos(a x) - 2 +--R (1) --------------------------- +--R a sin(a x) - a cos(a x) - a +--R Type: Union(Expression Integer,...) +--E + +--S 49 +bb:=1/a*tan(%pi/4+(a*x)/2) +--R +--R 2a x + %pi +--R tan(----------) +--R 4 +--R (2) --------------- +--R a +--R Type: Expression Integer +--E + +--S 50 +cc:=aa-bb +--R +--R 2a x + %pi +--R (- sin(a x) + cos(a x) + 1)tan(----------) - 2cos(a x) - 2 +--R 4 +--R (3) ---------------------------------------------------------- +--R a sin(a x) - a cos(a x) - a +--R Type: Expression Integer +--E + +--S 51 14:354 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 1 +--R (4) - +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.355~~~~~$\displaystyle +\int{\frac{x~dx}{1-\sin{ax}}}$} +$$\int{\frac{x}{1-\sin{ax}}}= +\frac{x}{a}\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right) ++\frac{2}{a^2}\ln~\sin\left(\frac{\pi}{4}-\frac{ax}{2}\right) +$$ +<<*>>= +)clear all + +--S 52 +aa:=integrate(x/(1-sin(ax)),x) +--R +--R +--R 2 +--R x +--R (1) - ------------ +--R 2sin(ax) - 2 +--R Type: Union(Expression Integer,...) +--E + +--S 53 +bb:=x/a*tan(%pi/4+(a*x)/2)+2/a^2*log(sin(%pi/4-(a*x)/2)) +--R +--R 2a x - %pi 2a x + %pi +--R 2log(- sin(----------)) + a x tan(----------) +--R 4 4 +--R (2) --------------------------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 54 14:355 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 2a x - %pi +--R (- 4sin(ax) + 4)log(- sin(----------)) +--R 4 +--R + +--R 2a x + %pi 2 2 +--R (- 2a x sin(ax) + 2a x)tan(----------) - a x +--R 4 +--R / +--R 2 2 +--R 2a sin(ax) - 2a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.356~~~~~$\displaystyle +\int{\frac{dx}{1+\sin{ax}}}$} +$$\int{\frac{1}{1+\sin{ax}}}= +-\frac{1}{a}\tan\left(\frac{\pi}{4}-\frac{ax}{2}\right) +$$ +<<*>>= +)clear all + +--S 55 +aa:=integrate(1/(1+sin(ax)),x) +--R +--R +--R x +--R (1) ----------- +--R sin(ax) + 1 +--R Type: Union(Expression Integer,...) +--E + +--S 56 +bb:=-1/a*tan(%pi/4-(a*x)/2) +--R +--R 2a x - %pi +--R tan(----------) +--R 4 +--R (2) --------------- +--R a +--R Type: Expression Integer +--E + +--S 57 +cc:=aa-bb +--R +--R 2a x - %pi +--R (- sin(ax) - 1)tan(----------) + a x +--R 4 +--R (3) ------------------------------------ +--R a sin(ax) + a +--R Type: Expression Integer +--E + +--S 58 +tanrule:=rule(tan(a/b) == sin(a)/cos(b)) +--R +--R a sin(a) +--R (4) tan(-) == ------ +--R b cos(b) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 59 14:356 Axiom cannot simplify this expression +dd:=tanrule cc +--R +--R (- sin(ax) - 1)sin(2a x - %pi) + a x cos(4) +--R (5) ------------------------------------------- +--R a cos(4)sin(ax) + a cos(4) +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.357~~~~~$\displaystyle +\int{\frac{x~dx}{1+\sin{ax}}}$} +$$\int{\frac{x}{1+\sin{ax}}}= +-\frac{x}{a}\tan\left(\frac{\pi}{4}-\frac{ax}{2}\right) ++\frac{2}{a^2}\ln~\sin\left(\frac{\pi}{4}+\frac{ax}{2}\right) +$$ +<<*>>= +)clear all + +--S 60 +aa:=integrate(x/(1+sin(a*x)),x) +--R +--R +--R (1) +--R sin(a x) + cos(a x) + 1 +--R (2sin(a x) + 2cos(a x) + 2)log(-----------------------) +--R cos(a x) + 1 +--R + +--R 2 +--R (- sin(a x) - cos(a x) - 1)log(------------) + a x sin(a x) +--R cos(a x) + 1 +--R + +--R - a x cos(a x) - a x +--R / +--R 2 2 2 +--R a sin(a x) + a cos(a x) + a +--R Type: Union(Expression Integer,...) +--E + +--S 61 +bb:=-x/a*tan(%pi/4-(a*x)/2)+2/a^2*log(sin(%pi/4+(a*x)/2)) +--R +--R 2a x + %pi 2a x - %pi +--R 2log(sin(----------)) + a x tan(----------) +--R 4 4 +--R (2) ------------------------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 62 14:257 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R sin(a x) + cos(a x) + 1 +--R (2sin(a x) + 2cos(a x) + 2)log(-----------------------) +--R cos(a x) + 1 +--R + +--R 2a x + %pi +--R (- 2sin(a x) - 2cos(a x) - 2)log(sin(----------)) +--R 4 +--R + +--R 2 +--R (- sin(a x) - cos(a x) - 1)log(------------) +--R cos(a x) + 1 +--R + +--R 2a x - %pi +--R (- a x sin(a x) - a x cos(a x) - a x)tan(----------) + a x sin(a x) +--R 4 +--R + +--R - a x cos(a x) - a x +--R / +--R 2 2 2 +--R a sin(a x) + a cos(a x) + a +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.358~~~~~$\displaystyle +\int{\frac{dx}{(1-\sin{ax})^2}}$} +$$\int{\frac{1}{(1-\sin{ax})^2}}= +\frac{1}{2a}\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right) ++\frac{1}{6a}\tan^3\left(\frac{\pi}{4}+\frac{ax}{2}\right) +$$ +<<*>>= +)clear all + +--S 63 +aa:=integrate(1/(1-sin(a*x))^2,x) +--R +--R 2 +--R (3cos(a x) + 3)sin(a x) + cos(a x) - 4cos(a x) - 5 +--R (1) ------------------------------------------------------------ +--R 2 +--R (3a cos(a x) + 6a)sin(a x) + 3a cos(a x) - 3a cos(a x) - 6a +--R Type: Union(Expression Integer,...) +--E + +--S 64 +bb:=1/(2*a)*tan(%pi/4+(a*x)/2)+1/(6*a)*tan(%pi/4+(a*x)/2)^3 +--R +--R 2a x + %pi 3 2a x + %pi +--R tan(----------) + 3tan(----------) +--R 4 4 +--R (2) ----------------------------------- +--R 6a +--R Type: Expression Integer +--E + +--S 65 +cc:=aa-bb +--R +--R (3) +--R 2 2a x + %pi 3 +--R ((- cos(a x) - 2)sin(a x) - cos(a x) + cos(a x) + 2)tan(----------) +--R 4 +--R + +--R 2 2a x + %pi +--R ((- 3cos(a x) - 6)sin(a x) - 3cos(a x) + 3cos(a x) + 6)tan(----------) +--R 4 +--R + +--R 2 +--R (6cos(a x) + 6)sin(a x) + 2cos(a x) - 8cos(a x) - 10 +--R / +--R 2 +--R (6a cos(a x) + 12a)sin(a x) + 6a cos(a x) - 6a cos(a x) - 12a +--R Type: Expression Integer +--E + +--S 66 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 67 +dd:=tanrule cc +--R +--R (5) +--R 2a x + %pi 3 +--R (- cos(a x) - 2)sin(----------) +--R 4 +--R + +--R 2a x + %pi 2 2a x + %pi 2 2a x + %pi +--R (- 3cos(----------) cos(a x) - 6cos(----------) )sin(----------) +--R 4 4 4 +--R + +--R 2a x + %pi 3 2a x + %pi 3 +--R 6cos(----------) cos(a x) + 6cos(----------) +--R 4 4 +--R * +--R sin(a x) +--R + +--R 2 2a x + %pi 3 +--R (- cos(a x) + cos(a x) + 2)sin(----------) +--R 4 +--R + +--R 2a x + %pi 2 2 2a x + %pi 2 +--R - 3cos(----------) cos(a x) + 3cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x + %pi 2 +--R 6cos(----------) +--R 4 +--R * +--R 2a x + %pi +--R sin(----------) +--R 4 +--R + +--R 2a x + %pi 3 2 2a x + %pi 3 2a x + %pi 3 +--R 2cos(----------) cos(a x) - 8cos(----------) cos(a x) - 10cos(----------) +--R 4 4 4 +--R / +--R 2a x + %pi 3 2a x + %pi 3 +--R (6a cos(----------) cos(a x) + 12a cos(----------) )sin(a x) +--R 4 4 +--R + +--R 2a x + %pi 3 2 2a x + %pi 3 +--R 6a cos(----------) cos(a x) - 6a cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x + %pi 3 +--R - 12a cos(----------) +--R 4 +--R Type: Expression Integer +--E + +--S 68 +sindiffrule2:=rule(sin((a-b)/4) == sin(a/4)*cos(b/4)-cos(a/4)*sin(b/4)) +--R +--R b - a a b b a +--I (6) - %K sin(-----) == - %K cos(-)sin(-) + %K cos(-)sin(-) +--R 4 4 4 4 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 69 +ee:=sindiffrule2 dd +--R +--R (7) +--R 2a x + %pi 3 +--R (- cos(a x) - 2)sin(----------) +--R 4 +--R + +--R 2a x + %pi 2 2a x + %pi 2 2a x + %pi +--R (- 3cos(----------) cos(a x) - 6cos(----------) )sin(----------) +--R 4 4 4 +--R + +--R 2a x + %pi 3 2a x + %pi 3 +--R 6cos(----------) cos(a x) + 6cos(----------) +--R 4 4 +--R * +--R sin(a x) +--R + +--R 2 2a x + %pi 3 +--R (- cos(a x) + cos(a x) + 2)sin(----------) +--R 4 +--R + +--R 2a x + %pi 2 2 2a x + %pi 2 +--R - 3cos(----------) cos(a x) + 3cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x + %pi 2 +--R 6cos(----------) +--R 4 +--R * +--R 2a x + %pi +--R sin(----------) +--R 4 +--R + +--R 2a x + %pi 3 2 2a x + %pi 3 2a x + %pi 3 +--R 2cos(----------) cos(a x) - 8cos(----------) cos(a x) - 10cos(----------) +--R 4 4 4 +--R / +--R 2a x + %pi 3 2a x + %pi 3 +--R (6a cos(----------) cos(a x) + 12a cos(----------) )sin(a x) +--R 4 4 +--R + +--R 2a x + %pi 3 2 2a x + %pi 3 +--R 6a cos(----------) cos(a x) - 6a cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x + %pi 3 +--R - 12a cos(----------) +--R 4 +--R Type: Expression Integer +--E + +--S 70 +sincuberule:=rule(sin(a)^3 == 3/4*sin(a)-1/4*sin(3*a)) +--R +--R 3 - sin(3a) + 3sin(a) +--R (8) sin(a) == ------------------- +--R 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 71 +ff:=sincuberule ee +--R +--R (9) +--R 2 6a x + 3%pi +--R ((cos(a x) + 2)sin(a x) + cos(a x) - cos(a x) - 2)sin(-----------) +--R 4 +--R + +--R 2a x + %pi 2 2a x + %pi 2 +--R ((- 12cos(----------) - 3)cos(a x) - 24cos(----------) - 6) +--R 4 4 +--R * +--R 2a x + %pi +--R sin(----------) +--R 4 +--R + +--R 2a x + %pi 3 2a x + %pi 3 +--R 24cos(----------) cos(a x) + 24cos(----------) +--R 4 4 +--R * +--R sin(a x) +--R + +--R 2a x + %pi 2 2 +--R (- 12cos(----------) - 3)cos(a x) +--R 4 +--R + +--R 2a x + %pi 2 2a x + %pi 2 +--R (12cos(----------) + 3)cos(a x) + 24cos(----------) + 6 +--R 4 4 +--R * +--R 2a x + %pi +--R sin(----------) +--R 4 +--R + +--R 2a x + %pi 3 2 2a x + %pi 3 +--R 8cos(----------) cos(a x) - 32cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x + %pi 3 +--R - 40cos(----------) +--R 4 +--R / +--R 2a x + %pi 3 2a x + %pi 3 +--R (24a cos(----------) cos(a x) + 48a cos(----------) )sin(a x) +--R 4 4 +--R + +--R 2a x + %pi 3 2 2a x + %pi 3 +--R 24a cos(----------) cos(a x) - 24a cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x + %pi 3 +--R - 48a cos(----------) +--R 4 +--R Type: Expression Integer +--E + +--S 72 14:358 Schaums and Axiom differ by a constant +complexNormalize % +--R +--R 2 +--R (10) -- +--R 3a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.359~~~~~$\displaystyle +\int{\frac{dx}{(1+\sin{ax})^2}}$} +$$\int{\frac{1}{(1+\sin{ax})^2}}= +-\frac{1}{2a}\tan\left(\frac{\pi}{4}-\frac{ax}{2}\right) +-\frac{1}{6a}\tan^3\left(\frac{\pi}{4}-\frac{ax}{2}\right) +$$ +<<*>>= +)clear all + +--S 73 +aa:=integrate(1/(1+sin(a*x))^2,x) +--R +--R 2 +--R (- 3cos(a x) - 3)sin(a x) + cos(a x) - 4cos(a x) - 5 +--R (1) ------------------------------------------------------------ +--R 2 +--R (3a cos(a x) + 6a)sin(a x) - 3a cos(a x) + 3a cos(a x) + 6a +--R Type: Union(Expression Integer,...) +--E + +--S 74 +bb:=-1/(2*a)*tan(%pi/4-(a*x)/2)-1/(6*a)*tan(%pi/4-(a*x)/2)^3 +--R +--R 2a x - %pi 3 2a x - %pi +--R tan(----------) + 3tan(----------) +--R 4 4 +--R (2) ----------------------------------- +--R 6a +--R Type: Expression Integer +--E + +--S 75 +cc:=aa-bb +--R +--R (3) +--R 2 2a x - %pi 3 +--R ((- cos(a x) - 2)sin(a x) + cos(a x) - cos(a x) - 2)tan(----------) +--R 4 +--R + +--R 2 2a x - %pi +--R ((- 3cos(a x) - 6)sin(a x) + 3cos(a x) - 3cos(a x) - 6)tan(----------) +--R 4 +--R + +--R 2 +--R (- 6cos(a x) - 6)sin(a x) + 2cos(a x) - 8cos(a x) - 10 +--R / +--R 2 +--R (6a cos(a x) + 12a)sin(a x) - 6a cos(a x) + 6a cos(a x) + 12a +--R Type: Expression Integer +--E + +--S 76 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 77 +dd:=tanrule cc +--R +--R (5) +--R 2a x - %pi 3 +--R (- cos(a x) - 2)sin(----------) +--R 4 +--R + +--R 2a x - %pi 2 2a x - %pi 2 2a x - %pi +--R (- 3cos(----------) cos(a x) - 6cos(----------) )sin(----------) +--R 4 4 4 +--R + +--R 2a x - %pi 3 2a x - %pi 3 +--R - 6cos(----------) cos(a x) - 6cos(----------) +--R 4 4 +--R * +--R sin(a x) +--R + +--R 2 2a x - %pi 3 +--R (cos(a x) - cos(a x) - 2)sin(----------) +--R 4 +--R + +--R 2a x - %pi 2 2 2a x - %pi 2 +--R 3cos(----------) cos(a x) - 3cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x - %pi 2 +--R - 6cos(----------) +--R 4 +--R * +--R 2a x - %pi +--R sin(----------) +--R 4 +--R + +--R 2a x - %pi 3 2 2a x - %pi 3 2a x - %pi 3 +--R 2cos(----------) cos(a x) - 8cos(----------) cos(a x) - 10cos(----------) +--R 4 4 4 +--R / +--R 2a x - %pi 3 2a x - %pi 3 +--R (6a cos(----------) cos(a x) + 12a cos(----------) )sin(a x) +--R 4 4 +--R + +--R 2a x - %pi 3 2 2a x - %pi 3 +--R - 6a cos(----------) cos(a x) + 6a cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x - %pi 3 +--R 12a cos(----------) +--R 4 +--R Type: Expression Integer +--E + +--S 78 +sindiffrule2:=rule(sin((a-b)/4) == sin(a/4)*cos(b/4)-cos(a/4)*sin(b/4)) +--R +--R +--I b - a a b b a +--I (6) - %U sin(-----) == - %U cos(-)sin(-) + %U cos(-)sin(-) +--I 4 4 4 4 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 79 +ee:=sindiffrule2 dd +--R +--R (7) +--R +-+ 2a x - %pi 2 +-+ 2a x - %pi 2 a x +--R (- 3\|2 cos(----------) cos(a x) - 6\|2 cos(----------) )sin(---) +--R 4 4 2 +--R + +--R 2a x - %pi 3 +--R (- 2cos(a x) - 4)sin(----------) +--R 4 +--R + +--R +-+ 2a x - %pi 2 a x 2a x - %pi 3 +--R (3\|2 cos(----------) cos(---) - 12cos(----------) )cos(a x) +--R 4 2 4 +--R + +--R +-+ 2a x - %pi 2 a x 2a x - %pi 3 +--R 6\|2 cos(----------) cos(---) - 12cos(----------) +--R 4 2 4 +--R * +--R sin(a x) +--R + +--R +-+ 2a x - %pi 2 +-+ 2a x - %pi 2 a x +--R (- 3\|2 cos(----------) cos(a x) - 6\|2 cos(----------) )sin(---) +--R 4 4 2 +--R + +--R 2 2a x - %pi 3 +--R (2cos(a x) - 2cos(a x) - 4)sin(----------) +--R 4 +--R + +--R 2a x - %pi 2 2 2a x - %pi 2a x - %pi 3 2 +--R 6cos(----------) cos(a x) sin(----------) + 4cos(----------) cos(a x) +--R 4 4 4 +--R + +--R +-+ 2a x - %pi 2 a x 2a x - %pi 3 +--R (3\|2 cos(----------) cos(---) - 16cos(----------) )cos(a x) +--R 4 2 4 +--R + +--R +-+ 2a x - %pi 2 a x 2a x - %pi 3 +--R 6\|2 cos(----------) cos(---) - 20cos(----------) +--R 4 2 4 +--R / +--R 2a x - %pi 3 2a x - %pi 3 +--R (12a cos(----------) cos(a x) + 24a cos(----------) )sin(a x) +--R 4 4 +--R + +--R 2a x - %pi 3 2 2a x - %pi 3 +--R - 12a cos(----------) cos(a x) + 12a cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x - %pi 3 +--R 24a cos(----------) +--R 4 +--R Type: Expression Integer +--E + +--S 80 +sincuberule:=rule(sin(a)^3 == 3/4*sin(a)-1/4*sin(3*a)) +--R +--R 3 - sin(3a) + 3sin(a) +--R (8) sin(a) == ------------------- +--R 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 81 +ff:=sincuberule ee +--R +--R (9) +--R 2 6a x - 3%pi +--R ((cos(a x) + 2)sin(a x) - cos(a x) + cos(a x) + 2)sin(-----------) +--R 4 +--R + +--R +-+ 2a x - %pi 2 +-+ 2a x - %pi 2 a x +--R (- 6\|2 cos(----------) cos(a x) - 12\|2 cos(----------) )sin(---) +--R 4 4 2 +--R + +--R 2a x - %pi +--R (- 3cos(a x) - 6)sin(----------) +--R 4 +--R + +--R +-+ 2a x - %pi 2 a x 2a x - %pi 3 +--R (6\|2 cos(----------) cos(---) - 24cos(----------) )cos(a x) +--R 4 2 4 +--R + +--R +-+ 2a x - %pi 2 a x 2a x - %pi 3 +--R 12\|2 cos(----------) cos(---) - 24cos(----------) +--R 4 2 4 +--R * +--R sin(a x) +--R + +--R +-+ 2a x - %pi 2 +-+ 2a x - %pi 2 a x +--R (- 6\|2 cos(----------) cos(a x) - 12\|2 cos(----------) )sin(---) +--R 4 4 2 +--R + +--R 2a x - %pi 2 2 2a x - %pi +--R ((12cos(----------) + 3)cos(a x) - 3cos(a x) - 6)sin(----------) +--R 4 4 +--R + +--R 2a x - %pi 3 2 +--R 8cos(----------) cos(a x) +--R 4 +--R + +--R +-+ 2a x - %pi 2 a x 2a x - %pi 3 +--R (6\|2 cos(----------) cos(---) - 32cos(----------) )cos(a x) +--R 4 2 4 +--R + +--R +-+ 2a x - %pi 2 a x 2a x - %pi 3 +--R 12\|2 cos(----------) cos(---) - 40cos(----------) +--R 4 2 4 +--R / +--R 2a x - %pi 3 2a x - %pi 3 +--R (24a cos(----------) cos(a x) + 48a cos(----------) )sin(a x) +--R 4 4 +--R + +--R 2a x - %pi 3 2 2a x - %pi 3 +--R - 24a cos(----------) cos(a x) + 24a cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x - %pi 3 +--R 48a cos(----------) +--R 4 +--R Type: Expression Integer +--E + +--S 82 14:359 Schaums and Axiom differ by a constant +complexNormalize % +--R +--R 2 +--R (10) - -- +--R 3a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.360~~~~~$\displaystyle +\int{\frac{dx}{p+q\sin{ax}}}$} +$$\int{\frac{1}{p+q\sin{ax}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{2}{a\sqrt{p^2-q^q}} +\tan^{-1}\frac{p\tan{\frac{1}{2}ax}+q}{\sqrt{p^2-q^2}}\\ +\\ +\displaystyle +\frac{1}{a\sqrt{q^2-p^2}}\ln\left(\frac{p\tan{\frac{1}{2}ax}+q-\sqrt{q^2-p^2}} +{p\tan{\frac{1}{2}ax}+q+\sqrt{q^2-p^2}}\right) +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 83 +aa:=integrate(1/(p+q*sin(a*x)),x) +--R +--R (1) +--R [ +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R / +--R +-------+ +--R | 2 2 +--R a\|q - p +--R , +--R +---------+ +--R | 2 2 +--R (p sin(a x) + q cos(a x) + q)\|- q + p +--R 2atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R - ------------------------------------------------] +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Union(List Expression Integer,...) +--E + +--S 84 +bb1:=2/(a*sqrt(p^2-q^2))*atan((p*tan(a*x/2)+q)/sqrt(p^2-q^2)) +--R +--R a x +--R p tan(---) + q +--R 2 +--R 2atan(--------------) +--R +---------+ +--R | 2 2 +--R \|- q + p +--R (2) --------------------- +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Expression Integer +--E + +--S 85 +bb2:=1/(a*sqrt(q^2-p^2))*log((p*tan((a*x)/2)+q-sqrt(q^2-p^2))/(p*tan((a*x)/2)+q+sqrt(q^2-p^2))) +--R +--R +-------+ +--R | 2 2 a x +--R - \|q - p + p tan(---) + q +--R 2 +--R log(-----------------------------) +--R +-------+ +--R | 2 2 a x +--R \|q - p + p tan(---) + q +--R 2 +--R (3) ---------------------------------- +--R +-------+ +--R | 2 2 +--R a\|q - p +--R Type: Expression Integer +--E + +--S 86 +cc1:=aa.1-bb1 +--R +--R (4) +--R +---------+ +--R | 2 2 +--R \|- q + p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R + +--R a x +--R +-------+ p tan(---) + q +--R | 2 2 2 +--R - 2\|q - p atan(--------------) +--R +---------+ +--R | 2 2 +--R \|- q + p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 87 +cc2:=aa.2-bb1 +--R +--R (5) +--R +---------+ a x +--R | 2 2 p tan(---) + q +--R (p sin(a x) + q cos(a x) + q)\|- q + p 2 +--R - 2atan(-----------------------------------------) - 2atan(--------------) +--R 2 2 2 2 +---------+ +--R (q - p )cos(a x) + q - p | 2 2 +--R \|- q + p +--R -------------------------------------------------------------------------- +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Expression Integer +--E + +--S 88 +cc3:=aa.1-bb2 +--R +--R (6) +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R + +--R +-------+ +--R | 2 2 a x +--R - \|q - p + p tan(---) + q +--R 2 +--R - log(-----------------------------) +--R +-------+ +--R | 2 2 a x +--R \|q - p + p tan(---) + q +--R 2 +--R / +--R +-------+ +--R | 2 2 +--R a\|q - p +--R Type: Expression Integer +--E + +--S 89 +cc4:=aa.2-bb2 +--R +--R (7) +--R +-------+ +--R | 2 2 a x +--R +---------+ - \|q - p + p tan(---) + q +--R | 2 2 2 +--R - \|- q + p log(-----------------------------) +--R +-------+ +--R | 2 2 a x +--R \|q - p + p tan(---) + q +--R 2 +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p +--R - 2\|q - p atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 90 +dd2:=ratDenom cc2 +--R +--R (8) +--R +---------+ +--R a x | 2 2 +--R +---------+ (p tan(---) + q)\|- q + p +--R | 2 2 2 +--R - 2\|- q + p atan(----------------------------) +--R 2 2 +--R q - p +--R + +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p +--R 2\|- q + p atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R / +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 91 +atanrule2:=rule(atan(x) == 1/2*%i*(log(1-%i*x)-log(1+%i*x))) +--R +--R 1 1 +--R (9) atan(x) == - - %i log(%i x + 1) + - %i log(- %i x + 1) +--R 2 2 +--RType: RewriteRule(Integer,Complex Fraction Integer,Expression Complex Fraction Integer) +--E + +--S 92 +ee2:=atanrule2 dd2 +--R +--R (10) +--R +---------+ +--R 1 | 2 2 2 2 +--R +---------+ (%i p tan(- a x) + %i q)\|- q + p + q - p +--R | 2 2 2 +--R %i\|- q + p log(----------------------------------------------) +--R 2 2 +--R q - p +--R + +--R - +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 +--R (%i p sin(a x) + %i q cos(a x) + %i q)\|- q + p +--R + +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R / +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R + +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 +--R (- %i p sin(a x) - %i q cos(a x) - %i q)\|- q + p +--R + +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R / +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R + +--R +---------+ +--R 1 | 2 2 2 2 +--R +---------+ (- %i p tan(- a x) - %i q)\|- q + p + q - p +--R | 2 2 2 +--R - %i\|- q + p log(------------------------------------------------) +--R 2 2 +--R q - p +--R / +--R 2 2 +--R a q - a p +--R Type: Expression Complex Fraction Integer +--E + +--S 93 +ff2:=expandLog ee2 +--R +--R (11) +--R +---------+ +---------+ +--R | 2 2 1 | 2 2 2 2 +--R - %i\|- q + p log((p tan(- a x) + q)\|- q + p + %i q - %i p ) +--R 2 +--R + +--R +---------+ +---------+ +--R | 2 2 1 | 2 2 2 2 +--R %i\|- q + p log((p tan(- a x) + q)\|- q + p - %i q + %i p ) +--R 2 +--R + +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 +--R (p sin(a x) + q cos(a x) + q)\|- q + p +--R + +--R 2 2 2 2 +--R (%i q - %i p )cos(a x) + %i q - %i p +--R + +--R - +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 +--R (p sin(a x) + q cos(a x) + q)\|- q + p +--R + +--R 2 2 2 2 +--R (- %i q + %i p )cos(a x) - %i q + %i p +--R / +--R 2 2 +--R a q - a p +--R Type: Expression Complex Fraction Integer +--E + +--S 94 +gg2:=numer(ff2)/denom(ff2) +--R +--R (12) +--R +---------+ +---------+ +--R | 2 2 1 | 2 2 2 2 +--R - %i\|- q + p log((p tan(- a x) + q)\|- q + p + %i q - %i p ) +--R 2 +--R + +--R +---------+ +---------+ +--R | 2 2 1 | 2 2 2 2 +--R %i\|- q + p log((p tan(- a x) + q)\|- q + p - %i q + %i p ) +--R 2 +--R + +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 +--R (p sin(a x) + q cos(a x) + q)\|- q + p +--R + +--R 2 2 2 2 +--R (%i q - %i p )cos(a x) + %i q - %i p +--R + +--R - +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 +--R (p sin(a x) + q cos(a x) + q)\|- q + p +--R + +--R 2 2 2 2 +--R (- %i q + %i p )cos(a x) - %i q + %i p +--R / +--R 2 2 +--R a q - a p +--RType: Fraction SparseMultivariatePolynomial(Complex Fraction Integer,Kernel Expression Complex Fraction Integer) +--E + +--S 95 +hh2:=gg2::Expression Complex Fraction Integer +--R +--R (13) +--R +---------+ +---------+ +--R | 2 2 1 | 2 2 2 2 +--R - %i\|- q + p log((p tan(- a x) + q)\|- q + p + %i q - %i p ) +--R 2 +--R + +--R +---------+ +---------+ +--R | 2 2 1 | 2 2 2 2 +--R %i\|- q + p log((p tan(- a x) + q)\|- q + p - %i q + %i p ) +--R 2 +--R + +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 +--R (p sin(a x) + q cos(a x) + q)\|- q + p +--R + +--R 2 2 2 2 +--R (%i q - %i p )cos(a x) + %i q - %i p +--R + +--R - +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 +--R (p sin(a x) + q cos(a x) + q)\|- q + p +--R + +--R 2 2 2 2 +--R (- %i q + %i p )cos(a x) - %i q + %i p +--R / +--R 2 2 +--R a q - a p +--R Type: Expression Complex Fraction Integer +--E + +--S 96 14:360 Schaums and Axiom agree +complexNormalize hh2 +--R +--R (14) 0 +--R Type: Expression Complex Fraction Integer +--E +@ + +\section{\cite{1}:14.361~~~~~$\displaystyle +\int{\frac{dx}{(p+q\sin{ax})^2}}$} +$$\int{\frac{1}{(p+q\sin{ax})^2}}= +\frac{q\cos{ax}}{a(p^2-q^2)(p+q\sin{ax})} ++\frac{p}{p^2-q^2}\int{\frac{1}{p+q\sin{ax}}} +$$ +<<*>>= +)clear all + +--S 97 +aa:=integrate(1/(p+q*sin(a*x))^2,x) +--R +--R +--R (1) +--R [ +--R 2 3 +--R (p q sin(a x) + p ) +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (p q - p )sin(a x) + (q - p q)cos(a x) + q - p q +--R / +--R q sin(a x) + p +--R + +--R +-------+ +--R 2 | 2 2 +--R (- q sin(a x) - p q cos(a x) - p q)\|q - p +--R / +--R +-------+ +--R 3 3 2 2 4 | 2 2 +--R ((a p q - a p q)sin(a x) + a p q - a p )\|q - p +--R , +--R +--R +---------+ +--R | 2 2 +--R 2 3 (p sin(a x) + q cos(a x) + q)\|- q + p +--R (2p q sin(a x) + 2p )atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R + +--R +---------+ +--R 2 | 2 2 +--R (- q sin(a x) - p q cos(a x) - p q)\|- q + p +--R / +--R +---------+ +--R 3 3 2 2 4 | 2 2 +--R ((a p q - a p q)sin(a x) + a p q - a p )\|- q + p +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 98 +t1:=integrate(1/(p+q*sin(a*x)),x) +--R +--R (2) +--R [ +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R / +--R +-------+ +--R | 2 2 +--R a\|q - p +--R , +--R +---------+ +--R | 2 2 +--R (p sin(a x) + q cos(a x) + q)\|- q + p +--R 2atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R - ------------------------------------------------] +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Union(List Expression Integer,...) +--E + +--S 99 +bb1:=(q*cos(a*x))/(a*(p^2-q^2)*(p+q*sin(a*x)))+p/(p^2-q^2)*t1.1 +--R +--R (3) +--R 2 +--R (- p q sin(a x) - p ) +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R + +--R +-------+ +--R | 2 2 +--R - q cos(a x)\|q - p +--R / +--R +-------+ +--R 3 2 2 3 | 2 2 +--R ((a q - a p q)sin(a x) + a p q - a p )\|q - p +--R Type: Expression Integer +--E + +--S 100 +bb2:=(q*cos(a*x))/(a*(p^2-q^2)*(p+q*sin(a*x)))+p/(p^2-q^2)*t1.2 +--R +--R (4) +--R +---------+ +--R | 2 2 +--R 2 (p sin(a x) + q cos(a x) + q)\|- q + p +--R (2p q sin(a x) + 2p )atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R + +--R +---------+ +--R | 2 2 +--R - q cos(a x)\|- q + p +--R / +--R +---------+ +--R 3 2 2 3 | 2 2 +--R ((a q - a p q)sin(a x) + a p q - a p )\|- q + p +--R Type: Expression Integer +--E + +--S 101 +cc1:=aa.1-bb1 +--R +--R (5) +--R 2 +--R p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (p q - p )sin(a x) + (q - p q)cos(a x) + q - p q +--R / +--R q sin(a x) + p +--R + +--R 2 +--R p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R + +--R +-------+ +--R | 2 2 +--R - q\|q - p +--R / +--R +-------+ +--R 2 3 | 2 2 +--R (a p q - a p )\|q - p +--R Type: Expression Integer +--E + +--S 102 +cc2:=aa.2-bb1 +--R +--R (6) +--R +---------+ +--R 2 | 2 2 +--R p \|- q + p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R 2 | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p +--R 2p \|q - p atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R + +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R - q\|- q + p \|q - p +--R / +--R +---------+ +-------+ +--R 2 3 | 2 2 | 2 2 +--R (a p q - a p )\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 103 +cc3:=aa.1-bb2 +--R +--R (7) +--R +---------+ +--R 2 | 2 2 +--R p \|- q + p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (p q - p )sin(a x) + (q - p q)cos(a x) + q - p q +--R / +--R q sin(a x) + p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R 2 | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p +--R - 2p \|q - p atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R + +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R - q\|- q + p \|q - p +--R / +--R +---------+ +-------+ +--R 2 3 | 2 2 | 2 2 +--R (a p q - a p )\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 104 14:361 Schaums and Axiom differ by a constant +cc4:=aa.2-bb2 +--R +--R q +--R (8) - ------------- +--R 2 3 +--R a p q - a p +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.362~~~~~$\displaystyle +\int{\frac{dx}{p^2+q^2\sin^2{ax}}}$} +$$\int{\frac{1}{p^2+q^2\sin^2{ax}}}= +\frac{1}{ap\sqrt{p^2+q^2}}\tan^{-1}\frac{\sqrt{p^2+q^2}\tan{ax}}{p} +$$ +<<*>>= +)clear all + +--S 105 +aa:=integrate(1/(p^2+q^2*sin(a*x)^2),x) +--R +--R (1) +--R +-------+ +--R | 2 2 +--R p sin(a x)\|q + p +--R atan(-------------------------------) +--R 2 2 2 2 +--R (2q + 2p )cos(a x) + 2q + 2p +--R + +--R 2 2 2 2 +--R ((2q + p )cos(a x) + 2q + 2p )sin(a x) +--R atan(-----------------------------------------) +--R +-------+ +--R 2 | 2 2 +--R (p cos(a x) + 2p cos(a x) + p)\|q + p +--R / +--R +-------+ +--R | 2 2 +--R a p\|q + p +--R Type: Union(Expression Integer,...) +--E + +--S 106 +bb:=1/(a*p*sqrt(p^2+q^2))*atan((sqrt(p^2+q^2)*tan(a*x))/p) +--R +--R +-------+ +--R | 2 2 +--R tan(a x)\|q + p +--R atan(------------------) +--R p +--R (2) ------------------------ +--R +-------+ +--R | 2 2 +--R a p\|q + p +--R Type: Expression Integer +--E + +--S 107 +cc:=aa-bb +--R +--R (3) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R tan(a x)\|q + p p sin(a x)\|q + p +--R - atan(------------------) + atan(-------------------------------) +--R p 2 2 2 2 +--R (2q + 2p )cos(a x) + 2q + 2p +--R + +--R 2 2 2 2 +--R ((2q + p )cos(a x) + 2q + 2p )sin(a x) +--R atan(-----------------------------------------) +--R +-------+ +--R 2 | 2 2 +--R (p cos(a x) + 2p cos(a x) + p)\|q + p +--R / +--R +-------+ +--R | 2 2 +--R a p\|q + p +--R Type: Expression Integer +--E + +--S 108 +dd:=ratDenom cc +--R +--R (4) +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 tan(a x)\|q + p +--R - \|q + p atan(------------------) +--R p +--R + +--R +-------+ +--R +-------+ 2 2 2 2 | 2 2 +--R | 2 2 ((2q + p )cos(a x) + 2q + 2p )sin(a x)\|q + p +--R \|q + p atan(--------------------------------------------------------) +--R 2 3 2 2 3 2 3 +--R (p q + p )cos(a x) + (2p q + 2p )cos(a x) + p q + p +--R + +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 p sin(a x)\|q + p +--R \|q + p atan(-------------------------------) +--R 2 2 2 2 +--R (2q + 2p )cos(a x) + 2q + 2p +--R / +--R 2 3 +--R a p q + a p +--R Type: Expression Integer +--E + +--S 109 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (5) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 110 +ee:=atanrule dd +--R +--R (6) +--R - +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R | 2 2 2 2 2 +--R - p sin(a x)\|q + p + (2%i q + 2%i p )cos(a x) + 2%i q +--R + +--R 2 +--R 2%i p +--R / +--R +-------+ +--R | 2 2 2 2 2 +--R p sin(a x)\|q + p + (2%i q + 2%i p )cos(a x) + 2%i q +--R + +--R 2 +--R 2%i p +--R + +--R - +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R 2 2 2 2 | 2 2 +--R ((- 2q - p )cos(a x) - 2q - 2p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 +--R (%i p q + %i p )cos(a x) + (2%i p q + 2%i p )cos(a x) +--R + +--R 2 3 +--R %i p q + %i p +--R / +--R +-------+ +--R 2 2 2 2 | 2 2 +--R ((2q + p )cos(a x) + 2q + 2p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 +--R (%i p q + %i p )cos(a x) + (2%i p q + 2%i p )cos(a x) +--R + +--R 2 3 +--R %i p q + %i p +--R + +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 - tan(a x)\|q + p + %i p +--R %i\|q + p log(---------------------------) +--R +-------+ +--R | 2 2 +--R tan(a x)\|q + p + %i p +--R / +--R 2 3 +--R 2a p q + 2a p +--R Type: Expression Complex Integer +--E + +--S 111 +ff:=expandLog ee +--R +--R (7) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - %i\|q + p log(tan(a x)\|q + p + %i p) +--R + +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R %i\|q + p log(tan(a x)\|q + p - %i p) +--R + +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R 2 2 2 2 | 2 2 +--R ((2q + p )cos(a x) + 2q + 2p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 2 +--R (%i p q + %i p )cos(a x) + (2%i p q + 2%i p )cos(a x) + %i p q +--R + +--R 3 +--R %i p +--R + +--R - +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R 2 2 2 2 | 2 2 +--R ((2q + p )cos(a x) + 2q + 2p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 +--R (- %i p q - %i p )cos(a x) + (- 2%i p q - 2%i p )cos(a x) +--R + +--R 2 3 +--R - %i p q - %i p +--R + +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R +-------+ +--R | 2 2 2 2 2 2 +--R log(p sin(a x)\|q + p + (2%i q + 2%i p )cos(a x) + 2%i q + 2%i p ) +--R + +--R - +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R | 2 2 2 2 2 +--R p sin(a x)\|q + p + (- 2%i q - 2%i p )cos(a x) - 2%i q +--R + +--R 2 +--R - 2%i p +--R + +--R +-------+ +--R | 2 2 +--R - %i log(- 1)\|q + p +--R / +--R 2 3 +--R 2a p q + 2a p +--R Type: Expression Complex Integer +--E + +--S 112 +tanrule2:RewriteRule(INT,COMPLEX(INT),EXPR(COMPLEX(INT))):=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (8) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 113 +gg:=tanrule2 ff +--R +--R (9) +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R 2 2 2 2 | 2 2 +--R ((2q + p )cos(a x) + 2q + 2p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 2 +--R (%i p q + %i p )cos(a x) + (2%i p q + 2%i p )cos(a x) + %i p q +--R + +--R 3 +--R %i p +--R + +--R - +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R 2 2 2 2 | 2 2 +--R ((2q + p )cos(a x) + 2q + 2p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 +--R (- %i p q - %i p )cos(a x) + (- 2%i p q - 2%i p )cos(a x) +--R + +--R 2 3 +--R - %i p q - %i p +--R + +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R +-------+ +--R | 2 2 2 2 2 2 +--R log(p sin(a x)\|q + p + (2%i q + 2%i p )cos(a x) + 2%i q + 2%i p ) +--R + +--R - +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R | 2 2 2 2 2 +--R p sin(a x)\|q + p + (- 2%i q - 2%i p )cos(a x) - 2%i q +--R + +--R 2 +--R - 2%i p +--R + +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 sin(a x)\|q + p + %i p cos(a x) +--R - %i\|q + p log(----------------------------------) +--R cos(a x) +--R + +--R +-------+ +--R +-------+ | 2 2 +-------+ +--R | 2 2 sin(a x)\|q + p - %i p cos(a x) | 2 2 +--R %i\|q + p log(----------------------------------) - %i log(- 1)\|q + p +--R cos(a x) +--R / +--R 2 3 +--R 2a p q + 2a p +--R Type: Expression Complex Integer +--E + +--S 114 +hh:=expandLog gg +--R +--R (10) +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R 2 2 2 2 | 2 2 +--R ((2q + p )cos(a x) + 2q + 2p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 2 +--R (%i p q + %i p )cos(a x) + (2%i p q + 2%i p )cos(a x) + %i p q +--R + +--R 3 +--R %i p +--R + +--R - +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R 2 2 2 2 | 2 2 +--R ((2q + p )cos(a x) + 2q + 2p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 +--R (- %i p q - %i p )cos(a x) + (- 2%i p q - 2%i p )cos(a x) +--R + +--R 2 3 +--R - %i p q - %i p +--R + +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R +-------+ +--R | 2 2 2 2 2 2 +--R log(p sin(a x)\|q + p + (2%i q + 2%i p )cos(a x) + 2%i q + 2%i p ) +--R + +--R - +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R | 2 2 2 2 2 +--R p sin(a x)\|q + p + (- 2%i q - 2%i p )cos(a x) - 2%i q +--R + +--R 2 +--R - 2%i p +--R + +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - %i\|q + p log(sin(a x)\|q + p + %i p cos(a x)) +--R + +--R +-------+ +-------+ +-------+ +--R | 2 2 | 2 2 | 2 2 +--R %i\|q + p log(sin(a x)\|q + p - %i p cos(a x)) - %i log(- 1)\|q + p +--R / +--R 2 3 +--R 2a p q + 2a p +--R Type: Expression Complex Integer +--E + +--S 115 14:362 Schaums and Axiom differ by a constant +ii:=complexNormalize hh +--R +--R +-------+ +--R | 2 2 +--R (%i log(%i) - %i log(- %i) - %i log(- 1))\|q + p +--R (11) --------------------------------------------------- +--R 2 3 +--R 2a p q + 2a p +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.363~~~~~$\displaystyle +\int{\frac{dx}{p^2-q^2\sin^2{ax}}}$} +$$\int{\frac{1}{p^2-q^2\sin^2{ax}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{ap\sqrt{p^2-q^2}}\tan^{-1}\frac{\sqrt{p^2-q^2}\tan{ax}}{p}\\ +\\ +\displaystyle +\frac{1}{2ap\sqrt{q^2-p^2}}\ln\left(\frac{\sqrt{q^2-p^2}\tan{ax}+p} +{\sqrt{q^2-p^2}\tan{ax}-p}\right) +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 116 +aa:=integrate(1/(p^2-q^2*sin(a*x)^2),x) +--R +--R (1) +--R [ +--R log +--R +-------+ +--R 2 2 2 2 2 | 2 2 +--R ((- q + 2p )cos(a x) + q - p )\|q - p +--R + +--R 2 3 +--R (2p q - 2p )cos(a x)sin(a x) +--R / +--R 2 2 2 2 +--R q cos(a x) - q + p +--R / +--R +-------+ +--R | 2 2 +--R 2a p\|q - p +--R , +--R +--R +---------+ +--R | 2 2 +--R p sin(a x)\|- q + p +--R - atan(-------------------------------) +--R 2 2 2 2 +--R (2q - 2p )cos(a x) + 2q - 2p +--R + +--R 2 2 2 2 +--R ((2q - p )cos(a x) + 2q - 2p )sin(a x) +--R - atan(-------------------------------------------) +--R +---------+ +--R 2 | 2 2 +--R (p cos(a x) + 2p cos(a x) + p)\|- q + p +--R / +--R +---------+ +--R | 2 2 +--R a p\|- q + p +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 117 +bb1:=1/(a*p*sqrt(p^2-q^2))*atan((sqrt(p^2-q^2)*tan(a*x))/p) +--R +--R +---------+ +--R | 2 2 +--R tan(a x)\|- q + p +--R atan(--------------------) +--R p +--R (2) -------------------------- +--R +---------+ +--R | 2 2 +--R a p\|- q + p +--R Type: Expression Integer +--E + +--S 118 +bb2:=1/(2*a*p*sqrt(q^2-p^2))*log((sqrt(q^2-p^2)*tan(a*x)+p)/(sqrt(q^2-p^2)*tan(a*x)-p)) +--R +--R +-------+ +--R | 2 2 +--R tan(a x)\|q - p + p +--R log(----------------------) +--R +-------+ +--R | 2 2 +--R tan(a x)\|q - p - p +--R (3) --------------------------- +--R +-------+ +--R | 2 2 +--R 2a p\|q - p +--R Type: Expression Integer +--E + +--S 119 +cc1:=aa.1-bb1 +--R +--R (4) +--R +---------+ +--R | 2 2 +--R \|- q + p +--R * +--R log +--R +-------+ +--R 2 2 2 2 2 | 2 2 +--R ((- q + 2p )cos(a x) + q - p )\|q - p +--R + +--R 2 3 +--R (2p q - 2p )cos(a x)sin(a x) +--R / +--R 2 2 2 2 +--R q cos(a x) - q + p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 tan(a x)\|- q + p +--R - 2\|q - p atan(--------------------) +--R p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 120 +cc2:=aa.2-bb1 +--R +--R (5) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R tan(a x)\|- q + p p sin(a x)\|- q + p +--R - atan(--------------------) - atan(-------------------------------) +--R p 2 2 2 2 +--R (2q - 2p )cos(a x) + 2q - 2p +--R + +--R 2 2 2 2 +--R ((2q - p )cos(a x) + 2q - 2p )sin(a x) +--R - atan(-------------------------------------------) +--R +---------+ +--R 2 | 2 2 +--R (p cos(a x) + 2p cos(a x) + p)\|- q + p +--R / +--R +---------+ +--R | 2 2 +--R a p\|- q + p +--R Type: Expression Integer +--E + +--S 121 +cc3:=aa.1-bb2 +--R +--R (6) +--R +-------+ +--R | 2 2 +--R tan(a x)\|q - p + p +--R - log(----------------------) +--R +-------+ +--R | 2 2 +--R tan(a x)\|q - p - p +--R + +--R log +--R +-------+ +--R 2 2 2 2 2 | 2 2 +--R ((- q + 2p )cos(a x) + q - p )\|q - p +--R + +--R 2 3 +--R (2p q - 2p )cos(a x)sin(a x) +--R / +--R 2 2 2 2 +--R q cos(a x) - q + p +--R / +--R +-------+ +--R | 2 2 +--R 2a p\|q - p +--R Type: Expression Integer +--E + +--S 122 +cc4:=aa.2-bb2 +--R +--R (7) +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 tan(a x)\|q - p + p +--R - \|- q + p log(----------------------) +--R +-------+ +--R | 2 2 +--R tan(a x)\|q - p - p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 p sin(a x)\|- q + p +--R - 2\|q - p atan(-------------------------------) +--R 2 2 2 2 +--R (2q - 2p )cos(a x) + 2q - 2p +--R + +--R +-------+ 2 2 2 2 +--R | 2 2 ((2q - p )cos(a x) + 2q - 2p )sin(a x) +--R - 2\|q - p atan(-------------------------------------------) +--R +---------+ +--R 2 | 2 2 +--R (p cos(a x) + 2p cos(a x) + p)\|- q + p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 123 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (8) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 124 +dd2:=tanrule cc2 +--R +--R (9) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R sin(a x)\|- q + p p sin(a x)\|- q + p +--R - atan(--------------------) - atan(-------------------------------) +--R p cos(a x) 2 2 2 2 +--R (2q - 2p )cos(a x) + 2q - 2p +--R + +--R 2 2 2 2 +--R ((2q - p )cos(a x) + 2q - 2p )sin(a x) +--R - atan(-------------------------------------------) +--R +---------+ +--R 2 | 2 2 +--R (p cos(a x) + 2p cos(a x) + p)\|- q + p +--R / +--R +---------+ +--R | 2 2 +--R a p\|- q + p +--R Type: Expression Integer +--E + +--S 125 +ee2:=ratDenom dd2 +--R +--R (10) +--R - +--R +---------+ +--R | 2 2 +--R \|- q + p +--R * +--R +---------+ +--R 2 2 2 2 | 2 2 +--R ((2q - p )cos(a x) + 2q - 2p )sin(a x)\|- q + p +--R atan(--------------------------------------------------------) +--R 2 3 2 2 3 2 3 +--R (p q - p )cos(a x) + (2p q - 2p )cos(a x) + p q - p +--R + +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 sin(a x)\|- q + p +--R \|- q + p atan(--------------------) +--R p cos(a x) +--R + +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 p sin(a x)\|- q + p +--R \|- q + p atan(-------------------------------) +--R 2 2 2 2 +--R (2q - 2p )cos(a x) + 2q - 2p +--R / +--R 2 3 +--R a p q - a p +--R Type: Expression Integer +--E + +--S 126 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (11) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 127 +ff2:=atanrule ee2 +--R +--R (12) +--R - +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 2 2 2 +--R - p sin(a x)\|- q + p + (2%i q - 2%i p )cos(a x) + 2%i q +--R + +--R 2 +--R - 2%i p +--R / +--R +---------+ +--R | 2 2 2 2 2 +--R p sin(a x)\|- q + p + (2%i q - 2%i p )cos(a x) + 2%i q +--R + +--R 2 +--R - 2%i p +--R + +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 - sin(a x)\|- q + p + %i p cos(a x) +--R - %i\|- q + p log(--------------------------------------) +--R +---------+ +--R | 2 2 +--R sin(a x)\|- q + p + %i p cos(a x) +--R + +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R 2 2 2 2 | 2 2 +--R ((- 2q + p )cos(a x) - 2q + 2p )sin(a x)\|- q + p +--R + +--R 2 3 2 2 3 +--R (%i p q - %i p )cos(a x) + (2%i p q - 2%i p )cos(a x) +--R + +--R 2 3 +--R %i p q - %i p +--R / +--R +---------+ +--R 2 2 2 2 | 2 2 +--R ((2q - p )cos(a x) + 2q - 2p )sin(a x)\|- q + p +--R + +--R 2 3 2 2 3 +--R (%i p q - %i p )cos(a x) + (2%i p q - 2%i p )cos(a x) +--R + +--R 2 3 +--R %i p q - %i p +--R / +--R 2 3 +--R 2a p q - 2a p +--R Type: Expression Complex Integer +--E + +--S 128 +gg2:=expandLog ff2 +--R +--R (13) +--R - +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R 2 2 2 2 | 2 2 +--R ((2q - p )cos(a x) + 2q - 2p )sin(a x)\|- q + p +--R + +--R 2 3 2 2 3 +--R (%i p q - %i p )cos(a x) + (2%i p q - 2%i p )cos(a x) +--R + +--R 2 3 +--R %i p q - %i p +--R + +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R 2 2 2 2 | 2 2 +--R ((2q - p )cos(a x) + 2q - 2p )sin(a x)\|- q + p +--R + +--R 2 3 2 2 3 +--R (- %i p q + %i p )cos(a x) + (- 2%i p q + 2%i p )cos(a x) +--R + +--R 2 3 +--R - %i p q + %i p +--R + +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R +---------+ +--R | 2 2 2 2 2 2 +--R log(p sin(a x)\|- q + p + (2%i q - 2%i p )cos(a x) + 2%i q - 2%i p ) +--R + +--R - +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 2 2 2 +--R p sin(a x)\|- q + p + (- 2%i q + 2%i p )cos(a x) - 2%i q +--R + +--R 2 +--R 2%i p +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R %i\|- q + p log(sin(a x)\|- q + p + %i p cos(a x)) +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R - %i\|- q + p log(sin(a x)\|- q + p - %i p cos(a x)) +--R + +--R +---------+ +--R | 2 2 +--R - %i log(- 1)\|- q + p +--R / +--R 2 3 +--R 2a p q - 2a p +--R Type: Expression Complex Integer +--E + +--S 129 +rootrule4a:RewriteRule(INT,COMPLEX(INT),EXPR(COMPLEX(INT))):=rule(sqrt(p^2-q^2)==sqrt(p-q)*sqrt(q+p)) +--R +--R +---------+ +--R | 2 2 +-------+ +-----+ +--R (14) \|- q + p == \|- q + p \|q + p +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 130 +hh2:=rootrule4a gg2 +--R +--R (15) +--R - +--R +-------+ +-----+ +--R %i\|- q + p \|q + p +--R * +--R log +--R 2 2 2 2 +-------+ +-----+ +--R ((2q - p )cos(a x) + 2q - 2p )sin(a x)\|- q + p \|q + p +--R + +--R 2 3 2 2 3 +--R (%i p q - %i p )cos(a x) + (2%i p q - 2%i p )cos(a x) +--R + +--R 2 3 +--R %i p q - %i p +--R + +--R +-------+ +-----+ +--R %i\|- q + p \|q + p +--R * +--R log +--R 2 2 2 2 +-------+ +-----+ +--R ((2q - p )cos(a x) + 2q - 2p )sin(a x)\|- q + p \|q + p +--R + +--R 2 3 2 2 3 +--R (- %i p q + %i p )cos(a x) + (- 2%i p q + 2%i p )cos(a x) +--R + +--R 2 3 +--R - %i p q + %i p +--R + +--R +-------+ +-----+ +--R %i\|- q + p \|q + p +--R * +--R log +--R +-------+ +-----+ 2 2 2 +--R p sin(a x)\|- q + p \|q + p + (2%i q - 2%i p )cos(a x) + 2%i q +--R + +--R 2 +--R - 2%i p +--R + +--R - +--R +-------+ +-----+ +--R %i\|- q + p \|q + p +--R * +--R log +--R +-------+ +-----+ 2 2 +--R p sin(a x)\|- q + p \|q + p + (- 2%i q + 2%i p )cos(a x) +--R + +--R 2 2 +--R - 2%i q + 2%i p +--R + +--R +-------+ +-----+ +-------+ +-----+ +--R %i\|- q + p \|q + p log(sin(a x)\|- q + p \|q + p + %i p cos(a x)) +--R + +--R +-------+ +-----+ +-------+ +-----+ +--R - %i\|- q + p \|q + p log(sin(a x)\|- q + p \|q + p - %i p cos(a x)) +--R + +--R +-------+ +-----+ +--R - %i log(- 1)\|- q + p \|q + p +--R / +--R 2 3 +--R 2a p q - 2a p +--R Type: Expression Complex Integer +--E + +--S 131 14:363 Schaums and Axiom differ by a constant +ii2:=complexNormalize hh2 +--R +--R +-------+ +-----+ +--R (%i log(%i) - %i log(- %i) - %i log(- 1))\|- q + p \|q + p +--R (16) ----------------------------------------------------------- +--R 2 3 +--R 2a p q - 2a p +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.364~~~~~$\displaystyle +\int{x^m\sin{ax}}~dx$} +$$\int{x^m\sin{ax}}= +-\frac{x^m\cos{ax}}{a}+\frac{mx^{m-1}\sin{ax}}{a^2} +-\frac{m(m-1)}{a^2}\int{x^{m-2}\sin{ax}} +$$ +<<*>>= +)clear all + +--S 132 14:364 Axiom cannot compute this integral +aa:=integrate(x^m*sin(a*x),x) +--R +--R +--R x +--R ++ m +--I (1) | sin(%I a)%I d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.365~~~~~$\displaystyle +\int{\frac{\sin{ax}}{x^n}}~dx$} +$$\int{\frac{\sin{ax}}{x^n}}= +-\frac{\sin{ax}}{(n-1)x^{n-1}}+\frac{a}{n-1}\int{\frac{\cos{ax}}{x^{n-1}}} +$$ +<<*>>= +)clear all + +--S 133 14:365 Axiom cannot compute this integral +aa:=integrate(sin(a*x)/x^n,x) +--R +--R +--R x +--I ++ sin(%I a) +--I (1) | --------- d%I +--R ++ n +--I %I +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.366~~~~~$\displaystyle +\int{\sin^n{ax}}~dx$} +$$\int{\sin^n{ax}}= +-\frac{\sin^{n-1}{ax}\cos{ax}}{an}+\frac{n-1}{n}\int{\sin^{n-2}{ax}} +$$ +<<*>>= +)clear all + +--S 134 14:366 Axiom cannot compute this integral +aa:=integrate(sin(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | sin(%I a) d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.367~~~~~$\displaystyle +\int{\frac{1}{\sin^n{ax}}}~dx$} +$$\int{\frac{1}{\sin^n{ax}}}= +\frac{-\cos{ax}}{a(n-1)\sin^{n-1}{ax}} ++\frac{n-2}{n-1}\int{\frac{1}{\sin^{n-2}{ax}}} +$$ +<<*>>= +)clear all + +--S 135 14:367 Axiom cannot compute this integral +aa:=integrate(1/(sin(a*x))^n,x) +--R +--R +--R x +--R ++ 1 +--I (1) | ---------- d%I +--R ++ n +--I sin(%I a) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.368~~~~~$\displaystyle +\int{\frac{x~dx}{sin^n{ax}}}$} +$$\int{\frac{x}{sin^n{ax}}}= +\frac{-x\cos{ax}}{a(n-1)\sin^{n-1}{ax}} +-\frac{1}{a^2(n-1)(n-2)\sin^{n-2}{ax}} ++\frac{n-2}{n-1}\int{\frac{x}{\sin^{n-2}{ax}}} +$$ +<<*>>= +)clear all + +--S 136 14:368 Axiom cannot compute this integral +aa:=integrate(x/sin(a*x)^n,x) +--R +--R +--R x +--I ++ %I +--I (1) | ---------- d%I +--R ++ n +--I sin(%I a) +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp75-76 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum17.input.pdf b/src/axiom-website/CATS/schaum17.input.pdf new file mode 100644 index 0000000..7d72c08 --- /dev/null +++ b/src/axiom-website/CATS/schaum17.input.pdf @@ -0,0 +1,4492 @@ +%PDF-1.2 +7 0 obj 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all + +--S 1 +aa:=integrate(cos(a*x),x) +--R +--R +--R sin(a x) +--R (1) -------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=sin(a*x)/a +--R +--R sin(a x) +--R (2) -------- +--R a +--R Type: Expression Integer +--E + +--S 3 14:369 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.370~~~~~$\displaystyle +\int{x\cos{ax}~dx}$} +$$\int{x\cos{ax}}= +\frac{\cos{ax}}{a^2}+\frac{x\sin{ax}}{a} +$$ +<<*>>= +)clear all + +--S 4 +aa:=integrate(x*cos(a*x),x) +--R +--R +--R a x sin(a x) + cos(a x) +--R (1) ----------------------- +--R 2 +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 5 +bb:=cos(a*x)/a^2+(x*sin(a*x))/a +--R +--R a x sin(a x) + cos(a x) +--R (2) ----------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 6 14:370 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.371~~~~~$\displaystyle +\int{x^2\cos{ax}~dx}$} +$$\int{x^2\cos{ax}}= +\frac{2x}{a^2}\cos{ax}+\left(\frac{x^2}{a}-\frac{2}{a^3}\right)\sin{ax} +$$ +<<*>>= +)clear all + +--S 7 +aa:=integrate(x^2*cos(a*x),x) +--R +--R +--R 2 2 +--R (a x - 2)sin(a x) + 2a x cos(a x) +--R (1) ---------------------------------- +--R 3 +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 8 +bb:=(2*x)/a^2*cos(a*x)+(x^2/a-2/a^3)*sin(a*x) +--R +--R 2 2 +--R (a x - 2)sin(a x) + 2a x cos(a x) +--R (2) ---------------------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 9 14:371 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.372~~~~~$\displaystyle +\int{x^3\cos{ax}~dx}$} +$$\int{x^3\cos{ax}}= +\left(\frac{3x^2}{a^2}-\frac{6}{a^4}\right)\cos{ax} ++\left(\frac{x^3}{a}-\frac{6x}{a^3}\right)\sin{ax} +$$ +<<*>>= +)clear all + +--S 10 +aa:=integrate(x^3*cos(a*x),x) +--R +--R +--R 3 3 2 2 +--R (a x - 6a x)sin(a x) + (3a x - 6)cos(a x) +--R (1) ------------------------------------------- +--R 4 +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 11 +bb:=((3*x^2)/a^2-6/a^4)*cos(a*x)+(x^3/a-(6*x)/a^3)*sin(a*x) +--R +--R 3 3 2 2 +--R (a x - 6a x)sin(a x) + (3a x - 6)cos(a x) +--R (2) ------------------------------------------- +--R 4 +--R a +--R Type: Expression Integer +--E + +--S 12 14:372 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.373~~~~~$\displaystyle +\int{\frac{\cos{ax}}{x}}~dx$} +$$\int{\frac{\cos{ax}}{x}}= +\ln{x}-\frac{(ax)^2}{2\cdot 2!}+\frac{(ax)^4}{4\cdot 4!} +-\frac{(ax)^6}{6\cdot 6!}+\cdots +$$ +<<*>>= +)clear all + +--S 13 14:373 Schaums and Axiom agree by definition +aa:=integrate(cos(x)/x,x) +--R +--R +--R (1) Ci(x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.374~~~~~$\displaystyle +\int{\frac{\cos{ax}}{x^2}}~dx$} +$$\int{\frac{\cos{ax}}{x^2}}= +-\frac{\cos{ax}}{x}-a\int{\frac{\sin{ax}}{x}} +$$ +<<*>>= +)clear all + +--S 14 14:374 Axiom cannot compute this integral +aa:=integrate(cos(a*x)/x^2,x) +--R +--R +--R x +--I ++ cos(%I a) +--I (1) | --------- d%I +--R ++ 2 +--I %I +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.375~~~~~$\displaystyle +\int{\frac{dx}{\cos{ax}}}$} +$$\int{\frac{1}{\cos{ax}}}= +\frac{1}{a}\ln(\sec{ax}-\tan{ax})= +\frac{1}{a}\ln\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right) +$$ +<<*>>= +)clear all + +--S 15 +aa:=integrate(1/cos(a*x),x) +--R +--R +--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1 +--R log(-----------------------) - log(-----------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (1) ----------------------------------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 16 +bb1:=1/a*log(sec(a*x)+tan(a*x)) +--R +--R log(tan(a x) + sec(a x)) +--R (2) ------------------------ +--R a +--R Type: Expression Integer +--E + +--S 17 +bb2:=1/a*log(tan(%pi/4+(a*x)/2)) +--R +--R 2a x + %pi +--R log(tan(----------)) +--R 4 +--R (3) -------------------- +--R a +--R Type: Expression Integer +--E + +--S 18 +cc1:=aa-bb1 +--R +--R (4) +--R sin(a x) + cos(a x) + 1 +--R - log(tan(a x) + sec(a x)) + log(-----------------------) +--R cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R a +--R Type: Expression Integer +--E + +--S 19 +cc2:=aa-bb2 +--R +--R (5) +--R 2a x + %pi sin(a x) + cos(a x) + 1 +--R - log(tan(----------)) + log(-----------------------) +--R 4 cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R a +--R Type: Expression Integer +--E + +--S 20 14:375 Schaums and Axiom differ by a constant +complexNormalize cc1 +--R +--R log(- 1) +--R (6) -------- +--R a +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.376~~~~~$\displaystyle +\int{\frac{x~dx}{\cos{ax}}}$} +$$\int{\frac{x}{\cos{ax}}}= +\frac{1}{a^2}\left\{ +\frac{(ax)^2}{2}+\frac{(ax)^4}{8}+\frac{5(ax)^6}{144}+\cdots+ +\frac{E_n(ax)^{2n+2}}{(2n+2)(2n)!}+\cdots +\right\} +$$ +<<*>>= +)clear all + +--S 21 14:376 Axiom cannot compute this integral +aa:=integrate(x/cos(a*x),x) +--R +--R +--R x +--I ++ %I +--I (1) | --------- d%I +--I ++ cos(%I a) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.377~~~~~$\displaystyle +\int{\cos^2{ax}}~dx$} +$$\int{\cos^2{ax}}= +\frac{x}{2}+\frac{\sin{2ax}}{4a} +$$ +<<*>>= +)clear all + +--S 22 +aa:=integrate(cos(a*x)^2,x) +--R +--R +--R cos(a x)sin(a x) + a x +--R (1) ---------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 23 +bb:=x/2+sin(2*a*x)/(4*a) +--R +--R sin(2a x) + 2a x +--R (2) ---------------- +--R 4a +--R Type: Expression Integer +--E + +--S 24 +cc:=aa-bb +--R +--R - sin(2a x) + 2cos(a x)sin(a x) +--R (3) ------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 25 +cossinrule:=rule(cos(b)*sin(a) == 1/2*(sin(a-b)+sin(a+b))) +--R +--R +--I %M sin(b + a) - %M sin(b - a) +--I (4) %M cos(b)sin(a) == ----------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 26 14:377 Schaums and Axiom agree +dd:=cossinrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.378~~~~~$\displaystyle +\int{x\cos^2{ax}}~dx$} +$$\int{x\cos^2{ax}}= +\frac{x^2}{4}+\frac{x\sin{2ax}}{4a}+\frac{\cos{2ax}}{8a^2} +$$ +<<*>>= +)clear all + +--S 27 +aa:=integrate(x*cos(a*x)^2,x) +--R +--R +--R 2 2 2 +--R 2a x cos(a x)sin(a x) + cos(a x) + a x +--R (1) ---------------------------------------- +--R 2 +--R 4a +--R Type: Union(Expression Integer,...) +--E + +--S 28 +bb:=x^2/4+(x*sin(2*a*x))/(4*a)+cos(2*a*x)/(8*a^2) +--R +--R 2 2 +--R 2a x sin(2a x) + cos(2a x) + 2a x +--R (2) ---------------------------------- +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 29 +cc:=aa-bb +--R +--R 2 +--R - 2a x sin(2a x) + 4a x cos(a x)sin(a x) - cos(2a x) + 2cos(a x) +--R (3) ----------------------------------------------------------------- +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 30 +cossinrule:=rule(cos(b)*sin(a) == 1/2*(sin(a-b)+sin(a+b))) +--R +--R +--I %N sin(b + a) - %N sin(b - a) +--I (4) %N cos(b)sin(a) == ----------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 31 +dd:=cossinrule cc +--R +--R 2 +--R - cos(2a x) + 2cos(a x) +--R (5) ------------------------ +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 32 +coscosrule:=rule(cos(a)*cos(b) == 1/2*(cos(a-b)+cos(a+b))) +--R +--R +--I %O cos(b + a) + %O cos(b - a) +--I (6) %O cos(a)cos(b) == ----------------------------- +--I 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 33 +ee:=coscosrule dd +--R +--R 2 +--R - cos(2a x) + 2cos(a x) +--R (7) ------------------------ +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 34 +cossqrrule1:=rule(cos(a)^2 == 1/2+1/2*cos(2*a)) +--R +--R 2 cos(2a) + 1 +--R (8) cos(a) == ----------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 35 14:378 Schaums and Axiom differ by a constant +ff:=cossqrrule1 ee +--R +--R 1 +--R (9) --- +--R 2 +--R 8a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.379~~~~~$\displaystyle +\int{\cos^3{ax}}~dx$} +$$\int{\cos^3{ax}}= +\frac{\sin{ax}}{a}-\frac{\sin^3{ax}}{3a} +$$ +<<*>>= +)clear all + +--S 36 +aa:=integrate(cos(a*x)^3,x) +--R +--R +--R 2 +--R (cos(a x) + 2)sin(a x) +--R (1) ----------------------- +--R 3a +--R Type: Union(Expression Integer,...) +--E + +--S 37 +bb:=sin(a*x)/a-sin(a*x)^3/(3*a) +--R +--R 3 +--R - sin(a x) + 3sin(a x) +--R (2) ----------------------- +--R 3a +--R Type: Expression Integer +--E + +--S 38 +cc:=aa-bb +--R +--R 3 2 +--R sin(a x) + (cos(a x) - 1)sin(a x) +--R (3) ----------------------------------- +--R 3a +--R Type: Expression Integer +--E + +--S 39 +cossqrrule:=rule(cos(a)^2 == 1-sin(a)^2) +--R +--R 2 2 +--R (4) cos(a) == - sin(a) + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 40 14:379 Schaums and Axiom agree +dd:=cossqrrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.380~~~~~$\displaystyle +\int{\cos^4{ax}}~dx$} +$$\int{\cos^4{ax}}= +\frac{3x}{8}+\frac{\sin{2ax}}{4a}+\frac{\sin{4ax}}{32a} +$$ +<<*>>= +)clear all + +--S 41 +aa:=integrate(cos(a*x)^4,x) +--R +--R +--R 3 +--R (2cos(a x) + 3cos(a x))sin(a x) + 3a x +--R (1) --------------------------------------- +--R 8a +--R Type: Union(Expression Integer,...) +--E + +--S 42 +bb:=(3*x)/8+sin(2*a*x)/(4*a)+sin(4*a*x)/(32*a) +--R +--R sin(4a x) + 8sin(2a x) + 12a x +--R (2) ------------------------------ +--R 32a +--R Type: Expression Integer +--E + +--S 43 +cc:=aa-bb +--R +--R 3 +--R - sin(4a x) - 8sin(2a x) + (8cos(a x) + 12cos(a x))sin(a x) +--R (3) ------------------------------------------------------------ +--R 32a +--R Type: Expression Integer +--E + +--S 44 14:380 Schaums and Axiom agree +complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.381~~~~~$\displaystyle +\int{\frac{dx}{\cos^2{ax}}}$} +$$\int{\frac{1}{\cos^2{ax}}}= +\frac{\tan{ax}}{a} +$$ +<<*>>= +)clear all + +--S 45 +aa:=integrate(1/cos(a*x)^2,x) +--R +--R +--R sin(a x) +--R (1) ---------- +--R a cos(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 46 +bb:=tan(a*x)/a +--R +--R tan(a x) +--R (2) -------- +--R a +--R Type: Expression Integer +--E + +--S 47 +cc:=aa-bb +--R +--R - cos(a x)tan(a x) + sin(a x) +--R (3) ----------------------------- +--R a cos(a x) +--R Type: Expression Integer +--E + +--S 48 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 49 14:381 Schaums and Axiom agree +dd:=tanrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.382~~~~~$\displaystyle +\int{\frac{dx}{\cos^3{ax}}}$} +$$\int{\frac{1}{\cos^3{ax}}}= +\frac{\sin{ax}}{2a\cos^2{ax}} ++\frac{1}{2a}\ln\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right) +$$ +<<*>>= +)clear all + +--S 50 +aa:=integrate(1/cos(a*x)^3,x) +--R +--R +--R (1) +--R 2 sin(a x) + cos(a x) + 1 +--R cos(a x) log(-----------------------) +--R cos(a x) + 1 +--R + +--R 2 sin(a x) - cos(a x) - 1 +--R - cos(a x) log(-----------------------) + sin(a x) +--R cos(a x) + 1 +--R / +--R 2 +--R 2a cos(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 51 +bb:=sin(a*x)/(2*a*cos(a*x)^2)+1/(2*a)*log(tan(%pi/4+(a*x)/2)) +--R +--R 2 2a x + %pi +--R cos(a x) log(tan(----------)) + sin(a x) +--R 4 +--R (2) ---------------------------------------- +--R 2 +--R 2a cos(a x) +--R Type: Expression Integer +--E + +--S 52 +cc:=aa-bb +--R +--R (3) +--R 2a x + %pi sin(a x) + cos(a x) + 1 +--R - log(tan(----------)) + log(-----------------------) +--R 4 cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 53 14:382 Schaums and Axiom differ by a constant +complexNormalize cc +--R +--R log(- 1) +--R (4) -------- +--R 2a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.383~~~~~$\displaystyle +\int{\cos{px}\cos{qx}}~dx$} +$$\int{\cos{ax}\cos{px}}= +\frac{\sin{(a-p)x}}{2(a-p)}+\frac{\sin{(a+p)x}}{2(a+p)} +$$ +<<*>>= +)clear all + +--S 54 +aa:=integrate(cos(a*x)*cos(p*x),x) +--R +--R p cos(a x)sin(p x) - a cos(p x)sin(a x) +--R (1) --------------------------------------- +--R 2 2 +--R p - a +--R Type: Union(Expression Integer,...) +--E + +--S 55 +bb:=(sin((a-p)*x))/(2*(a-p))+(sin((a+p)*x))/(2*(a+p)) +--R +--R (p - a)sin((p + a)x) + (p + a)sin((p - a)x) +--R (2) ------------------------------------------- +--R 2 2 +--R 2p - 2a +--R Type: Expression Integer +--E + +--S 56 +cc:=aa-bb +--R +--R (3) +--R (- p + a)sin((p + a)x) + 2p cos(a x)sin(p x) + (- p - a)sin((p - a)x) +--R + +--R - 2a cos(p x)sin(a x) +--R / +--R 2 2 +--R 2p - 2a +--R Type: Expression Integer +--E + +--S 57 14:383 Schaums and Axiom agree +complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.384~~~~~$\displaystyle +\int{\frac{dx}{1-\cos{ax}}}$} +$$\int{\frac{1}{1-\cos{ax}}}= +-\frac{1}{a}\cot\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 58 +aa:=integrate(1/(1-cos(a*x)),x) +--R +--R +--R - cos(a x) - 1 +--R (1) -------------- +--R a sin(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 59 +bb:=-1/a*cot((a*x)/2) +--R +--R a x +--R cot(---) +--R 2 +--R (2) - -------- +--R a +--R Type: Expression Integer +--E + +--S 60 +cc:=aa-bb +--R +--R a x +--R cot(---)sin(a x) - cos(a x) - 1 +--R 2 +--R (3) ------------------------------- +--R a sin(a x) +--R Type: Expression Integer +--E + +--S 61 14:384 Schaums and Axiom agree +dd:=complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.385~~~~~$\displaystyle +\int{\frac{x~dx}{1-\cos{ax}}}$} +$$\int{\frac{x}{1-\cos{ax}}}= +-\frac{x}{a}\cot\frac{ax}{2} ++\frac{2}{a^2}\ln~\sin\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 62 +aa:=integrate(x/(1-cos(a*x)),x) +--R +--R (1) +--R sin(a x) 2 +--R 2sin(a x)log(------------) - sin(a x)log(------------) - a x cos(a x) - a x +--R cos(a x) + 1 cos(a x) + 1 +--R --------------------------------------------------------------------------- +--R 2 +--R a sin(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 63 +bb:=-x/a*cot((a*x)/2)+2/a^2*log(sin((a*x)/2)) +--R +--R a x a x +--R 2log(sin(---)) - a x cot(---) +--R 2 2 +--R (2) ----------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 64 +cc:=aa-bb +--R +--R (3) +--R sin(a x) a x +--R 2sin(a x)log(------------) - 2sin(a x)log(sin(---)) +--R cos(a x) + 1 2 +--R + +--R 2 a x +--R - sin(a x)log(------------) + a x cot(---)sin(a x) - a x cos(a x) - a x +--R cos(a x) + 1 2 +--R / +--R 2 +--R a sin(a x) +--R Type: Expression Integer +--E + +--S 65 +cotrule:=rule(cot(a) == cos(a)/sin(a)) +--R +--R cos(a) +--R (4) cot(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 66 +dd:=cotrule cc +--R +--R (5) +--R a x sin(a x) a x a x +--R 2sin(---)sin(a x)log(------------) - 2sin(---)sin(a x)log(sin(---)) +--R 2 cos(a x) + 1 2 2 +--R + +--R a x 2 a x +--R - sin(---)sin(a x)log(------------) + a x cos(---)sin(a x) +--R 2 cos(a x) + 1 2 +--R + +--R a x +--R (- a x cos(a x) - a x)sin(---) +--R 2 +--R / +--R 2 a x +--R a sin(---)sin(a x) +--R 2 +--R Type: Expression Integer +--E + +--S 67 +ee:=expandLog dd +--R +--R (6) +--R a x a x a x +--R 2sin(---)sin(a x)log(sin(a x)) - 2sin(---)sin(a x)log(sin(---)) +--R 2 2 2 +--R + +--R a x +--R - sin(---)sin(a x)log(cos(a x) + 1) +--R 2 +--R + +--R a x a x a x +--R (- log(2)sin(---) + a x cos(---))sin(a x) + (- a x cos(a x) - a x)sin(---) +--R 2 2 2 +--R / +--R 2 a x +--R a sin(---)sin(a x) +--R 2 +--R Type: Expression Integer +--E + +--S 68 14:385 Schaums and Axiom agree +complexNormalize ee +--R +--R (7) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.386~~~~~$\displaystyle +\int{\frac{dx}{1+\cos{ax}}}$} +$$\int{\frac{1}{1+\cos{ax}}}= +\frac{1}{a}\tan\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 69 +aa:=integrate(1/(1+cos(a*x)),x) +--R +--R sin(a x) +--R (1) -------------- +--R a cos(a x) + a +--R Type: Union(Expression Integer,...) +--E + +--S 70 +bb:=1/a*tan((a*x)/2) +--R +--R a x +--R tan(---) +--R 2 +--R (2) -------- +--R a +--R Type: Expression Integer +--E + +--S 71 +cc:=aa-bb +--R +--R a x +--R (- cos(a x) - 1)tan(---) + sin(a x) +--R 2 +--R (3) ----------------------------------- +--R a cos(a x) + a +--R Type: Expression Integer +--E + +--S 72 14:386 Schaums and Axiom agree +complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.387~~~~~$\displaystyle +\int{\frac{x~dx}{1+\cos{ax}}}$} +$$\int{\frac{x}{1+\cos{ax}}}= +\frac{x}{a}\tan\frac{ax}{2} ++\frac{2}{a^2}\ln~\cos\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 73 +aa:=integrate(x/(1+cos(a*x)),x) +--R +--R +--R 2 +--R (- cos(a x) - 1)log(------------) + a x sin(a x) +--R cos(a x) + 1 +--R (1) ------------------------------------------------ +--R 2 2 +--R a cos(a x) + a +--R Type: Union(Expression Integer,...) +--E + +--S 74 +bb:=x/a*tan((a*x)/2)+2/a^2*log(cos((a*x)/2)) +--R +--R a x a x +--R 2log(cos(---)) + a x tan(---) +--R 2 2 +--R (2) ----------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 75 +cc:=aa-bb +--R +--R (3) +--R a x 2 +--R (- 2cos(a x) - 2)log(cos(---)) + (- cos(a x) - 1)log(------------) +--R 2 cos(a x) + 1 +--R + +--R a x +--R (- a x cos(a x) - a x)tan(---) + a x sin(a x) +--R 2 +--R / +--R 2 2 +--R a cos(a x) + a +--R Type: Expression Integer +--E + +--S 76 +dd:=expandLog cc +--R +--R (4) +--R a x +--R (cos(a x) + 1)log(cos(a x) + 1) + (- 2cos(a x) - 2)log(cos(---)) +--R 2 +--R + +--R a x +--R (- a x cos(a x) - a x)tan(---) + a x sin(a x) - log(2)cos(a x) - log(2) +--R 2 +--R / +--R 2 2 +--R a cos(a x) + a +--R Type: Expression Integer +--E + +--S 77 14:387 Schaums and Axiom agree +complexNormalize dd +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.388~~~~~$\displaystyle +\int{\frac{dx}{(1-\cos{ax})^2}}$} +$$\int{\frac{1}{(1-\cos{ax})^2}}= +-\frac{1}{2a}\cot\frac{ax}{2} +-\frac{1}{6a}\cot^3\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 78 +aa:=integrate(1/(1-cos(a*x))^2,x) +--R +--R +--R 2 +--R - cos(a x) + cos(a x) + 2 +--R (1) -------------------------- +--R (3a cos(a x) - 3a)sin(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 79 +bb:=-1/(2*a)*cot((a*x)/2)-1/(6*a)*cot((a*x)/2)^3 +--R +--R a x 3 a x +--R - cot(---) - 3cot(---) +--R 2 2 +--R (2) ----------------------- +--R 6a +--R Type: Expression Integer +--E + +--S 80 +cc:=aa-bb +--R +--R (3) +--R a x 3 a x 2 +--R ((cos(a x) - 1)cot(---) + (3cos(a x) - 3)cot(---))sin(a x) - 2cos(a x) +--R 2 2 +--R + +--R 2cos(a x) + 4 +--R / +--R (6a cos(a x) - 6a)sin(a x) +--R Type: Expression Integer +--E + +--S 81 14:388 Schaums and Axiom agree +complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.389~~~~~$\displaystyle +\int{\frac{dx}{(1+\cos{ax})^2}}$} +$$\int{\frac{1}{(1+\cos{ax})^2}}= +\frac{1}{2a}\tan\frac{ax}{2} ++\frac{1}{6a}\tan^3\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 82 +aa:=integrate(1/(1+cos(a*x))^2,x) +--R +--R +--R (cos(a x) + 2)sin(a x) +--R (1) ------------------------------- +--R 2 +--R 3a cos(a x) + 6a cos(a x) + 3a +--R Type: Union(Expression Integer,...) +--E + +--S 83 +bb:=1/(2*a)*tan((a*x)/2)+1/(6*a)*tan((a*x)/2)^3 +--R +--R a x 3 a x +--R tan(---) + 3tan(---) +--R 2 2 +--R (2) --------------------- +--R 6a +--R Type: Expression Integer +--E + +--S 84 +cc:=aa-bb +--R +--R (3) +--R 2 a x 3 +--R (- cos(a x) - 2cos(a x) - 1)tan(---) +--R 2 +--R + +--R 2 a x +--R (- 3cos(a x) - 6cos(a x) - 3)tan(---) + (2cos(a x) + 4)sin(a x) +--R 2 +--R / +--R 2 +--R 6a cos(a x) + 12a cos(a x) + 6a +--R Type: Expression Integer +--E + +--S 85 14:389 Schaums and Axiom agree +complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.390~~~~~$\displaystyle +\int{\frac{dx}{p+q\cos{ax}}}$} +$$\int{\frac{1}{p+q\cos{ax}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{2}{a\sqrt{p^2-q^q}} +\tan^{-1}\sqrt{(p-q)/(p+q)}\tan\frac{1}{2}ax +\\ +\displaystyle +\frac{1}{a\sqrt{q^2-p^2}}\ln\left( +\frac{\tan\frac{1}{2}ax+\sqrt{(q+p)/(q-p)}} +{\tan\frac{1}{2}ax-\sqrt{(q+p)(q-p)}}\right) +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 86 +aa:=integrate(1/(p+q*cos(a*x)),x) +--R +--R (1) +--R +-------+ +--R | 2 2 2 2 +--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) +--R log(--------------------------------------------------) +--R q cos(a x) + p +--R [-------------------------------------------------------, +--R +-------+ +--R | 2 2 +--R a\|q - p +--R +---------+ +--R | 2 2 +--R sin(a x)\|- q + p +--R 2atan(-----------------------) +--R (q + p)cos(a x) + q + p +--R ------------------------------] +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Union(List Expression Integer,...) +--E + +--S 87 +bb1:=2/(a*sqrt(p^2-q^2))*atan(sqrt((p-q)/(p+q))*tan(1/2*a*x)) +--R +--R +--R +-------+ +--R a x |- q + p +--R 2atan(tan(---) |------- ) +--R 2 \| q + p +--R (2) ------------------------- +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Expression Integer +--E + +--S 88 +bb2:=1/(a*sqrt(q^2-p^2))*log((tan(1/2*a*x)+sqrt((q+p)/(q-p)))/(tan(1/2*a*x)-sqrt((q+p)/(q-p)))) +--R +--R +-----+ +--R |q + p a x +--R - |----- - tan(---) +--R \|q - p 2 +--R log(---------------------) +--R +-----+ +--R |q + p a x +--R |----- - tan(---) +--R \|q - p 2 +--R (3) -------------------------- +--R +-------+ +--R | 2 2 +--R a\|q - p +--R Type: Expression Integer +--E + +--S 89 +cc1:=aa.1-bb1 +--R +--R +--R (4) +--R +-------+ +--R +---------+ | 2 2 2 2 +--R | 2 2 (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) +--R \|- q + p log(--------------------------------------------------) +--R q cos(a x) + p +--R + +--R +-------+ +-------+ +--R | 2 2 a x |- q + p +--R - 2\|q - p atan(tan(---) |------- ) +--R 2 \| q + p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 90 +cc2:=aa.2-bb1 +--R +--R +--R +---------+ +--R +-------+ | 2 2 +--R a x |- q + p sin(a x)\|- q + p +--R - 2atan(tan(---) |------- ) + 2atan(-----------------------) +--R 2 \| q + p (q + p)cos(a x) + q + p +--R (5) ------------------------------------------------------------ +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Expression Integer +--E + +--S 91 +cc3:=aa.1-bb2 +--R +--R (6) +--R +-----+ +--R |q + p a x +--R - |----- - tan(---) +--R \|q - p 2 +--R - log(---------------------) +--R +-----+ +--R |q + p a x +--R |----- - tan(---) +--R \|q - p 2 +--R + +--R +-------+ +--R | 2 2 2 2 +--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) +--R log(--------------------------------------------------) +--R q cos(a x) + p +--R / +--R +-------+ +--R | 2 2 +--R a\|q - p +--R Type: Expression Integer +--E + +--S 92 14:390 Axiom cannot simplify these expressions +cc4:=aa.2-bb2 +--R +--R (7) +--R +-----+ +--R |q + p a x +--R +---------+ - |----- - tan(---) +--R | 2 2 \|q - p 2 +--R - \|- q + p log(---------------------) +--R +-----+ +--R |q + p a x +--R |----- - tan(---) +--R \|q - p 2 +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 sin(a x)\|- q + p +--R 2\|q - p atan(-----------------------) +--R (q + p)cos(a x) + q + p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a\|- q + p \|q - p +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.391~~~~~$\displaystyle +\int{\frac{dx}{(p+q\cos{ax})^2}}$} +$$\int{\frac{1}{(p+q\cos{ax})^2}}= +\frac{q\sin{ax}}{a(q^2-p^2)(p+q\cos{ax})} +-\frac{p}{q^2-p^2}\int{\frac{1}{p+q\cos{ax}}} +$$ +<<*>>= +)clear all + +--S 93 +aa:=integrate(1/(p+q*cos(a*x))^2,x) +--R +--R +--R (1) +--R [ +--R 2 +--R (p q cos(a x) + p ) +--R * +--R +-------+ +--R | 2 2 2 2 +--R (- p cos(a x) - q)\|q - p + (q - p )sin(a x) +--R log(------------------------------------------------) +--R q cos(a x) + p +--R + +--R +-------+ +--R | 2 2 +--R q sin(a x)\|q - p +--R / +--R +-------+ +--R 3 2 2 3 | 2 2 +--R ((a q - a p q)cos(a x) + a p q - a p )\|q - p +--R , +--R +--R +---------+ +--R | 2 2 +--R 2 sin(a x)\|- q + p +--R (- 2p q cos(a x) - 2p )atan(-----------------------) +--R (q + p)cos(a x) + q + p +--R + +--R +---------+ +--R | 2 2 +--R q sin(a x)\|- q + p +--R / +--R +---------+ +--R 3 2 2 3 | 2 2 +--R ((a q - a p q)cos(a x) + a p q - a p )\|- q + p +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 94 +t1:=integrate(1/(p+q*cos(a*x)),x) +--R +--R (2) +--R +-------+ +--R | 2 2 2 2 +--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) +--R log(--------------------------------------------------) +--R q cos(a x) + p +--R [-------------------------------------------------------, +--R +-------+ +--R | 2 2 +--R a\|q - p +--R +---------+ +--R | 2 2 +--R sin(a x)\|- q + p +--R 2atan(-----------------------) +--R (q + p)cos(a x) + q + p +--R ------------------------------] +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Union(List Expression Integer,...) +--E + +--S 95 +bb1:=(q*sin(a*x))/(a*(q^2-p^2)*(p+q*cos(a*x)))-p/(q^2-p^2)*t1.1 +--R +--R (3) +--R 2 +--R (- p q cos(a x) - p ) +--R * +--R +-------+ +--R | 2 2 2 2 +--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) +--R log(--------------------------------------------------) +--R q cos(a x) + p +--R + +--R +-------+ +--R | 2 2 +--R q sin(a x)\|q - p +--R / +--R +-------+ +--R 3 2 2 3 | 2 2 +--R ((a q - a p q)cos(a x) + a p q - a p )\|q - p +--R Type: Expression Integer +--E + +--S 96 +bb2:=(q*sin(a*x))/(a*(q^2-p^2)*(p+q*cos(a*x)))-p/(q^2-p^2)*t1.2 +--R +--R (4) +--R +---------+ +--R | 2 2 +---------+ +--R 2 sin(a x)\|- q + p | 2 2 +--R (- 2p q cos(a x) - 2p )atan(-----------------------) + q sin(a x)\|- q + p +--R (q + p)cos(a x) + q + p +--R ----------------------------------------------------------------------------- +--R +---------+ +--R 3 2 2 3 | 2 2 +--R ((a q - a p q)cos(a x) + a p q - a p )\|- q + p +--R Type: Expression Integer +--E + +--S 97 +cc1:=aa.1-bb1 +--R +--R (5) +--R +-------+ +--R | 2 2 2 2 +--R (- p cos(a x) - q)\|q - p + (q - p )sin(a x) +--R p log(------------------------------------------------) +--R q cos(a x) + p +--R + +--R +-------+ +--R | 2 2 2 2 +--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) +--R p log(--------------------------------------------------) +--R q cos(a x) + p +--R / +--R +-------+ +--R 2 2 | 2 2 +--R (a q - a p )\|q - p +--R Type: Expression Integer +--E + +--S 98 +cc2:=aa.2-bb1 +--R +--R (6) +--R +-------+ +--R +---------+ | 2 2 2 2 +--R | 2 2 (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) +--R p\|- q + p log(--------------------------------------------------) +--R q cos(a x) + p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 sin(a x)\|- q + p +--R - 2p\|q - p atan(-----------------------) +--R (q + p)cos(a x) + q + p +--R / +--R +---------+ +-------+ +--R 2 2 | 2 2 | 2 2 +--R (a q - a p )\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 99 +cc3:=aa.1-bb2 +--R +--R (7) +--R +-------+ +--R +---------+ | 2 2 2 2 +--R | 2 2 (- p cos(a x) - q)\|q - p + (q - p )sin(a x) +--R p\|- q + p log(------------------------------------------------) +--R q cos(a x) + p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 sin(a x)\|- q + p +--R 2p\|q - p atan(-----------------------) +--R (q + p)cos(a x) + q + p +--R / +--R +---------+ +-------+ +--R 2 2 | 2 2 | 2 2 +--R (a q - a p )\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 100 14:391 Schaums and Axiom agree +cc4:=aa.2-bb2 +--R +--R (8) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.392~~~~~$\displaystyle +\int{\frac{dx}{p^2+q^2\cos^2{ax}}}$} +$$\int{\frac{1}{p^2+q^2\cos^2{ax}}}= +\frac{1}{ap\sqrt{p^2+q^2}}\tan^{-1}\frac{p\tan{ax}}{\sqrt{p^2+q^2}} +$$ +<<*>>= +)clear all + +--S 101 +aa:=integrate(1/(p^2+q^2*cos(a*x)^2),x) +--R +--R +--R (1) +--R +-------+ +--R | 2 2 2 2 2 +--R sin(a x)\|q + p ((q - p )cos(a x) - 2p )sin(a x) +--R atan(------------------) - atan(-----------------------------------------) +--R 2p cos(a x) + 2p +-------+ +--R 2 | 2 2 +--R (p cos(a x) + 2p cos(a x) + p)\|q + p +--R -------------------------------------------------------------------------- +--R +-------+ +--R | 2 2 +--R a p\|q + p +--R Type: Union(Expression Integer,...) +--E + +--S 102 +bb:=1/(a*p*sqrt(p^2+q^2))*atan((p*tan(a*x))/sqrt(p^2+q^2)) +--R +--R p tan(a x) +--R atan(----------) +--R +-------+ +--R | 2 2 +--R \|q + p +--R (2) ---------------- +--R +-------+ +--R | 2 2 +--R a p\|q + p +--R Type: Expression Integer +--E + +--S 103 +cc:=aa-bb +--R +--R (3) +--R +-------+ +--R | 2 2 +--R sin(a x)\|q + p p tan(a x) +--R atan(------------------) - atan(----------) +--R 2p cos(a x) + 2p +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 2 2 2 +--R ((q - p )cos(a x) - 2p )sin(a x) +--R - atan(-----------------------------------------) +--R +-------+ +--R 2 | 2 2 +--R (p cos(a x) + 2p cos(a x) + p)\|q + p +--R / +--R +-------+ +--R | 2 2 +--R a p\|q + p +--R Type: Expression Integer +--E + +--S 104 +dd:=ratDenom cc +--R +--R (4) +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 p tan(a x)\|q + p +--R - \|q + p atan(--------------------) +--R 2 2 +--R q + p +--R + +--R - +--R +-------+ +--R | 2 2 +--R \|q + p +--R * +--R +-------+ +--R 2 2 2 | 2 2 +--R ((q - p )cos(a x) - 2p )sin(a x)\|q + p +--R atan(--------------------------------------------------------) +--R 2 3 2 2 3 2 3 +--R (p q + p )cos(a x) + (2p q + 2p )cos(a x) + p q + p +--R + +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 sin(a x)\|q + p +--R \|q + p atan(------------------) +--R 2p cos(a x) + 2p +--R / +--R 2 3 +--R a p q + a p +--R Type: Expression Integer +--E + +--S 105 +atanrule2:=rule(atan(x) == 1/2*%i*(log(1-%i*x)-log(1+%i*x))) +--R +--R 1 1 +--R (5) atan(x) == - - %i log(%i x + 1) + - %i log(- %i x + 1) +--R 2 2 +--RType: RewriteRule(Integer,Complex Fraction Integer,Expression Complex Fraction Integer) +--E + +--S 106 +ee:=atanrule2 dd +--R +--R (6) +--R +-------+ +--R +-------+ | 2 2 2 2 +--R 1 | 2 2 %i p tan(a x)\|q + p + q + p +--R - %i\|q + p log(---------------------------------) +--R 2 2 2 +--R q + p +--R + +--R +-------+ +--R 1 | 2 2 +--R - %i\|q + p +--R 2 +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R ((%i q - %i p )cos(a x) - 2%i p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 2 3 +--R (p q + p )cos(a x) + (2p q + 2p )cos(a x) + p q + p +--R / +--R 2 3 2 2 3 2 3 +--R (p q + p )cos(a x) + (2p q + 2p )cos(a x) + p q + p +--R + +--R +-------+ +--R 1 | 2 2 +--R +-------+ - %i sin(a x)\|q + p + p cos(a x) + p +--R 1 | 2 2 2 +--R - - %i\|q + p log(----------------------------------------) +--R 2 p cos(a x) + p +--R + +--R +-------+ +--R 1 | 2 2 +--R +-------+ - - %i sin(a x)\|q + p + p cos(a x) + p +--R 1 | 2 2 2 +--R - %i\|q + p log(------------------------------------------) +--R 2 p cos(a x) + p +--R + +--R - +--R +-------+ +--R 1 | 2 2 +--R - %i\|q + p +--R 2 +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R ((- %i q + %i p )cos(a x) + 2%i p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 2 3 +--R (p q + p )cos(a x) + (2p q + 2p )cos(a x) + p q + p +--R / +--R 2 3 2 2 3 2 3 +--R (p q + p )cos(a x) + (2p q + 2p )cos(a x) + p q + p +--R + +--R +-------+ +--R +-------+ | 2 2 2 2 +--R 1 | 2 2 - %i p tan(a x)\|q + p + q + p +--R - - %i\|q + p log(-----------------------------------) +--R 2 2 2 +--R q + p +--R / +--R 2 3 +--R a p q + a p +--R Type: Expression Complex Fraction Integer +--E + +--S 107 +ff:=expandLog ee +--R +--R (7) +--R +-------+ +-------+ +--R 1 | 2 2 | 2 2 2 2 +--R - - %i\|q + p log(p tan(a x)\|q + p + %i q + %i p ) +--R 2 +--R + +--R +-------+ +-------+ +--R 1 | 2 2 | 2 2 2 2 +--R - %i\|q + p log(p tan(a x)\|q + p - %i q - %i p ) +--R 2 +--R + +--R - +--R +-------+ +--R 1 | 2 2 +--R - %i\|q + p +--R 2 +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R ((q - p )cos(a x) - 2p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 +--R (%i p q + %i p )cos(a x) + (2%i p q + 2%i p )cos(a x) +--R + +--R 2 3 +--R %i p q + %i p +--R + +--R +-------+ +--R 1 | 2 2 +--R - %i\|q + p +--R 2 +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R ((q - p )cos(a x) - 2p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 +--R (- %i p q - %i p )cos(a x) + (- 2%i p q - 2%i p )cos(a x) +--R + +--R 2 3 +--R - %i p q - %i p +--R + +--R +-------+ +-------+ +--R 1 | 2 2 | 2 2 +--R - %i\|q + p log(sin(a x)\|q + p + 2%i p cos(a x) + 2%i p) +--R 2 +--R + +--R +-------+ +-------+ +--R 1 | 2 2 | 2 2 +--R - - %i\|q + p log(sin(a x)\|q + p - 2%i p cos(a x) - 2%i p) +--R 2 +--R + +--R +-------+ +--R 1 1 1 1 | 2 2 +--R (%i log(%i) - - %i log(- %i) + - %i log(- - %i) - %i log(- %i))\|q + p +--R 2 2 2 2 +--R / +--R 2 3 +--R a p q + a p +--R Type: Expression Complex Fraction Integer +--E + +--S 108 14:392 Schaums and Axiom differ by a constant +complexNormalize ff +--R +--R (8) +--R 1 1 1 1 +--R %i log(%i) - - %i log(- %i) + - %i log(- - %i) - %i log(- %i) +--R 2 2 2 2 +--R + +--R 1 +--R - - %i log(- 1) +--R 2 +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R / +--R 2 3 +--R a p q + a p +--R Type: Expression Complex Fraction Integer +--E + +@ + +\section{\cite{1}:14.393~~~~~$\displaystyle +\int{\frac{dx}{p^2-q^2\cos^2{ax}}}$} +$$\int{\frac{1}{p^2-q^2\cos^2{ax}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{ap\sqrt{p^2-q^2}}\tan^{-1}\frac{p\tan{ax}}{\sqrt{p^2-q^2}}\\ +\\ +\displaystyle +\frac{1}{2ap\sqrt{q^2-p^2}}\ln\left(\frac{p\tan{ax}-\sqrt{q^2-p^2}} +{p\tan{ax}+\sqrt{q^2-p^2}}\right) +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 109 +aa:=integrate(1/(p^2-q^2*cos(a*x)^2),x) +--R +--R +--R (1) +--R +-------+ +--R 2 2 2 2 | 2 2 2 3 +--R ((q - 2p )cos(a x) + p )\|q - p + (- 2p q + 2p )cos(a x)sin(a x) +--R log(----------------------------------------------------------------------) +--R 2 2 2 +--R q cos(a x) - p +--R [---------------------------------------------------------------------------, +--R +-------+ +--R | 2 2 +--R 2a p\|q - p +--R +--R +---------+ +--R | 2 2 +--R sin(a x)\|- q + p +--R atan(--------------------) +--R 2p cos(a x) + 2p +--R + +--R 2 2 2 +--R ((q + p )cos(a x) + 2p )sin(a x) +--R atan(-------------------------------------------) +--R +---------+ +--R 2 | 2 2 +--R (p cos(a x) + 2p cos(a x) + p)\|- q + p +--R / +--R +---------+ +--R | 2 2 +--R a p\|- q + p +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 110 +bb1:=1/(a*p*sqrt(p^2-q^2))*atan((p*tan(a*x))/sqrt(p^2-q^2)) +--R +--R p tan(a x) +--R atan(------------) +--R +---------+ +--R | 2 2 +--R \|- q + p +--R (2) ------------------ +--R +---------+ +--R | 2 2 +--R a p\|- q + p +--R Type: Expression Integer +--E + +--S 111 +bb2:=1/(2*a*p*sqrt(q^2-p^2))*log((p*tan(a*x)-sqrt(q^2-p^2))/(p*tan(a*x)+sqrt(q^2-p^2))) +--R +--R +-------+ +--R | 2 2 +--R - \|q - p + p tan(a x) +--R log(-------------------------) +--R +-------+ +--R | 2 2 +--R \|q - p + p tan(a x) +--R (3) ------------------------------ +--R +-------+ +--R | 2 2 +--R 2a p\|q - p +--R Type: Expression Integer +--E + +--S 112 +cc1:=aa.1-bb1 +--R +--R (4) +--R +---------+ +--R | 2 2 +--R \|- q + p +--R * +--R log +--R +-------+ +--R 2 2 2 2 | 2 2 +--R ((q - 2p )cos(a x) + p )\|q - p +--R + +--R 2 3 +--R (- 2p q + 2p )cos(a x)sin(a x) +--R / +--R 2 2 2 +--R q cos(a x) - p +--R + +--R +-------+ +--R | 2 2 p tan(a x) +--R - 2\|q - p atan(------------) +--R +---------+ +--R | 2 2 +--R \|- q + p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 113 +cc2:=aa.2-bb1 +--R +--R (5) +--R +---------+ +--R | 2 2 +--R sin(a x)\|- q + p p tan(a x) +--R atan(--------------------) - atan(------------) +--R 2p cos(a x) + 2p +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 2 2 2 +--R ((q + p )cos(a x) + 2p )sin(a x) +--R atan(-------------------------------------------) +--R +---------+ +--R 2 | 2 2 +--R (p cos(a x) + 2p cos(a x) + p)\|- q + p +--R / +--R +---------+ +--R | 2 2 +--R a p\|- q + p +--R Type: Expression Integer +--E + +--S 114 +cc3:=aa.1-bb2 +--R +--R (6) +--R log +--R +-------+ +--R 2 2 2 2 | 2 2 2 3 +--R ((q - 2p )cos(a x) + p )\|q - p + (- 2p q + 2p )cos(a x)sin(a x) +--R ---------------------------------------------------------------------- +--R 2 2 2 +--R q cos(a x) - p +--R + +--R +-------+ +--R | 2 2 +--R - \|q - p + p tan(a x) +--R - log(-------------------------) +--R +-------+ +--R | 2 2 +--R \|q - p + p tan(a x) +--R / +--R +-------+ +--R | 2 2 +--R 2a p\|q - p +--R Type: Expression Integer +--E + +--S 115 +cc4:=aa.2-bb2 +--R +--R (7) +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 - \|q - p + p tan(a x) +--R - \|- q + p log(-------------------------) +--R +-------+ +--R | 2 2 +--R \|q - p + p tan(a x) +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 sin(a x)\|- q + p +--R 2\|q - p atan(--------------------) +--R 2p cos(a x) + 2p +--R + +--R +-------+ 2 2 2 +--R | 2 2 ((q + p )cos(a x) + 2p )sin(a x) +--R 2\|q - p atan(-------------------------------------------) +--R +---------+ +--R 2 | 2 2 +--R (p cos(a x) + 2p cos(a x) + p)\|- q + p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 116 +dd2:=ratDenom cc2 +--R +--R (8) +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 p tan(a x)\|- q + p +--R - \|- q + p atan(----------------------) +--R 2 2 +--R q - p +--R + +--R +---------+ +--R | 2 2 +--R \|- q + p +--R * +--R +---------+ +--R 2 2 2 | 2 2 +--R ((q + p )cos(a x) + 2p )sin(a x)\|- q + p +--R atan(--------------------------------------------------------) +--R 2 3 2 2 3 2 3 +--R (p q - p )cos(a x) + (2p q - 2p )cos(a x) + p q - p +--R + +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 sin(a x)\|- q + p +--R - \|- q + p atan(--------------------) +--R 2p cos(a x) + 2p +--R / +--R 2 3 +--R a p q - a p +--R Type: Expression Integer +--E + +--S 117 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (9) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 118 +ee2:=tanrule dd2 +--R +--R (10) +--R +---------+ +--R | 2 2 +--R \|- q + p +--R * +--R +---------+ +--R 2 2 2 | 2 2 +--R ((q + p )cos(a x) + 2p )sin(a x)\|- q + p +--R atan(--------------------------------------------------------) +--R 2 3 2 2 3 2 3 +--R (p q - p )cos(a x) + (2p q - 2p )cos(a x) + p q - p +--R + +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 sin(a x)\|- q + p +--R - \|- q + p atan(--------------------) +--R 2p cos(a x) + 2p +--R + +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 p sin(a x)\|- q + p +--R - \|- q + p atan(----------------------) +--R 2 2 +--R (q - p )cos(a x) +--R / +--R 2 3 +--R a p q - a p +--R Type: Expression Integer +--E + +--S 119 +atanrule2:=rule(atan(x) == 1/2*%i*(log(1-%i*x)-log(1+%i*x))) +--R +--R 1 1 +--R (11) atan(x) == - - %i log(%i x + 1) + - %i log(- %i x + 1) +--R 2 2 +--RType: RewriteRule(Integer,Complex Fraction Integer,Expression Complex Fraction Integer) +--E + +--S 120 +ff2:=atanrule2 ee2 +--R +--R (12) +--R - +--R +---------+ +--R 1 | 2 2 +--R - %i\|- q + p +--R 2 +--R * +--R log +--R +---------+ +--R 2 2 2 | 2 2 +--R ((%i q + %i p )cos(a x) + 2%i p )sin(a x)\|- q + p +--R + +--R 2 3 2 2 3 2 3 +--R (p q - p )cos(a x) + (2p q - 2p )cos(a x) + p q - p +--R / +--R 2 3 2 2 3 2 3 +--R (p q - p )cos(a x) + (2p q - 2p )cos(a x) + p q - p +--R + +--R +---------+ +--R 1 | 2 2 +--R +---------+ - %i sin(a x)\|- q + p + p cos(a x) + p +--R 1 | 2 2 2 +--R - %i\|- q + p log(------------------------------------------) +--R 2 p cos(a x) + p +--R + +--R +---------+ +--R +---------+ | 2 2 2 2 +--R 1 | 2 2 %i p sin(a x)\|- q + p + (q - p )cos(a x) +--R - %i\|- q + p log(---------------------------------------------) +--R 2 2 2 +--R (q - p )cos(a x) +--R + +--R +---------+ +--R +---------+ | 2 2 2 2 +--R 1 | 2 2 - %i p sin(a x)\|- q + p + (q - p )cos(a x) +--R - - %i\|- q + p log(-----------------------------------------------) +--R 2 2 2 +--R (q - p )cos(a x) +--R + +--R +---------+ +--R 1 | 2 2 +--R +---------+ - - %i sin(a x)\|- q + p + p cos(a x) + p +--R 1 | 2 2 2 +--R - - %i\|- q + p log(--------------------------------------------) +--R 2 p cos(a x) + p +--R + +--R +---------+ +--R 1 | 2 2 +--R - %i\|- q + p +--R 2 +--R * +--R log +--R +---------+ +--R 2 2 2 | 2 2 +--R ((- %i q - %i p )cos(a x) - 2%i p )sin(a x)\|- q + p +--R + +--R 2 3 2 2 3 2 3 +--R (p q - p )cos(a x) + (2p q - 2p )cos(a x) + p q - p +--R / +--R 2 3 2 2 3 2 3 +--R (p q - p )cos(a x) + (2p q - 2p )cos(a x) + p q - p +--R / +--R 2 3 +--R a p q - a p +--R Type: Expression Complex Fraction Integer +--E + +--S 121 +gg2:=expandLog ff2 +--R +--R (13) +--R +---------+ +--R 1 | 2 2 +--R - %i\|- q + p +--R 2 +--R * +--R log +--R +---------+ +--R 2 2 2 | 2 2 +--R ((q + p )cos(a x) + 2p )sin(a x)\|- q + p +--R + +--R 2 3 2 2 3 2 +--R (%i p q - %i p )cos(a x) + (2%i p q - 2%i p )cos(a x) + %i p q +--R + +--R 3 +--R - %i p +--R + +--R - +--R +---------+ +--R 1 | 2 2 +--R - %i\|- q + p +--R 2 +--R * +--R log +--R +---------+ +--R 2 2 2 | 2 2 +--R ((q + p )cos(a x) + 2p )sin(a x)\|- q + p +--R + +--R 2 3 2 2 3 +--R (- %i p q + %i p )cos(a x) + (- 2%i p q + 2%i p )cos(a x) +--R + +--R 2 3 +--R - %i p q + %i p +--R + +--R +---------+ +---------+ +--R 1 | 2 2 | 2 2 2 2 +--R - - %i\|- q + p log(p sin(a x)\|- q + p + (%i q - %i p )cos(a x)) +--R 2 +--R + +--R +---------+ +---------+ +--R 1 | 2 2 | 2 2 2 2 +--R - %i\|- q + p log(p sin(a x)\|- q + p + (- %i q + %i p )cos(a x)) +--R 2 +--R + +--R +---------+ +---------+ +--R 1 | 2 2 | 2 2 +--R - - %i\|- q + p log(sin(a x)\|- q + p + 2%i p cos(a x) + 2%i p) +--R 2 +--R + +--R +---------+ +---------+ +--R 1 | 2 2 | 2 2 +--R - %i\|- q + p log(sin(a x)\|- q + p - 2%i p cos(a x) - 2%i p) +--R 2 +--R + +--R +---------+ +--R 1 1 1 1 | 2 2 +--R (- %i log(- %i) - - %i log(- - %i))\|- q + p +--R 2 2 2 2 +--R / +--R 2 3 +--R a p q - a p +--R Type: Expression Complex Fraction Integer +--E + +--S 122 14:393 Schaums and Axiom differ by a constant +hh2:=complexNormalize gg2 +--R +--R (14) +--R 1 1 1 1 1 1 +--R (- - %i log(%i) + - %i log(- %i) - - %i log(- - %i) + - %i log(- %i)) +--R 2 2 2 2 2 2 +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R / +--R 2 3 +--R a p q - a p +--R Type: Expression Complex Fraction Integer +--E +@ + +\section{\cite{1}:14.394~~~~~$\displaystyle +\int{x^m\cos{ax}}~dx$} +$$\int{x^m\cos{ax}}= +\frac{x^m\sin{ax}}{a}+\frac{mx^{m-1}}{a^2}\cos{ax} +-\frac{m(m-1)}{a^2}\int{x^{m-2}\cos{ax}} +$$ +<<*>>= +)clear all + +--S 123 14:394 Axiom cannot compute this integral +aa:=integrate(x^m*cos(a*x),x) +--R +--R +--R x +--R ++ m +--I (1) | cos(%I a)%I d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.395~~~~~$\displaystyle +\int{\frac{\cos{ax}}{x^n}}~dx$} +$$\int{\frac{\cos{ax}}{x^n}}= +-\frac{\cos{ax}}{(n-1)x^{n-1}}-\frac{a}{n-1}\int{\frac{\sin{ax}}{x^{n-1}}} +$$ +<<*>>= +)clear all + +--S 124 14:395 Axiom cannot compute this integral +aa:=integrate(cos(a*x)/x^n,x) +--R +--R +--R x +--I ++ cos(%I a) +--I (1) | --------- d%I +--R ++ n +--I %I +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.396~~~~~$\displaystyle +\int{\cos^n{ax}}~dx$} +$$\int{\cos^n{ax}}= +\frac{\sin{ax}\cos^{n-1}{ax}}{an}+\frac{n-1}{n}\int{\cos^{n-2}{ax}} +$$ +<<*>>= +)clear all + +--S 125 14:396 Axiom cannot compute this integral +aa:=integrate(cos(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | cos(%I a) d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.397~~~~~$\displaystyle +\int{\frac{1}{\cos^n{ax}}}~dx$} +$$\int{\frac{1}{\cos^n{ax}}}= +\frac{\sin{ax}}{a(n-1)\cos^{n-1}{ax}} ++\frac{n-2}{n-1}\int{\frac{1}{\cos^{n-2}{ax}}} +$$ +<<*>>= +)clear all + +--S 126 14:397 Axiom cannot compute this integral +aa:=integrate(1/(cos(a*x))^n,x) +--R +--R +--R x +--R ++ 1 +--I (1) | ---------- d%I +--R ++ n +--I cos(%I a) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.398~~~~~$\displaystyle +\int{\frac{x~dx}{cos^n{ax}}}$} +$$\int{\frac{x}{cos^n{ax}}}= +\frac{x\sin{ax}}{a(n-1)\cos^{n-1}{ax}} +-\frac{1}{a^2(n-1)(n-2)\cos^{n-2}{ax}} ++\frac{n-2}{n-1}\int{\frac{x}{\cos^{n-2}{ax}}} +$$ +<<*>>= +)clear all + +--S 127 14:398 Axiom cannot compute this integral +aa:=integrate(x/cos(a*x)^n,x) +--R +--R +--R x +--I ++ %I +--I (1) | ---------- d%I +--R ++ n +--I cos(%I a) +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp77-78 +\end{thebibliography} +\end{document} diff 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+--S 2 +bb:=sin(a*x)^2/(2*a) +--R +--R 2 +--R sin(a x) +--R (2) --------- +--R 2a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R 2 2 +--R - sin(a x) - cos(a x) +--R (3) ----------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 4 +cossqrrule:=rule(cos(a)^2 == 1-sin(a)^2) +--R +--R 2 2 +--R (4) cos(a) == - sin(a) + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 5 14:399 Schaums and Axiom differ by a constant +dd:=cossqrrule cc +--R +--R 1 +--R (5) - -- +--R 2a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.400~~~~~$\displaystyle +\int{\sin{px}\cos{qx}}~dx$} +$$\int{\sin{px}\cos{qx}}= +-\frac{cos(p-q)x}{2(p-q)}-\frac{cos(p+q)x}{2(p+q)} +$$ +<<*>>= +)clear all + +--S 6 +aa:=integrate(sin(p*x)*cos(q*x),x) +--R +--R +--R q sin(p x)sin(q x) + p cos(p x)cos(q x) +--R (1) --------------------------------------- +--R 2 2 +--R q - p +--R Type: Union(Expression Integer,...) +--E + +--S 7 +bb:=-cos((p-q)*x)/(2*(p-q))-cos((p+q)*x)/(2*(p+q)) +--R +--R (- q + p)cos((q + p)x) + (q + p)cos((q - p)x) +--R (2) --------------------------------------------- +--R 2 2 +--R 2q - 2p +--R Type: Expression Integer +--E + +--S 8 +cc:=aa-bb +--R +--R (3) +--R 2q sin(p x)sin(q x) + (q - p)cos((q + p)x) + 2p cos(p x)cos(q x) +--R + +--R (- q - p)cos((q - p)x) +--R / +--R 2 2 +--R 2q - 2p +--R Type: Expression Integer +--E + +--S 9 14:400 Schaums and Axiom agree +complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.401~~~~~$\displaystyle +\int{\sin^n{ax}\cos{ax}}~dx$ provided $n \ne -1$} +$$\int{\sin^n{ax}\cos{ax}}= +\frac{\sin^{n+1}{ax}}{(n+1)a} +$$ +<<*>>= +)clear all + +--S 10 +aa:=integrate(sin(a*x)^n*cos(a*x),x) +--R +--R +--R n log(sin(a x)) +--R sin(a x)%e +--R (1) ------------------------- +--R a n + a +--R Type: Union(Expression Integer,...) +--E + +--S 11 +bb:=sin(a*x)^(n+1)/((n+1)*a) +--R +--R n + 1 +--R sin(a x) +--R (2) ------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 12 +cc:=aa-bb +--R +--R n log(sin(a x)) n + 1 +--R sin(a x)%e - sin(a x) +--R (3) ----------------------------------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 13 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 14 +dd:=explog cc +--R +--R n + 1 n +--R - sin(a x) + sin(a x)sin(a x) +--R (5) ----------------------------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 15 14:401 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.402~~~~~$\displaystyle +\int{\cos^n{ax}*sin{ax}}~dx$ provided $n \ne -1$} +$$\int{\cos^n{ax}*sin{ax}}= +-\frac{\cos^{n+1}{ax}}{(n+1)a} +$$ +<<*>>= +)clear all + +--S 16 +aa:=integrate(cos(a*x)^n*sin(a*x),x) +--R +--R +--R n log(cos(a x)) +--R cos(a x)%e +--R (1) - ------------------------- +--R a n + a +--R Type: Union(Expression Integer,...) +--E + +--S 17 +bb:=-cos(a*x)^(n+1)/((n+1)*a) +--R +--R n + 1 +--R cos(a x) +--R (2) - ------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 18 +cc:=aa-bb +--R +--R n log(cos(a x)) n + 1 +--R - cos(a x)%e + cos(a x) +--R (3) ------------------------------------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 19 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 20 +dd:=explog cc +--R +--R n + 1 n +--R cos(a x) - cos(a x)cos(a x) +--R (5) --------------------------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 21 14:402 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.403~~~~~$\displaystyle +\int{\sin^2{ax}\cos^2{ax}}$} +$$\int{\sin^2{ax}\cos^2{ax}}= +\frac{x}{8}-\frac{\sin{4ax}}{32a} +$$ +<<*>>= +)clear all + +--S 22 +aa:=integrate(sin(a*x)^2*cos(a*x)^2,x) +--R +--R +--R 3 +--R (- 2cos(a x) + cos(a x))sin(a x) + a x +--R (1) --------------------------------------- +--R 8a +--R Type: Union(Expression Integer,...) +--E + +--S 23 +bb:=x/8-sin(4*a*x)/(32*a) +--R +--R - sin(4a x) + 4a x +--R (2) ------------------ +--R 32a +--R Type: Expression Integer +--E + +--S 24 +cc:=aa-bb +--R +--R 3 +--R sin(4a x) + (- 8cos(a x) + 4cos(a x))sin(a x) +--R (3) ---------------------------------------------- +--R 32a +--R Type: Expression Integer +--E + +--S 25 14:403 Schaums and Axiom agree +dd:=complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.404~~~~~$\displaystyle +\int{\frac{dx}{\sin{ax}\cos{ax}}}$} +$$\int{\frac{1}{\sin{ax}\cos{ax}}}= +\frac{1}{a}\ln~\tan{ax} +$$ +<<*>>= +)clear all + +--S 26 +aa:=integrate(1/(sin(a*x)*cos(a*x)),x) +--R +--R +--R sin(a x) 2cos(a x) +--R log(------------) - log(- ------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (1) --------------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 27 +bb:=1/a*log(tan(a*x)) +--R +--R log(tan(a x)) +--R (2) ------------- +--R a +--R Type: Expression Integer +--E + +--S 28 +cc:=aa-bb +--R +--R sin(a x) 2cos(a x) +--R - log(tan(a x)) + log(------------) - log(- ------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (3) --------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 29 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 30 +dd:=tanrule cc +--R +--R sin(a x) sin(a x) 2cos(a x) +--R - log(--------) + log(------------) - log(- ------------) +--R cos(a x) cos(a x) + 1 cos(a x) + 1 +--R (5) --------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 31 14:404 Schaums and Axiom differ by a constant +ee:=expandLog dd +--R +--R log(- 2) +--R (6) - -------- +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.405~~~~~$\displaystyle +\int{\frac{dx}{\sin^2{ax}\cos{ax}}}$} +$$\int{\frac{1}{\sin^2{ax}\cos{ax}}}= +\frac{1}{a}\ln~\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right)-\frac{1}{a\sin{ax}} +$$ +<<*>>= +)clear all + +--S 32 +aa:=integrate(1/(sin(a*x)^2*cos(a*x)),x) +--R +--R +--R (1) +--R sin(a x) + cos(a x) + 1 +--R sin(a x)log(-----------------------) +--R cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - sin(a x)log(-----------------------) - 1 +--R cos(a x) + 1 +--R / +--R a sin(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 33 +bb:=1/a*log(tan(%pi/4+(a*x)/2))-1/(a*sin(a*x)) +--R +--R 2a x + %pi +--R sin(a x)log(tan(----------)) - 1 +--R 4 +--R (2) -------------------------------- +--R a sin(a x) +--R Type: Expression Integer +--E + +--S 34 +cc:=aa-bb +--R +--R (3) +--R 2a x + %pi sin(a x) + cos(a x) + 1 +--R - log(tan(----------)) + log(-----------------------) +--R 4 cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R a +--R Type: Expression Integer +--E + +--S 35 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 36 +dd:=tanrule cc +--R +--R (5) +--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1 +--R log(-----------------------) - log(-----------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R + +--R 2a x + %pi +--R sin(----------) +--R 4 +--R - log(---------------) +--R 2a x + %pi +--R cos(----------) +--R 4 +--R / +--R a +--R Type: Expression Integer +--E + +--S 37 +ee:=expandLog dd +--R +--R (6) +--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) - cos(a x) - 1) +--R + +--R 2a x + %pi 2a x + %pi +--R - log(sin(----------)) + log(cos(----------)) +--R 4 4 +--R / +--R a +--R Type: Expression Integer +--E + +--S 38 14:405 Schaums and Axiom differ by a constant +ff:=complexNormalize % +--R +--R log(- 1) +--R (7) -------- +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.406~~~~~$\displaystyle +\int{\frac{dx}{\sin{ax}\cos^2{ax}}}$} +$$\int{\frac{1}{\sin{ax}\cos^2{ax}}}= +\frac{1}{a}\ln~\tan\frac{ax}{2}+\frac{1}{a\cos{ax}} +$$ +<<*>>= +)clear all + +--S 39 +aa:=integrate(1/(sin(a*x)*cos(a*x)^2),x) +--R +--R +--R sin(a x) +--R cos(a x)log(------------) + cos(a x) + 1 +--R cos(a x) + 1 +--R (1) ---------------------------------------- +--R a cos(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 40 +bb:=1/a*log(tan((a*x)/2))+1/(a*cos(a*x)) +--R +--R a x +--R cos(a x)log(tan(---)) + 1 +--R 2 +--R (2) ------------------------- +--R a cos(a x) +--R Type: Expression Integer +--E + +--S 41 +cc:=aa-bb +--R +--R a x sin(a x) +--R - log(tan(---)) + log(------------) + 1 +--R 2 cos(a x) + 1 +--R (3) --------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 42 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 43 +dd:=tanrule cc +--R +--R a x +--R sin(---) +--R sin(a x) 2 +--R log(------------) - log(--------) + 1 +--R cos(a x) + 1 a x +--R cos(---) +--R 2 +--R (5) ------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 44 +ee:=expandLog dd +--R +--R a x a x +--R log(sin(a x)) - log(sin(---)) - log(cos(a x) + 1) + log(cos(---)) + 1 +--R 2 2 +--R (6) --------------------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 45 14:406 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R 1 +--R (7) - +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.407~~~~~$\displaystyle +\int{\frac{dx}{\sin^2{ax}\cos^2{ax}}}$} +$$\int{\frac{1}{\sin^2{ax}\cos^2{ax}}}= +-\frac{2\cot{2ax}}{a} +$$ +<<*>>= +)clear all + +--S 46 +aa:=integrate(1/(sin(a*x)^2*cos(a*x)^2),x) +--R +--R +--R 2 +--R - 2cos(a x) + 1 +--R (1) ------------------ +--R a cos(a x)sin(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 47 +bb:=-(2*cot(2*a*x))/a +--R +--R 2cot(2a x) +--R (2) - ---------- +--R a +--R Type: Expression Integer +--E + +--S 48 +cc:=aa-bb +--R +--R 2 +--R 2cos(a x)cot(2a x)sin(a x) - 2cos(a x) + 1 +--R (3) ------------------------------------------- +--R a cos(a x)sin(a x) +--R Type: Expression Integer +--E + +--S 49 +cotrule:=rule(cot(a) == cos(a)/sin(a)) +--R +--R cos(a) +--R (4) cot(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 50 +dd:=cotrule cc +--R +--R 2 +--R (- 2cos(a x) + 1)sin(2a x) + 2cos(a x)cos(2a x)sin(a x) +--R (5) -------------------------------------------------------- +--R a cos(a x)sin(a x)sin(2a x) +--R Type: Expression Integer +--E + +--S 51 14:407 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.408~~~~~$\displaystyle +\int{\frac{\sin^2{ax}}{\cos{ax}}}~dx$} +$$\int{\frac{\sin^2{ax}}{\cos{ax}}}= +-\frac{\sin{ax}}{a}+\frac{1}{a}\ln~\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right) +$$ +<<*>>= +)clear all + +--S 52 +aa:=integrate(sin(a*x)^2/cos(a*x),x) +--R +--R +--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1 +--R log(-----------------------) - log(-----------------------) - sin(a x) +--R cos(a x) + 1 cos(a x) + 1 +--R (1) ---------------------------------------------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 53 +bb:=-sin(a*x)/a+1/a*log(tan((a*x)/2+%pi/4)) +--R +--R 2a x + %pi +--R log(tan(----------)) - sin(a x) +--R 4 +--R (2) ------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 54 +cc:=aa-bb +--R +--R (3) +--R 2a x + %pi sin(a x) + cos(a x) + 1 +--R - log(tan(----------)) + log(-----------------------) +--R 4 cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R a +--R Type: Expression Integer +--E + +--S 55 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 56 +dd:=tanrule cc +--R +--R (5) +--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1 +--R log(-----------------------) - log(-----------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R + +--R 2a x + %pi +--R sin(----------) +--R 4 +--R - log(---------------) +--R 2a x + %pi +--R cos(----------) +--R 4 +--R / +--R a +--R Type: Expression Integer +--E + +--S 57 +ee:=expandLog dd +--R +--R (6) +--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) - cos(a x) - 1) +--R + +--R 2a x + %pi 2a x + %pi +--R - log(sin(----------)) + log(cos(----------)) +--R 4 4 +--R / +--R a +--R Type: Expression Integer +--E + +--S 58 14:408 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R log(- 1) +--R (7) -------- +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.409~~~~~$\displaystyle +\int{\frac{\cos^2{ax}}{\sin{ax}}}~dx$} +$$\int{\frac{\cos^2{ax}}{\sin{ax}}}= +\frac{\cos{ax}}{a}+\frac{1}{a}\ln~\tan{\frac{ax}{2}} +$$ +<<*>>= +)clear all + +--S 59 +aa:=integrate(cos(a*x)^2/sin(a*x),x) +--R +--R +--R sin(a x) +--R log(------------) + cos(a x) +--R cos(a x) + 1 +--R (1) ---------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 60 +bb:=cos(a*x)/a+1/a*log(tan((a*x)/2)) +--R +--R a x +--R log(tan(---)) + cos(a x) +--R 2 +--R (2) ------------------------ +--R a +--R Type: Expression Integer +--E + +--S 61 +cc:=aa-bb +--R +--R a x sin(a x) +--R - log(tan(---)) + log(------------) +--R 2 cos(a x) + 1 +--R (3) ----------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 62 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 63 +dd:=tanrule cc +--R +--R a x +--R sin(---) +--R sin(a x) 2 +--R log(------------) - log(--------) +--R cos(a x) + 1 a x +--R cos(---) +--R 2 +--R (5) --------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 64 +ee:=expandLog dd +--R +--R a x a x +--R log(sin(a x)) - log(sin(---)) - log(cos(a x) + 1) + log(cos(---)) +--R 2 2 +--R (6) ----------------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 65 14:409 Schaums and Axiom agree +ff:=complexNormalize ee +--R +--R (7) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.410~~~~~$\displaystyle +\int{\frac{dx}{\cos{ax}(1\pm\sin{ax})}}$} +$$\int{\frac{1}{\cos{ax}(1\pm\sin{ax})}}= +\mp\frac{1}{2a(1\pm\sin{ax})} ++\frac{1}{2a}\ln~\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right) +$$ +<<*>>= +)clear all + +--S 66 +aa:=integrate(1/(cos(a*x)*(1+sin(a*x))),x) +--R +--R +--R (1) +--R sin(a x) + cos(a x) + 1 +--R (sin(a x) + 1)log(-----------------------) +--R cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R (- sin(a x) - 1)log(-----------------------) + sin(a x) +--R cos(a x) + 1 +--R / +--R 2a sin(a x) + 2a +--R Type: Union(Expression Integer,...) +--E + +--S 67 +bb:=-1/(2*a*(1+sin(a*x)))+1/(2*a)*log(tan((a*x)/2+%pi/4)) +--R +--R 2a x + %pi +--R (sin(a x) + 1)log(tan(----------)) - 1 +--R 4 +--R (2) -------------------------------------- +--R 2a sin(a x) + 2a +--R Type: Expression Integer +--E + +--S 68 +cc:=aa-bb +--R +--R (3) +--R 2a x + %pi sin(a x) + cos(a x) + 1 +--R - log(tan(----------)) + log(-----------------------) +--R 4 cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) + 1 +--R cos(a x) + 1 +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 69 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 70 +dd:=tanrule cc +--R +--R (5) +--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1 +--R log(-----------------------) - log(-----------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R + +--R 2a x + %pi +--R sin(----------) +--R 4 +--R - log(---------------) + 1 +--R 2a x + %pi +--R cos(----------) +--R 4 +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 71 +ee:=expandLog dd +--R +--R (6) +--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) - cos(a x) - 1) +--R + +--R 2a x + %pi 2a x + %pi +--R - log(sin(----------)) + log(cos(----------)) + 1 +--R 4 4 +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 72 +ff:=complexNormalize ee +--R +--R log(- 1) + 1 +--R (7) ------------ +--R 2a +--R Type: Expression Integer +--E + +)clear all + +--S 73 +aa:=integrate(1/(cos(a*x)*(1-sin(a*x))),x) +--R +--R +--R (1) +--R sin(a x) + cos(a x) + 1 +--R (sin(a x) - 1)log(-----------------------) +--R cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R (- sin(a x) + 1)log(-----------------------) - sin(a x) +--R cos(a x) + 1 +--R / +--R 2a sin(a x) - 2a +--R Type: Union(Expression Integer,...) +--E + +--S 74 +bb:=1/(2*a*(1-sin(a*x)))+1/(2*a)*log(tan((a*x)/2+%pi/4)) +--R +--R 2a x + %pi +--R (sin(a x) - 1)log(tan(----------)) - 1 +--R 4 +--R (2) -------------------------------------- +--R 2a sin(a x) - 2a +--R Type: Expression Integer +--E + +--S 75 +cc:=aa-bb +--R +--R (3) +--R 2a x + %pi sin(a x) + cos(a x) + 1 +--R - log(tan(----------)) + log(-----------------------) +--R 4 cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) - 1 +--R cos(a x) + 1 +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 76 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 77 +dd:=tanrule cc +--R +--R (5) +--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1 +--R log(-----------------------) - log(-----------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R + +--R 2a x + %pi +--R sin(----------) +--R 4 +--R - log(---------------) - 1 +--R 2a x + %pi +--R cos(----------) +--R 4 +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 78 +ee:=expandLog dd +--R +--R (6) +--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) - cos(a x) - 1) +--R + +--R 2a x + %pi 2a x + %pi +--R - log(sin(----------)) + log(cos(----------)) - 1 +--R 4 4 +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 79 14:410 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R log(- 1) - 1 +--R (7) ------------ +--R 2a +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.411~~~~~$\displaystyle +\int{\frac{dx}{\sin{ax}(1\pm\cos{ax})}}$} +$$\int{\frac{1}{\sin{ax}(1\pm\cos{ax})}}= +\pm\frac{1}{2a(1\pm\cos{ax})}+\frac{1}{2a}\ln~\tan\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 80 +aa:=integrate(1/(sin(a*x)*(1+cos(a*x))),x) +--R +--R +--R sin(a x) +--R (2cos(a x) + 2)log(------------) - cos(a x) + 1 +--R cos(a x) + 1 +--R (1) ----------------------------------------------- +--R 4a cos(a x) + 4a +--R Type: Union(Expression Integer,...) +--E + +--S 81 +bb:=1/(2*a*(1+cos(a*x)))+1/(2*a)*log(tan((a*x)/2)) +--R +--R a x +--R (cos(a x) + 1)log(tan(---)) + 1 +--R 2 +--R (2) ------------------------------- +--R 2a cos(a x) + 2a +--R Type: Expression Integer +--E + +--S 82 +cc:=aa-bb +--R +--R a x sin(a x) +--R - 2log(tan(---)) + 2log(------------) - 1 +--R 2 cos(a x) + 1 +--R (3) ----------------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 83 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 84 +dd:=tanrule cc +--R +--R a x +--R sin(---) +--R sin(a x) 2 +--R 2log(------------) - 2log(--------) - 1 +--R cos(a x) + 1 a x +--R cos(---) +--R 2 +--R (5) --------------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 85 +ee:=expandLog dd +--R +--R (6) +--R a x a x +--R 2log(sin(a x)) - 2log(sin(---)) - 2log(cos(a x) + 1) + 2log(cos(---)) - 1 +--R 2 2 +--R ------------------------------------------------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 86 +ff:=complexNormalize ee +--R +--R 1 +--R (7) - -- +--R 4a +--R Type: Expression Integer +--E + +)clear all + +--S 87 +aa:=integrate(1/(sin(a*x)*(1-cos(a*x))),x) +--R +--R +--R sin(a x) +--R (2cos(a x) - 2)log(------------) + cos(a x) + 1 +--R cos(a x) + 1 +--R (1) ----------------------------------------------- +--R 4a cos(a x) - 4a +--R Type: Union(Expression Integer,...) +--E + +--S 88 +bb:=-1/(2*a*(1-cos(a*x)))+1/(2*a)*log(tan((a*x)/2)) +--R +--R a x +--R (cos(a x) - 1)log(tan(---)) + 1 +--R 2 +--R (2) ------------------------------- +--R 2a cos(a x) - 2a +--R Type: Expression Integer +--E + +--S 89 +cc:=aa-bb +--R +--R a x sin(a x) +--R - 2log(tan(---)) + 2log(------------) + 1 +--R 2 cos(a x) + 1 +--R (3) ----------------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 90 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 91 +dd:=tanrule cc +--R +--R a x +--R sin(---) +--R sin(a x) 2 +--R 2log(------------) - 2log(--------) + 1 +--R cos(a x) + 1 a x +--R cos(---) +--R 2 +--R (5) --------------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 92 +ee:=expandLog dd +--R +--R (6) +--R a x a x +--R 2log(sin(a x)) - 2log(sin(---)) - 2log(cos(a x) + 1) + 2log(cos(---)) + 1 +--R 2 2 +--R ------------------------------------------------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 93 14:411 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R 1 +--R (7) -- +--R 4a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.412~~~~~$\displaystyle +\int{\frac{dx}{\sin{ax}\pm\cos{ax}}}$} +$$\int{\frac{1}{\sin{ax}\pm\cos{ax}}}= +\frac{1}{a\sqrt{2}}\ln~\tan\left(\frac{ax}{2}\pm\frac{\pi}{8}\right) +$$ +<<*>>= +)clear all + +--S 94 +aa:=integrate(1/(sin(a*x)+cos(a*x)),x) +--R +--R +--R +-+ +-+ +-+ +--R +-+ (- \|2 + 1)sin(a x) + (\|2 - 1)cos(a x) + \|2 - 2 +--R \|2 log(----------------------------------------------------) +--R sin(a x) + cos(a x) +--R (1) ------------------------------------------------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 95 +bb:=1/(a*sqrt(2))*log(tan((a*x)/2+%pi/8)) +--R +--R +-+ 4a x + %pi +--R \|2 log(tan(----------)) +--R 8 +--R (2) ------------------------ +--R 2a +--R Type: Expression Integer +--E + +--S 96 +cc:=aa-bb +--R +--R (3) +--R +-+ 4a x + %pi +--R - \|2 log(tan(----------)) +--R 8 +--R + +--R +-+ +-+ +-+ +--R +-+ (- \|2 + 1)sin(a x) + (\|2 - 1)cos(a x) + \|2 - 2 +--R \|2 log(----------------------------------------------------) +--R sin(a x) + cos(a x) +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 97 +complexNormalize cc +--R +--R +-+ +--R +-+ \|2 - 2 +--R \|2 log(--------) +--R +-+ +--R \|2 +--R (4) ----------------- +--R 2a +--R Type: Expression Integer +--E + +)clear all + +--S 98 +aa:=integrate(1/(sin(a*x)-cos(a*x)),x) +--R +--R +--R +-+ +-+ +-+ +--R +-+ (- \|2 + 1)sin(a x) + (- \|2 + 1)cos(a x) - \|2 + 2 +--R \|2 log(------------------------------------------------------) +--R sin(a x) - cos(a x) +--R (1) --------------------------------------------------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 99 +bb:=1/(a*sqrt(2))*log(tan((a*x)/2-%pi/8)) +--R +--R +-+ 4a x - %pi +--R \|2 log(tan(----------)) +--R 8 +--R (2) ------------------------ +--R 2a +--R Type: Expression Integer +--E + +--S 100 +cc:=aa-bb +--R +--R (3) +--R +-+ 4a x - %pi +--R - \|2 log(tan(----------)) +--R 8 +--R + +--R +-+ +-+ +-+ +--R +-+ (- \|2 + 1)sin(a x) + (- \|2 + 1)cos(a x) - \|2 + 2 +--R \|2 log(------------------------------------------------------) +--R sin(a x) - cos(a x) +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 101 14:412 Schaums and Axiom differ by a constant +complexNormalize cc +--R +--R +-+ +-+ +--R \|2 log(\|2 - 1) +--R (4) ----------------- +--R 2a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.413~~~~~$\displaystyle +\int{\frac{\sin{ax}~dx}{\sin{ax}\pm\cos{ax}}}$} +$$\int{\frac{\sin{ax}}{\sin{ax}\pm\cos{ax}}}= +\frac{x}{2}\mp\frac{1}{2a}\ln(\sin{ax}\pm\cos{ax}) +$$ +<<*>>= +)clear all + +--S 102 +aa:=integrate(sin(a*x)/(sin(a*x)+cos(a*x)),x) +--R +--R +--R 2 - 2sin(a x) - 2cos(a x) +--R log(------------) - log(-----------------------) + a x +--R cos(a x) + 1 cos(a x) + 1 +--R (1) ------------------------------------------------------ +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 103 +bb:=x/2-1/(2*a)*log(sin(a*x)+cos(a*x)) +--R +--R - log(sin(a x) + cos(a x)) + a x +--R (2) -------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 104 +cc:=aa-bb +--R +--R (3) +--R 2 - 2sin(a x) - 2cos(a x) +--R log(sin(a x) + cos(a x)) + log(------------) - log(-----------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R --------------------------------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 105 +dd:=expandLog cc +--R +--R log(sin(a x) + cos(a x)) - log(- sin(a x) - cos(a x)) +--R (4) ----------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 106 +ee:=complexNormalize dd +--R +--R log(- 1) +--R (5) -------- +--R 2a +--R Type: Expression Integer +--E + +)clear all + +--S 107 +aa:=integrate(sin(a*x)/(sin(a*x)-cos(a*x)),x) +--R +--R +--R 2sin(a x) - 2cos(a x) 2 +--R log(---------------------) - log(------------) + a x +--R cos(a x) + 1 cos(a x) + 1 +--R (1) ---------------------------------------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 108 +bb:=x/2+1/(2*a)*log(sin(a*x)-cos(a*x)) +--R +--R log(sin(a x) - cos(a x)) + a x +--R (2) ------------------------------ +--R 2a +--R Type: Expression Integer +--E + +--S 109 +cc:=aa-bb +--R +--R (3) +--R 2sin(a x) - 2cos(a x) 2 +--R - log(sin(a x) - cos(a x)) + log(---------------------) - log(------------) +--R cos(a x) + 1 cos(a x) + 1 +--R --------------------------------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 110 14:413 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.414~~~~~$\displaystyle +\int{\frac{cos{ax}~dx}{\sin{ax}\pm{\cos{ax}}}}$} +$$\int{\frac{cos{ax}}{\sin{ax}\pm{\cos{ax}}}}= +\pm\frac{x}{2}+\frac{1}{2a}\ln(sin{ax}\pm\cos{ax}) +$$ +<<*>>= +)clear all + +--S 111 +aa:=integrate(cos(a*x)/(sin(a*x)+cos(a*x)),x) +--R +--R +--R 2 - 2sin(a x) - 2cos(a x) +--R - log(------------) + log(-----------------------) + a x +--R cos(a x) + 1 cos(a x) + 1 +--R (1) -------------------------------------------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 112 +bb:=x/2+1/(2*a)*log(sin(a*x)+cos(a*x)) +--R +--R log(sin(a x) + cos(a x)) + a x +--R (2) ------------------------------ +--R 2a +--R Type: Expression Integer +--E + +--S 113 +cc:=aa-bb +--R +--R (3) +--R 2 - 2sin(a x) - 2cos(a x) +--R - log(sin(a x) + cos(a x)) - log(------------) + log(-----------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R ----------------------------------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 114 +dd:=expandLog cc +--R +--R - log(sin(a x) + cos(a x)) + log(- sin(a x) - cos(a x)) +--R (4) ------------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 115 +ee:=complexNormalize dd +--R +--R log(- 1) +--R (5) - -------- +--R 2a +--R Type: Expression Integer +--E + +)clear all + +--S 116 +aa:=integrate(cos(a*x)/(sin(a*x)-cos(a*x)),x) +--R +--R +--R 2sin(a x) - 2cos(a x) 2 +--R log(---------------------) - log(------------) - a x +--R cos(a x) + 1 cos(a x) + 1 +--R (1) ---------------------------------------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 117 +bb:=-x/2+1/(2*a)*log(sin(a*x)-cos(a*x)) +--R +--R log(sin(a x) - cos(a x)) - a x +--R (2) ------------------------------ +--R 2a +--R Type: Expression Integer +--E + +--S 118 +cc:=aa-bb +--R +--R (3) +--R 2sin(a x) - 2cos(a x) 2 +--R - log(sin(a x) - cos(a x)) + log(---------------------) - log(------------) +--R cos(a x) + 1 cos(a x) + 1 +--R --------------------------------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 119 14:414 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.415~~~~~$\displaystyle +\int{\frac{\sin{ax}~dx}{p+q\cos{ax}}}$} +$$\int{\frac{\sin{ax}}{p+q\cos{ax}}}= +-\frac{1}{aq}\ln(p+q\cos{ax}) +$$ +<<*>>= +)clear all + +--S 120 +aa:=integrate(sin(a*x)/(p+q*cos(a*x)),x) +--R +--R +--R 2 - 2q cos(a x) - 2p +--R log(------------) - log(------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (1) ------------------------------------------- +--R a q +--R Type: Union(Expression Integer,...) +--E + +--S 121 +bb:=-1/(a*q)*log(p+q*cos(a*x)) +--R +--R log(q cos(a x) + p) +--R (2) - ------------------- +--R a q +--R Type: Expression Integer +--E + +--S 122 +cc:=aa-bb +--R +--R 2 - 2q cos(a x) - 2p +--R log(q cos(a x) + p) + log(------------) - log(------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (3) ----------------------------------------------------------------- +--R a q +--R Type: Expression Integer +--E + +--S 123 +dd:=expandLog cc +--R +--R log(q cos(a x) + p) - log(- q cos(a x) - p) +--R (4) ------------------------------------------- +--R a q +--R Type: Expression Integer +--E + +--S 124 14:415 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R log(- 1) +--R (5) -------- +--R a q +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.416~~~~~$\displaystyle +\int{\frac{\cos{ax}~dx}{p+q\sin{ax}}}$} +$$\int{\frac{\cos{ax}}{p+q\sin{ax}}}= +\frac{1}{aq}\ln(p+q\sin{ax}) +$$ +<<*>>= +)clear all + +--S 125 +aa:=integrate(cos(a*x)/(p+q*sin(a*x)),x) +--R +--R +--R 2q sin(a x) + 2p 2 +--R log(----------------) - log(------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (1) ----------------------------------------- +--R a q +--R Type: Union(Expression Integer,...) +--E + +--S 126 +bb:=1/(a*q)*log(p+q*sin(a*x)) +--R +--R log(q sin(a x) + p) +--R (2) ------------------- +--R a q +--R Type: Expression Integer +--E + +--S 127 +cc:=aa-bb +--R +--R 2q sin(a x) + 2p 2 +--R - log(q sin(a x) + p) + log(----------------) - log(------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (3) ----------------------------------------------------------------- +--R a q +--R Type: Expression Integer +--E + +--S 128 14:416 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.417~~~~~$\displaystyle +\int{\frac{\sin{ax}~dx}{(p+q\cos{ax})^n}}$} +$$\int{\frac{\sin{ax}}{(p+q\cos{ax})^n}}= +\frac{1}{aq(n-1)(p+q\cos{ax})^{n-1}} +$$ +<<*>>= +)clear all + +--S 129 +aa:=integrate(sin(a*x)/(p+q*cos(a*x))^n,x) +--R +--R +--R q cos(a x) + p +--R (1) ---------------------------------- +--R n log(q cos(a x) + p) +--R (a n - a)q %e +--R Type: Union(Expression Integer,...) +--E + +--S 130 +bb:=1/(a*q*(n-1)*(p+q*cos(a*x))^(n-1)) +--R +--R 1 +--R (2) -------------------------------- +--R n - 1 +--R (a n - a)q (q cos(a x) + p) +--R Type: Expression Integer +--E + +--S 131 +cc:=aa-bb +--R +--R n log(q cos(a x) + p) n - 1 +--R - %e + (q cos(a x) + p)(q cos(a x) + p) +--R (3) ----------------------------------------------------------------- +--R n - 1 n log(q cos(a x) + p) +--R (a n - a)q (q cos(a x) + p) %e +--R Type: Expression Integer +--E + +--S 132 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 133 +dd:=explog cc +--R +--R n n - 1 +--R - (q cos(a x) + p) + (q cos(a x) + p)(q cos(a x) + p) +--R (5) ----------------------------------------------------------- +--R n - 1 n +--R (a n - a)q (q cos(a x) + p) (q cos(a x) + p) +--R Type: Expression Integer +--E + +--S 134 14:417 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.418~~~~~$\displaystyle +\int{\frac{\cos{ax}~dx}{(p+q\sin{ax})^n}}$} +$$\int{\frac{\cos{ax}}{(p+q\sin{ax})^n}}= +\frac{-1}{aq(n-1)(p+q\sin{ax})^{n-1}} +$$ +<<*>>= +)clear all + +--S 135 +aa:=integrate(cos(a*x)/(p+q*sin(a*x))^n,x) +--R +--R +--R - q sin(a x) - p +--R (1) ---------------------------------- +--R n log(q sin(a x) + p) +--R (a n - a)q %e +--R Type: Union(Expression Integer,...) +--E + +--S 136 +bb:=-1/(a*q*(n-1)*(p+q*sin(a*x))^(n-1)) +--R +--R 1 +--R (2) - -------------------------------- +--R n - 1 +--R (a n - a)q (q sin(a x) + p) +--R Type: Expression Integer +--E + +--S 137 +cc:=aa-bb +--R +--R n log(q sin(a x) + p) n - 1 +--R %e + (- q sin(a x) - p)(q sin(a x) + p) +--R (3) ----------------------------------------------------------------- +--R n - 1 n log(q sin(a x) + p) +--R (a n - a)q (q sin(a x) + p) %e +--R Type: Expression Integer +--E + +--S 138 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 139 +dd:=explog cc +--R +--R n n - 1 +--R (q sin(a x) + p) + (- q sin(a x) - p)(q sin(a x) + p) +--R (5) ----------------------------------------------------------- +--R n - 1 n +--R (a n - a)q (q sin(a x) + p) (q sin(a x) + p) +--R Type: Expression Integer +--E + +--S 140 14:418 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.419~~~~~$\displaystyle +\int{\frac{dx}{p\sin{ax}+q\cos{ax}}}$} +$$\int{\frac{1}{p\sin{ax}+q\cos{ax}}}= +\frac{1}{a\sqrt{p^2+q^2}}\ln~\tan\left(\frac{ax+\tan^{-1}(q/p)}{2}\right) +$$ +<<*>>= +)clear all + +--S 141 +aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)),x) +--R +--R +--R (1) +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) - p cos(a x) - q - p )\|q + p +--R + +--R 3 2 2 3 2 3 +--R (- q - p q)sin(a x) + (p q + p )cos(a x) + p q + p +--R / +--R p sin(a x) + q cos(a x) +--R / +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Union(Expression Integer,...) +--E + +--S 142 +bb:=1/(a*sqrt(p^2+q^2))*log(tan((a*x+atan(q/p))/2)) +--R +--R q +--R atan(-) + a x +--R p +--R log(tan(-------------)) +--R 2 +--R (2) ----------------------- +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Expression Integer +--E + +--S 143 +cc:=aa-bb +--R +--R (3) +--R q +--R atan(-) + a x +--R p +--R - log(tan(-------------)) +--R 2 +--R + +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) - p cos(a x) - q - p )\|q + p +--R + +--R 3 2 2 3 2 3 +--R (- q - p q)sin(a x) + (p q + p )cos(a x) + p q + p +--R / +--R p sin(a x) + q cos(a x) +--R / +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Expression Integer +--E + +--S 144 +dd:=normalize cc +--R +--R +-------+ +--R | 2 2 2 2 +--R - 2p\|q + p + q + 2p +--R log(------------------------------------------) +--R +-------+ +--R 2 3 | 2 2 4 2 2 4 +--R (3p q + 4p )\|q + p - q - 5p q - 4p +--R (4) - ----------------------------------------------- +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Expression Integer +--E + +--S 145 14:419 Schaums and Axiom differ by a constant +ee:=ratDenom dd +--R +--R +-------+ +--R +-------+ | 2 2 2 2 +--R | 2 2 - p\|q + p - q - p +--R \|q + p log(-----------------------) +--R 4 2 2 +--R q + p q +--R (5) - -------------------------------------- +--R 2 2 +--R a q + a p +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.420~~~~~$\displaystyle +\int{\frac{dx}{p\sin{ax}+q\cos{ax}+r}}$} +$$\int{\frac{1}{p\sin{ax}+q\cos{ax}+r}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{2}{a\sqrt{r^2-p^2-q^q}} +\tan^{-1}\left(\frac{p+(r-q)\tan(ax/2)}{\sqrt{r^2-p^2-a^2}}\right)\\ +\\ +\displaystyle +\frac{1}{a\sqrt{p^2+q^2-r^2}}\ln\left( +\frac{p-\sqrt{p^2+q^2-r^2}+(r-q)\tan{(ax/2)}} +{p+\sqrt{p^2+q^2-r^2}+(r-q)\tan{(ax/2)}}\right) +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 146 +aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)+r),x) +--R +--R +--R (1) +--R [ +--R log +--R 2 2 2 +--R (p r - p q)sin(a x) + (- r + q r + p )cos(a x) - q r + q +--R + +--R 2 +--R p +--R * +--R +--------------+ +--R | 2 2 2 +--R \|- r + q + p +--R + +--R 3 2 2 2 3 2 +--R (r - q r + (- q - p )r + q + p q)sin(a x) +--R + +--R 2 2 3 2 2 3 +--R (p r - p q - p )cos(a x) + p r - p q - p +--R / +--R p sin(a x) + q cos(a x) + r +--R / +--R +--------------+ +--R | 2 2 2 +--R a\|- r + q + p +--R , +--R +------------+ +--R | 2 2 2 +--R ((r - q)sin(a x) + p cos(a x) + p)\|r - q - p +--R 2atan(-------------------------------------------------) +--R 2 2 2 2 2 2 +--R (r - q - p )cos(a x) + r - q - p +--R --------------------------------------------------------] +--R +------------+ +--R | 2 2 2 +--R a\|r - q - p +--R Type: Union(List Expression Integer,...) +--E + +--S 147 +bb1:=2/(a*sqrt(r^2-p^2-q^2))*atan((p+(r-q)*tan((a*x)/2))/sqrt(r^2-p^2-q^2)) +--R +--R a x +--R (r - q)tan(---) + p +--R 2 +--R 2atan(-------------------) +--R +------------+ +--R | 2 2 2 +--R \|r - q - p +--R (2) -------------------------- +--R +------------+ +--R | 2 2 2 +--R a\|r - q - p +--R Type: Expression Integer +--E + +--S 148 +bb2:=1/(a*sqrt(p^2+q^2-r^2))*log((p-sqrt(p^2+q^2-r^2)+(r-q)*tan((a*x)/2))/(p+sqrt(p^2+q^2-r^2)+(r-q)*tan((a*x)/2))) +--R +--R +--------------+ +--R | 2 2 2 a x +--R - \|- r + q + p + (r - q)tan(---) + p +--R 2 +--R log(-----------------------------------------) +--R +--------------+ +--R | 2 2 2 a x +--R \|- r + q + p + (r - q)tan(---) + p +--R 2 +--R (3) ---------------------------------------------- +--R +--------------+ +--R | 2 2 2 +--R a\|- r + q + p +--R Type: Expression Integer +--E + +--S 149 +cc1:=aa.1-bb1 +--R +--R (4) +--R +------------+ +--R | 2 2 2 +--R \|r - q - p +--R * +--R log +--R 2 2 2 +--R (p r - p q)sin(a x) + (- r + q r + p )cos(a x) - q r + q +--R + +--R 2 +--R p +--R * +--R +--------------+ +--R | 2 2 2 +--R \|- r + q + p +--R + +--R 3 2 2 2 3 2 +--R (r - q r + (- q - p )r + q + p q)sin(a x) +--R + +--R 2 2 3 2 2 3 +--R (p r - p q - p )cos(a x) + p r - p q - p +--R / +--R p sin(a x) + q cos(a x) + r +--R + +--R a x +--R +--------------+ (r - q)tan(---) + p +--R | 2 2 2 2 +--R - 2\|- r + q + p atan(-------------------) +--R +------------+ +--R | 2 2 2 +--R \|r - q - p +--R / +--R +--------------+ +------------+ +--R | 2 2 2 | 2 2 2 +--R a\|- r + q + p \|r - q - p +--R Type: Expression Integer +--E + +--S 150 +cc2:=aa.2-bb1 +--R +--R (5) +--R +------------+ +--R | 2 2 2 +--R ((r - q)sin(a x) + p cos(a x) + p)\|r - q - p +--R 2atan(-------------------------------------------------) +--R 2 2 2 2 2 2 +--R (r - q - p )cos(a x) + r - q - p +--R + +--R a x +--R (r - q)tan(---) + p +--R 2 +--R - 2atan(-------------------) +--R +------------+ +--R | 2 2 2 +--R \|r - q - p +--R / +--R +------------+ +--R | 2 2 2 +--R a\|r - q - p +--R Type: Expression Integer +--E + +--S 151 +cc3:=aa.1-bb2 +--R +--R (6) +--R log +--R 2 2 2 +--R (p r - p q)sin(a x) + (- r + q r + p )cos(a x) - q r + q +--R + +--R 2 +--R p +--R * +--R +--------------+ +--R | 2 2 2 +--R \|- r + q + p +--R + +--R 3 2 2 2 3 2 +--R (r - q r + (- q - p )r + q + p q)sin(a x) +--R + +--R 2 2 3 2 2 3 +--R (p r - p q - p )cos(a x) + p r - p q - p +--R / +--R p sin(a x) + q cos(a x) + r +--R + +--R +--------------+ +--R | 2 2 2 a x +--R - \|- r + q + p + (r - q)tan(---) + p +--R 2 +--R - log(-----------------------------------------) +--R +--------------+ +--R | 2 2 2 a x +--R \|- r + q + p + (r - q)tan(---) + p +--R 2 +--R / +--R +--------------+ +--R | 2 2 2 +--R a\|- r + q + p +--R Type: Expression Integer +--E + +--S 152 +cc4:=aa.2-bb2 +--R +--R (7) +--R +--------------+ +--R | 2 2 2 a x +--R +------------+ - \|- r + q + p + (r - q)tan(---) + p +--R | 2 2 2 2 +--R - \|r - q - p log(-----------------------------------------) +--R +--------------+ +--R | 2 2 2 a x +--R \|- r + q + p + (r - q)tan(---) + p +--R 2 +--R + +--R +------------+ +--R +--------------+ | 2 2 2 +--R | 2 2 2 ((r - q)sin(a x) + p cos(a x) + p)\|r - q - p +--R 2\|- r + q + p atan(-------------------------------------------------) +--R 2 2 2 2 2 2 +--R (r - q - p )cos(a x) + r - q - p +--R / +--R +--------------+ +------------+ +--R | 2 2 2 | 2 2 2 +--R a\|- r + q + p \|r - q - p +--R Type: Expression Integer +--E + +--S 153 14:420 Schaums and Axiom agree +dd2:=normalize cc2 +--R +--R (8) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.421~~~~~$\displaystyle +\int{\frac{dx}{p\sin{ax}+q(1+\cos{ax})}}$} +$$\int{\frac{1}{p\sin{ax}+q(1+\cos{ax})}}= +\frac{1}{ap}\ln\left(q+p\tan{\frac{ax}{2}}\right) +$$ +<<*>>= +)clear all + +--S 154 +aa:=integrate(1/(p*sin(a*x)+q*(1+cos(a*x))),x) +--R +--R +--R p sin(a x) + q cos(a x) + q +--R log(---------------------------) +--R cos(a x) + 1 +--R (1) -------------------------------- +--R a p +--R Type: Union(Expression Integer,...) +--E + +--S 155 +bb:=1/(a*p)*log(q+p*tan((a*x)/2)) +--R +--R a x +--R log(p tan(---) + q) +--R 2 +--R (2) ------------------- +--R a p +--R Type: Expression Integer +--E + +--S 156 +cc:=aa-bb +--R +--R a x p sin(a x) + q cos(a x) + q +--R - log(p tan(---) + q) + log(---------------------------) +--R 2 cos(a x) + 1 +--R (3) -------------------------------------------------------- +--R a p +--R Type: Expression Integer +--E + +--S 157 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 158 +dd:=tanrule cc +--R +--R a x a x +--R p sin(---) + q cos(---) +--R p sin(a x) + q cos(a x) + q 2 2 +--R log(---------------------------) - log(-----------------------) +--R cos(a x) + 1 a x +--R cos(---) +--R 2 +--R (5) --------------------------------------------------------------- +--R a p +--R Type: Expression Integer +--E + +--S 159 +ee:=expandLog dd +--R +--R (6) +--R a x a x +--R log(p sin(a x) + q cos(a x) + q) - log(p sin(---) + q cos(---)) +--R 2 2 +--R + +--R a x +--R - log(cos(a x) + 1) + log(cos(---)) +--R 2 +--R / +--R a p +--R Type: Expression Integer +--E + +--S 160 14:421 Schaums and Axiom agree +ff:=complexNormalize ee +--R +--R (7) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.422~~~~~$\displaystyle +\int{\frac{dx}{p\sin{ax}+q\cos{ax}\pm\sqrt{p^2+q^2}}}$} +$$\int{\frac{1}{p\sin{ax}+q\cos{ax}\pm\sqrt{p^2+q^2}}}= +\frac{-1}{a\sqrt{p^2+q^2}} +\tan\left(\frac{\pi}{4}\mp\frac{ax+\tan^{-1}{(q/p)}}{2}\right) +$$ +<<*>>= +)clear all + +--S 161 +aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)+sqrt(p^2+q^2)),x) +--R +--R +--R (1) +--R +-------+ +--R 5 2 3 4 5 2 3 4 | 2 2 +--R ((64q + 64p q + 12p q)cos(a x) + 64q + 64p q + 12p q)\|q + p +--R + +--R 6 2 4 4 2 6 6 2 4 4 2 6 +--R (- 64q - 96p q - 36p q - 2p )cos(a x) - 64q - 96p q - 36p q - 2p +--R / +--R 6 2 4 4 2 6 +--R (64a q + 80a p q + 24a p q + a p )sin(a x) +--R + +--R 5 3 3 5 5 3 3 5 +--R (- 32a p q - 32a p q - 6a p q)cos(a x) - 32a p q - 32a p q - 6a p q +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 7 2 5 4 3 6 +--R (- 64a q - 112a p q - 56a p q - 7a p q)sin(a x) +--R + +--R 6 3 4 5 2 7 6 3 4 +--R (32a p q + 48a p q + 18a p q + a p )cos(a x) + 32a p q + 48a p q +--R + +--R 5 2 7 +--R 18a p q + a p +--R Type: Union(Expression Integer,...) +--E + +--S 162 +bb:=-1/(a*sqrt(p^2+q^2))*tan(%pi/4-(a*x+atan(q/p))/2) +--R +--R q +--R 2atan(-) + 2a x - %pi +--R p +--R tan(---------------------) +--R 4 +--R (2) -------------------------- +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Expression Integer +--E + +--S 163 +cc:=aa-bb +--R +--R (3) +--R 6 2 4 4 2 6 +--R (64q + 80p q + 24p q + p )sin(a x) +--R + +--R 5 3 3 5 5 3 3 5 +--R (- 32p q - 32p q - 6p q)cos(a x) - 32p q - 32p q - 6p q +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 7 2 5 4 3 6 +--R (- 64q - 112p q - 56p q - 7p q)sin(a x) +--R + +--R 6 3 4 5 2 7 6 3 4 5 2 7 +--R (32p q + 48p q + 18p q + p )cos(a x) + 32p q + 48p q + 18p q + p +--R * +--R q +--R 2atan(-) + 2a x - %pi +--R p +--R tan(---------------------) +--R 4 +--R + +--R 6 2 4 4 2 6 6 2 4 4 2 6 +--R ((64q + 96p q + 36p q + 2p )cos(a x) + 64q + 96p q + 36p q + 2p ) +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 7 2 5 4 3 6 7 2 5 4 3 +--R (- 64q - 128p q - 76p q - 12p q)cos(a x) - 64q - 128p q - 76p q +--R + +--R 6 +--R - 12p q +--R / +--R 7 2 5 4 3 6 +--R (64a q + 112a p q + 56a p q + 7a p q)sin(a x) +--R + +--R 6 3 4 5 2 7 6 +--R (- 32a p q - 48a p q - 18a p q - a p )cos(a x) - 32a p q +--R + +--R 3 4 5 2 7 +--R - 48a p q - 18a p q - a p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 8 2 6 4 4 6 2 8 +--R (- 64a q - 144a p q - 104a p q - 25a p q - a p )sin(a x) +--R + +--R 7 3 5 5 3 7 7 3 5 +--R (32a p q + 64a p q + 38a p q + 6a p q)cos(a x) + 32a p q + 64a p q +--R + +--R 5 3 7 +--R 38a p q + 6a p q +--R Type: Expression Integer +--E + +--S 164 +dd:=normalize cc +--R +--R (4) +--R +-------+ +--R 6 2 5 3 4 4 3 5 2 6 7 | 2 2 +--R (- 32p q - 16p q - 48p q - 20p q - 18p q - 5p q - p )\|q + p +--R + +--R 7 2 6 3 5 4 4 5 3 6 2 7 8 +--R 32p q + 16p q + 64p q + 28p q + 38p q + 13p q + 6p q + p +--R / +--R 8 7 2 6 3 5 4 4 5 3 +--R 64a q + 32a p q + 144a p q + 64a p q + 104a p q + 38a p q +--R + +--R 6 2 7 8 +--R 25a p q + 6a p q + a p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 9 8 2 7 3 6 4 5 5 4 +--R - 64a q - 32a p q - 176a p q - 80a p q - 168a p q - 66a p q +--R + +--R 6 3 7 2 8 9 +--R - 63a p q - 19a p q - 7a p q - a p +--R Type: Expression Integer +--E + +--S 165 +ee:=ratDenom dd +--R +--R +-------+ +--R | 2 2 2 2 +--R - q\|q + p - q - p +--R (5) ----------------------- +--R 2 3 +--R a p q + a p +--R Type: Expression Integer +--E + +)clear all + +--S 166 +aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)-sqrt(p^2+q^2)),x) +--R +--R +--R (1) +--R +-------+ +--R 5 2 3 4 5 2 3 4 | 2 2 +--R ((64q + 64p q + 12p q)cos(a x) + 64q + 64p q + 12p q)\|q + p +--R + +--R 6 2 4 4 2 6 6 2 4 4 2 6 +--R (64q + 96p q + 36p q + 2p )cos(a x) + 64q + 96p q + 36p q + 2p +--R / +--R 6 2 4 4 2 6 +--R (64a q + 80a p q + 24a p q + a p )sin(a x) +--R + +--R 5 3 3 5 5 3 3 5 +--R (- 32a p q - 32a p q - 6a p q)cos(a x) - 32a p q - 32a p q - 6a p q +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 7 2 5 4 3 6 +--R (64a q + 112a p q + 56a p q + 7a p q)sin(a x) +--R + +--R 6 3 4 5 2 7 6 3 4 +--R (- 32a p q - 48a p q - 18a p q - a p )cos(a x) - 32a p q - 48a p q +--R + +--R 5 2 7 +--R - 18a p q - a p +--R Type: Union(Expression Integer,...) +--E + +--S 167 +bb:=-1/(a*sqrt(p^2+q^2))*tan(%pi/4+(a*x+atan(q/p))/2) +--R +--R q +--R 2atan(-) + 2a x + %pi +--R p +--R tan(---------------------) +--R 4 +--R (2) - -------------------------- +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Expression Integer +--E + +--S 168 +cc:=aa-bb +--R +--R (3) +--R 6 2 4 4 2 6 +--R (64q + 80p q + 24p q + p )sin(a x) +--R + +--R 5 3 3 5 5 3 3 5 +--R (- 32p q - 32p q - 6p q)cos(a x) - 32p q - 32p q - 6p q +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 7 2 5 4 3 6 +--R (64q + 112p q + 56p q + 7p q)sin(a x) +--R + +--R 6 3 4 5 2 7 6 3 4 5 2 +--R (- 32p q - 48p q - 18p q - p )cos(a x) - 32p q - 48p q - 18p q +--R + +--R 7 +--R - p +--R * +--R q +--R 2atan(-) + 2a x + %pi +--R p +--R tan(---------------------) +--R 4 +--R + +--R 6 2 4 4 2 6 6 2 4 4 2 6 +--R ((64q + 96p q + 36p q + 2p )cos(a x) + 64q + 96p q + 36p q + 2p ) +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 7 2 5 4 3 6 7 2 5 4 3 6 +--R (64q + 128p q + 76p q + 12p q)cos(a x) + 64q + 128p q + 76p q + 12p q +--R / +--R 7 2 5 4 3 6 +--R (64a q + 112a p q + 56a p q + 7a p q)sin(a x) +--R + +--R 6 3 4 5 2 7 6 +--R (- 32a p q - 48a p q - 18a p q - a p )cos(a x) - 32a p q +--R + +--R 3 4 5 2 7 +--R - 48a p q - 18a p q - a p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 8 2 6 4 4 6 2 8 +--R (64a q + 144a p q + 104a p q + 25a p q + a p )sin(a x) +--R + +--R 7 3 5 5 3 7 7 3 5 +--R (- 32a p q - 64a p q - 38a p q - 6a p q)cos(a x) - 32a p q - 64a p q +--R + +--R 5 3 7 +--R - 38a p q - 6a p q +--R Type: Expression Integer +--E + +--S 169 +dd:=normalize cc +--R +--R (4) +--R +-------+ +--R 6 2 5 3 4 4 3 5 2 6 7 | 2 2 +--R (- 32p q + 16p q - 48p q + 20p q - 18p q + 5p q - p )\|q + p +--R + +--R 7 2 6 3 5 4 4 5 3 6 2 7 8 +--R - 32p q + 16p q - 64p q + 28p q - 38p q + 13p q - 6p q + p +--R / +--R 8 7 2 6 3 5 4 4 5 3 +--R 64a q - 32a p q + 144a p q - 64a p q + 104a p q - 38a p q +--R + +--R 6 2 7 8 +--R 25a p q - 6a p q + a p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 9 8 2 7 3 6 4 5 5 4 +--R 64a q - 32a p q + 176a p q - 80a p q + 168a p q - 66a p q +--R + +--R 6 3 7 2 8 9 +--R 63a p q - 19a p q + 7a p q - a p +--R Type: Expression Integer +--E + +--S 170 14:422 Schaums and Axiom differ by a constant +ee:=ratDenom dd +--R +--R +-------+ +--R | 2 2 2 2 +--R q\|q + p - q - p +--R (5) --------------------- +--R 2 3 +--R a p q + a p +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.423~~~~~$\displaystyle +\int{\frac{dx}{p^2\sin^2{ax}+q^2\cos^2{ax}}}$} +$$\int{\frac{1}{p^2\sin^2{ax}+q^2\cos^2{ax}}}= +\frac{1}{apq}\tan^{-1}\left(\frac{p\tan{ax}}{q}\right) +$$ +<<*>>= +)clear all + +--S 171 +aa:=integrate(1/(p^2*sin(a*x)^2+q^2*cos(a*x)^2),x) +--R +--R +--R 2 2 2 +--R ((q - 2p )cos(a x) - 2p )sin(a x) q sin(a x) +--R - atan(-----------------------------------) + atan(----------------) +--R 2 2p cos(a x) + 2p +--R p q cos(a x) + 2p q cos(a x) + p q +--R (1) -------------------------------------------------------------------- +--R a p q +--R Type: Union(Expression Integer,...) +--E + +--S 172 +bb:=1/(a*p*q)*atan((p*tan(a*x))/q) +--R +--R p tan(a x) +--R atan(----------) +--R q +--R (2) ---------------- +--R a p q +--R Type: Expression Integer +--E + +--S 173 +cc:=aa-bb +--R +--R (3) +--R 2 2 2 +--R p tan(a x) ((q - 2p )cos(a x) - 2p )sin(a x) +--R - atan(----------) - atan(-----------------------------------) +--R q 2 +--R p q cos(a x) + 2p q cos(a x) + p q +--R + +--R q sin(a x) +--R atan(----------------) +--R 2p cos(a x) + 2p +--R / +--R a p q +--R Type: Expression Integer +--E + +--S 174 14:423 Schaums and Axiom agree +dd:=normalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E + + +@ + +\section{\cite{1}:14.424~~~~~$\displaystyle +\int{\frac{dx}{p^2\sin^2{ax}-q^2\cos^2{ax}}}$} +$$\int{\frac{1}{p^2\sin^2{ax}-q^2\cos^2{ax}}}= +\frac{1}{2apq}\ln\left(\frac{p\tan{ax}-q}{p\tan{ax}+q}\right) +$$ +<<*>>= +)clear all + +--S 175 +aa:=integrate(1/(p^2*sin(a*x)^2-q^2*cos(a*x)^2),x) +--R +--R +--R 2p sin(a x) - 2q cos(a x) - 2p sin(a x) - 2q cos(a x) +--R log(-------------------------) - log(---------------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (1) ----------------------------------------------------------------- +--R 2a p q +--R Type: Union(Expression Integer,...) +--E + +--S 176 +bb:=1/(2*a*p*q)*log((p*tan(a*x)-q)/(p*tan(a*x)+q)) +--R +--R p tan(a x) - q +--R log(--------------) +--R p tan(a x) + q +--R (2) ------------------- +--R 2a p q +--R Type: Expression Integer +--E + +--S 177 +cc:=aa-bb +--R +--R (3) +--R 2p sin(a x) - 2q cos(a x) p tan(a x) - q +--R log(-------------------------) - log(--------------) +--R cos(a x) + 1 p tan(a x) + q +--R + +--R - 2p sin(a x) - 2q cos(a x) +--R - log(---------------------------) +--R cos(a x) + 1 +--R / +--R 2a p q +--R Type: Expression Integer +--E + +--S 178 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 179 +dd:=tanrule cc +--R +--R (5) +--R 2p sin(a x) - 2q cos(a x) p sin(a x) - q cos(a x) +--R log(-------------------------) - log(-----------------------) +--R cos(a x) + 1 p sin(a x) + q cos(a x) +--R + +--R - 2p sin(a x) - 2q cos(a x) +--R - log(---------------------------) +--R cos(a x) + 1 +--R / +--R 2a p q +--R Type: Expression Integer +--E + +--S 180 +ee:=expandLog dd +--R +--R log(p sin(a x) + q cos(a x)) - log(- p sin(a x) - q cos(a x)) +--R (6) ------------------------------------------------------------- +--R 2a p q +--R Type: Expression Integer +--E + +--S 181 14:424 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R log(- 1) +--R (7) -------- +--R 2a p q +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.425~~~~~$\displaystyle +\int{\sin^m{ax}\cos^n{ax}}~dx$} +$$\int{\sin^m{ax}\cos^n{ax}}= +\left\{ +\begin{array}{l} +\displaystyle +-\frac{\sin^{m-1}{ax}\cos^{n+1}ax}{a(m+n)} ++\frac{m-1}{m+n}\int{\sin^{m-2}{ax}\cos^n{ax}}\\ +\\ +\displaystyle +\frac{\sin^{m+1}{ax}\cos^{n-1}{ax}}{a(m+n)} ++\frac{n-1}{m+n}\int{\sin^m{ax}\cos^{n-2}{ax}} +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 182 14:425 Axiom cannot compute this integral +aa:=integrate(sin(a*x)^m*cos(a*x)^n,x) +--R +--R +--R x +--R ++ n m +--I (1) | cos(%H a) sin(%H a) d%H +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.426~~~~~$\displaystyle +\int{\frac{\sin^m{ax}}{\cos^n{ax}}}~dx$} +$$\int{\frac{\sin^m{ax}}{\cos^n{ax}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{\sin^{m-1}{ax}}{a(n-1)\cos^{n-1}{ax}} +-\frac{m-1}{n-1}\int{\frac{\sin^{m-2}{ax}}{\cos^{n-2}{ax}}}\\ +\\ +\displaystyle +\frac{\sin^{m+1}{ax}}{a(n-1)\cos^{n-1}{ax}} +-\frac{m-n+2}{n-1}\int{\frac{\sin^m{ax}}{\cos^{n-2}{ax}}}\\ +\\ +\displaystyle +\frac{-\sin^{m-1}{ax}}{a(m-n)\cos^{n-1}{ax}} ++\frac{m-1}{m-n}\int{\frac{\sin^{m-2}{ax}}{\cos^n{ax}}} +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 183 14:426 Axiom cannot compute this integral +aa:=integrate(sin(a*x)^m/cos(a*x)^n,x) +--R +--R +--R x m +--I ++ sin(%H a) +--I (1) | ---------- d%H +--R ++ n +--I cos(%H a) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.427~~~~~$\displaystyle +\int{\frac{\cos^m{ax}}{\sin^n{ax}}}~dx$} +$$\int{\frac{\cos^m{ax}}{\sin^n{ax}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{-\cos^{m-1}{ax}}{a(n-1)\sin^{n-1}{ax}} +-\frac{m-1}{n-1}\int{\frac{\cos^{m-2}{ax}}{\sin^{n-2}{ax}}}\\ +\\ +\displaystyle +\frac{-\cos^{m+1}{ax}}{a(n-1)\sin^{n-1}{ax}} +-\frac{m-n+2}{n-1}\int{\frac{\cos^m{ax}}{\sin^{n-2}{ax}}}\\ +\\ +\displaystyle +\frac{\cos^{m-1}{ax}}{a(m-n)\sin^{n-1}{ax}} ++\frac{m-1}{m-n}\int{\frac{\cos^{m-2}{ax}}{\sin^n{ax}}} +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 184 14:427 Axiom cannot compute this integral +aa:=integrate(cos(a*x)^m/sin(a*x)^n,x) +--R +--R +--R x m +--I ++ cos(%H a) +--I (1) | ---------- d%H +--R ++ n +--I sin(%H a) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.428~~~~~$\displaystyle +\int{\frac{dx}{\sin^m{ax}\cos^n{ax}}}$} +$$\int{\frac{1}{\sin^m{ax}\cos^n{ax}}} +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{a(n-1)\sin^{m-1}{ax}\cos^{n-1}{ax}} ++\frac{m+n-2}{n-1}\int{\frac{1}{\sin^m{ax}\cos^{n-2}{ax}}}\\ +\\ +\displaystyle +\frac{-1}{a(m-1)\sin^{m-1}{ax}\cos^{n-1}{ax}} ++\frac{m+n-2}{m-1}\int{\frac{1}{\sin^{m-2}{ax}\cos^n{ax}}} +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 185 14:428 Axiom cannot compute this integral +aa:=integrate(1/(sin(a*x)^m*cos(a*x)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | -------------------- d%H +--R ++ n m +--I cos(%H a) sin(%H a) +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp78-80 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum19.input.pdf b/src/axiom-website/CATS/schaum19.input.pdf new file mode 100644 index 0000000..3543367 --- /dev/null +++ b/src/axiom-website/CATS/schaum19.input.pdf @@ -0,0 +1,4477 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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00000 n +0000171534 00000 n +0000172339 00000 n +0000172256 00000 n +0000172823 00000 n +0000172846 00000 n +0000172868 00000 n +trailer +<< +/Size 406 +/Root 2 0 R +/Info 1 0 R +>> +startxref +172989 +%%EOF diff --git a/src/axiom-website/CATS/schaum2.input.pamphlet b/src/axiom-website/CATS/schaum2.input.pamphlet new file mode 100644 index 0000000..7a4ef98 --- /dev/null +++ b/src/axiom-website/CATS/schaum2.input.pamphlet @@ -0,0 +1,1549 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum2.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.84~~~~~$\displaystyle +\int{\frac{dx}{\sqrt{ax+b}}}$} +$$\int{\frac{1}{\sqrt{ax+b}}}= +\frac{2\sqrt{ax+b}}{a} +$$ +<<*>>= +)spool schaum2.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(1/sqrt(a*x+b),x) +--R +--R +--R +-------+ +--R 2\|a x + b +--R (1) ----------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=(2*sqrt(a*x+b))/a +--R +--R +--R +-------+ +--R 2\|a x + b +--R (2) ----------- +--R a +--R Type: Expression Integer +--E + +--S 3 14:84 Schaums and Axiom agree +cc:=aa-bb +--R +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.85~~~~~$\displaystyle +\int{\frac{x~dx}{\sqrt{ax+b}}}$} +$$\int{\frac{x}{\sqrt{ax+b}}}= +\frac{2(ax-2b)}{3a^2}\sqrt{ax+b} +$$ +<<*>>= +)clear all + +--S 4 +aa:=integrate(x/sqrt(a*x+b),x) +--R +--R +--R +-------+ +--R (2a x - 4b)\|a x + b +--R (1) --------------------- +--R 2 +--R 3a +--R Type: Union(Expression Integer,...) +--E + +--S 5 +bb:=(2*(a*x-2*b))/(3*a^2)*sqrt(a*x+b) +--R +--R +--R +-------+ +--R (2a x - 4b)\|a x + b +--R (2) --------------------- +--R 2 +--R 3a +--R Type: Expression Integer +--E + +--S 6 14:85 Schaums and Axiom agree +cc:=aa-bb +--R +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.86~~~~~$\displaystyle +\int{\frac{x^2~dx}{\sqrt{ax+b}}}$} +$$\int{\frac{x}{\sqrt{ax+b}}}= +\frac{2(3a^2x^2-4abx+8b^2)}{15a^2}\sqrt{ax+b} +$$ +<<*>>= +)clear all + +--S 7 +aa:=integrate(x^2/sqrt(a*x+b),x) +--R +--R +--R 2 2 2 +-------+ +--R (6a x - 8a b x + 16b )\|a x + b +--R (1) --------------------------------- +--R 3 +--R 15a +--R Type: Union(Expression Integer,...) +--E + +--S 8 +bb:=(2*(3*a^2*x^2-4*a*b*x+8*b^2))/(15*a^3)*sqrt(a*x+b) +--R +--R +--R 2 2 2 +-------+ +--R (6a x - 8a b x + 16b )\|a x + b +--R (2) --------------------------------- +--R 3 +--R 15a +--R Type: Expression Integer +--E + +--S 9 14:86 Schaums and Axiom agree +cc:=aa-bb +--R +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.87~~~~~$\displaystyle +\int{\frac{dx}{x\sqrt{ax+b}}}$} +$$\int{\frac{1}{x\sqrt{ax+b}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{\sqrt{b}}~\ln +\left(\frac{\sqrt{ax+b}-\sqrt{b}}{\sqrt{ax+b}+\sqrt{b}}\right)\\ +\displaystyle +\frac{2}{\sqrt{-b}}~\tan^{-1}\sqrt{\frac{ax+b}{-b}} +\end{array} +\right. +$$ + +Note: the first answer assumes $b > 0$ and the second assumes $b < 0$. +<<*>>= +)clear all + +--S 10 +aa:=integrate(1/(x*sqrt(a*x+b)),x) +--R +--R +--R +-------+ +-+ +---+ +-------+ +--R - 2b\|a x + b + (a x + 2b)\|b \|- b \|a x + b +--R log(-------------------------------) 2atan(----------------) +--R x b +--R (1) [------------------------------------,- -----------------------] +--R +-+ +---+ +--R \|b \|- b +--R Type: Union(List Expression Integer,...) +--E +@ +Cleary Spiegel's first answer assumes $b > 0$: +<<*>>= +--S 11 +bb1:=1/sqrt(b)*log((sqrt(a*x+b)-sqrt(b))/(sqrt(a*x+b)+sqrt(b))) +--R +--R +--R +-------+ +-+ +--R \|a x + b - \|b +--R log(-----------------) +--R +-------+ +-+ +--R \|a x + b + \|b +--R (2) ---------------------- +--R +-+ +--R \|b +--R Type: Expression Integer +--E +@ +So we try the difference of the two results +<<*>>= +--S 12 +cc11:=aa.1-bb1 +--R +--R +-------+ +-+ +-------+ +-+ +--R \|a x + b - \|b - 2b\|a x + b + (a x + 2b)\|b +--R - log(-----------------) + log(-------------------------------) +--R +-------+ +-+ x +--R \|a x + b + \|b +--R (3) --------------------------------------------------------------- +--R +-+ +--R \|b +--R Type: Expression Integer +--E +@ +But the results don't simplify to 0. So we try some other tricks. + +Since both functions are of the form log(f(x))/sqrt(b) we extract +the f(x) from each. First we get the function from Axiom's first answer: +<<*>>= +--S 13 +ff:=exp(aa.1*sqrt(b)) +--R +--R +-------+ +-+ +--R - 2b\|a x + b + (a x + 2b)\|b +--R (4) ------------------------------- +--R x +--R Type: Expression Integer +--E +@ +and we get the same form from Spiegel's answer +<<*>>= +--S 14 +gg:=exp(bb1*sqrt(b)) +--R +--R +-------+ +-+ +--R \|a x + b - \|b +--R (5) ----------------- +--R +-------+ +-+ +--R \|a x + b + \|b +--R Type: Expression Integer +--E +@ +We can change Spiegel's form into Axiom's form because they differ by +the constant a*sqrt(b). To see this we multiply the numerator and +denominator by $1 == (sqrt(a*x+b) - sqrt(b))/(sqrt(a*x+b) - sqrt(b))$. + +First we multiply the numerator by $(sqrt(a*x+b) - sqrt(b))$ +<<*>>= +--S 15 +gg1:=gg*(sqrt(a*x+b) - sqrt(b)) +--R +--R +-+ +-------+ +--R - 2\|b \|a x + b + a x + 2b +--R (6) ---------------------------- +--R +-------+ +-+ +--R \|a x + b + \|b +--R Type: Expression Integer +--E +@ +Now we multiply the denominator by $(sqrt(a*x+b) - sqrt(b))$ +<<*>>= +--S 16 +gg2:=gg1/(sqrt(a*x+b) - sqrt(b)) +--R +--R +-+ +-------+ +--R - 2\|b \|a x + b + a x + 2b +--R (7) ---------------------------- +--R a x +--R Type: Expression Integer +--E +@ +and now we multiply by the integration constant $a*sqrt(b)$ +<<*>>= +--S 17 +gg3:=gg2*(a*sqrt(b)) +--R +--R +-------+ +-+ +--R - 2b\|a x + b + (a x + 2b)\|b +--R (8) ------------------------------- +--R x +--R Type: Expression Integer +--E +@ +and when we difference this with ff, the Axiom answer we get: +<<*>>= +--S 18 14:87a Schaums and Axiom differ by a constant +ff-gg3 +--R +--R (9) 0 +--R Type: Expression Integer +--E +@ +So the constant of integration difference is $a*sqrt(b)$ + +Now we look at the second equations. We difference Axiom's second answer +from Spiegel's answer: +<<*>>= +--S 19 +t1:=aa.2-bb1 +--R +--R +-------+ +-+ +---+ +-------+ +--R +---+ \|a x + b - \|b +-+ \|- b \|a x + b +--R - \|- b log(-----------------) - 2\|b atan(----------------) +--R +-------+ +-+ b +--R \|a x + b + \|b +--R (10) ------------------------------------------------------------ +--R +---+ +-+ +--R \|- b \|b +--R Type: Expression Integer +--E +@ +and again they do not simplify to zero. But we can show that both answers +differ by a constant because the derivative is zero: +<<*>>= +--S 20 +D(t1,x) +--R +--R (11) 0 +--R Type: Expression Integer +--E +@ + +Rather than find the constant this time we will differentiate both +answers and compare them with the original equation. +<<*>>= +--S 21 +target:=1/(x*sqrt(a*x+b)) +--R +--R 1 +--R (12) ----------- +--R +-------+ +--R x\|a x + b +--R Type: Expression Integer +--E +@ +and we select the second Axiom solution +<<*>>= +--S 22 +aa2:=aa.2 +--R +--R +---+ +-------+ +--R \|- b \|a x + b +--R 2atan(----------------) +--R b +--R (13) - ----------------------- +--R +---+ +--R \|- b +--R Type: Expression Integer +--E +@ +take its derivative +<<*>>= +--S 23 +ad2:=D(aa2,x) +--R +--R 1 +--R (14) ----------- +--R +-------+ +--R x\|a x + b +--R Type: Expression Integer +--E +@ +When we take the difference of Axiom's input and the derivative of the +output we see: +<<*>>= +--S 24 +ad2-target +--R +--R (15) 0 +--R Type: Expression Integer +--E +@ +Thus the original equation and Axiom's derivative of the integral are equal. + +Now we do the same with Spiegel's answer. We take the derivative of his +answer. +<<*>>= +--S 25 +ab1:=D(bb1,x) +--R +--R +-------+ +-+ +--R \|a x + b + \|b +--R (16) ---------------------------- +--R +-+ +-------+ 2 +--R x\|b \|a x + b + a x + b x +--R Type: Expression Integer +--E +@ +and we difference it from the original equation +<<*>>= +--S 26 14:87b Schaums and Axiom differ by a constant +ab1-target +--R +--R (17) 0 +--R Type: Expression Integer +--E +@ +Thus the original equation and Spiegel's derivative of the integral are equal. + +So we can conclude that both second answers are correct although they differ +by a constant of integration. + +\section{\cite{1}:14.88~~~~~$\displaystyle +\int{\frac{dx}{x^2\sqrt{ax+b}}}$} +$$\int{\frac{1}{x^2\sqrt{ax+b}}}= +-\frac{\sqrt{ax+b}}{bx}-\frac{a}{2b}~\int{\frac{1}{x\sqrt{ax+b}}} +$$ +<<*>>= +)clear all + +--S 27 +aa:=integrate(1/(x^2*sqrt(a*x+b)),x) +--R +--R +--R (1) +--R +-------+ +-+ +--R 2b\|a x + b + (a x + 2b)\|b +-+ +-------+ +--R a x log(-----------------------------) - 2\|b \|a x + b +--R x +--R [--------------------------------------------------------, +--R +-+ +--R 2b x\|b +--R +---+ +-------+ +--R \|- b \|a x + b +---+ +-------+ +--R a x atan(----------------) - \|- b \|a x + b +--R b +--R ---------------------------------------------] +--R +---+ +--R b x\|- b +--R Type: Union(List Expression Integer,...) +--E +@ + +In order to write down the book answer we need to first take the +integral which has two results +<<*>>= +--S 28 +dd:=integrate(1/(x*sqrt(a*x+b)),x) +--R +--R +--R +-------+ +-+ +---+ +-------+ +--R - 2b\|a x + b + (a x + 2b)\|b \|- b \|a x + b +--R log(-------------------------------) 2atan(----------------) +--R x b +--R (2) [------------------------------------,- -----------------------] +--R +-+ +---+ +--R \|b \|- b +--R Type: Union(List Expression Integer,...) +--E +@ +and derive two results for the book answer. The first result assumes +$b > 0$ +<<*>>= +--S 29 +bb1:=-sqrt(a*x+b)/(b*x)-a/(2*b)*dd.1 +--R +--R +--R +-------+ +-+ +--R - 2b\|a x + b + (a x + 2b)\|b +-+ +-------+ +--R - a x log(-------------------------------) - 2\|b \|a x + b +--R x +--R (3) ------------------------------------------------------------ +--R +-+ +--R 2b x\|b +--R Type: Expression Integer +--E +@ +and the second result assumes $b < 0$. +<<*>>= +--S 30 +bb2:=-sqrt(a*x+b)/(b*x)-a/(2*b)*dd.2 +--R +--R +--R +---+ +-------+ +--R \|- b \|a x + b +---+ +-------+ +--R a x atan(----------------) - \|- b \|a x + b +--R b +--R (4) --------------------------------------------- +--R +---+ +--R b x\|- b +--R Type: Expression Integer +--E +@ + +So we compute the difference of Axiom's first result with Spiegel's +first result +<<*>>= +--S 31 +cc11:=bb1-aa.1 +--R +--R (5) +--R +-------+ +-+ +--R 2b\|a x + b + (a x + 2b)\|b +--R - a log(-----------------------------) +--R x +--R + +--R +-------+ +-+ +--R - 2b\|a x + b + (a x + 2b)\|b +--R - a log(-------------------------------) +--R x +--R / +--R +-+ +--R 2b\|b +--R Type: Expression Integer +--E +@ +we compute its derivative +<<*>>= +--S 32 +D(cc11,x) +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ +and we can see that the answers differ by a constant, the constant of +integration. So Axiom's first answer should differentiate back to the target +equation. +<<*>>= +--S 33 +target:=1/(x^2*sqrt(a*x+b)) +--R +--R 1 +--R (7) ------------ +--R 2 +-------+ +--R x \|a x + b +--R Type: Expression Integer +--E +@ +We differentiate Axiom's first answer +<<*>>= +--S 34 +ad1:=D(aa.1,x) +--R +--R +-+ +-------+ 2 +--R (a x + 2b)\|b \|a x + b + 2a b x + 2b +--R (8) ---------------------------------------------------------- +--R 3 2 2 +-------+ 2 4 3 2 2 +-+ +--R (2a b x + 2b x )\|a x + b + (a x + 3a b x + 2b x )\|b +--R Type: Expression Integer +--E +@ +and subtract it from the target equation +<<*>>= +--S 35 +ad1-target +--R +--R (9) 0 +--R Type: Expression Integer +--E +@ +and now we do the same with first Spiegel's answer: +<<*>>= +--S 36 +bd1:=D(bb1,x) +--R +--R +-+ +-------+ 2 +--R (- a x - 2b)\|b \|a x + b + 2a b x + 2b +--R (10) ------------------------------------------------------------ +--R 3 2 2 +-------+ 2 4 3 2 2 +-+ +--R (2a b x + 2b x )\|a x + b + (- a x - 3a b x - 2b x )\|b +--R Type: Expression Integer +--E +@ +and we subtract it from the target +<<*>>= +--S 37 +bd1-target +--R +--R (11) 0 +--R Type: Expression Integer +--E +@ +so we know that the two first answers are both correct and that their +integrals differ by a constant. + +Now we look at the second answers. We difference the answers and can +see immediately that they are equal. +<<*>>= +--S 38 14:88 Schaums and Axiom differ by a constant +cc22:=bb2-aa.2 +--R +--R +--R (12) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.89~~~~~$\displaystyle +\int{\sqrt{ax+b}~dx}$} +$$\int{\sqrt{ax+b}}= +\frac{2\sqrt{(ax+b)^3}}{3a} +$$ +<<*>>= +)clear all + +--S 39 +aa:=integrate(sqrt(a*x+b),x) +--R +--R +--R +-------+ +--R (2a x + 2b)\|a x + b +--R (1) --------------------- +--R 3a +--R Type: Union(Expression Integer,...) +--E +@ +<<*>>= +--S 40 +bb:=(2*sqrt((a*x+b)^3))/(3*a) +--R +--R +--R +----------------------------+ +--R | 3 3 2 2 2 3 +--R 2\|a x + 3a b x + 3a b x + b +--R (2) -------------------------------- +--R 3a +--R Type: Expression Integer +--E +@ +<<*>>= +--S 41 +cc:=aa-bb +--R +--R +----------------------------+ +--R | 3 3 2 2 2 3 +-------+ +--R - 2\|a x + 3a b x + 3a b x + b + (2a x + 2b)\|a x + b +--R (3) ---------------------------------------------------------- +--R 3a +--R Type: Expression Integer +--E +@ +Since this didn't simplify we could check each answer using the derivative +<<*>>= +--S 42 +target:=sqrt(a*x+b) +--R +--R +-------+ +--R (4) \|a x + b +--R Type: Expression Integer +--E +@ +We take the derivative of Axiom's answer +<<*>>= +--S 43 +t1:=D(aa,x) +--R +--R a x + b +--R (5) ---------- +--R +-------+ +--R \|a x + b +--R Type: Expression Integer +--E +@ +And we subtract the target from the derivative of Axiom's answer +<<*>>= +--S 44 +t1-target +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ +So they are equal. Now we do the same with Spiegel's answer +<<*>>= +--S 45 +t2:=D(bb,x) +--R +--R 2 2 2 +--R a x + 2a b x + b +--R (7) ------------------------------- +--R +----------------------------+ +--R | 3 3 2 2 2 3 +--R \|a x + 3a b x + 3a b x + b +--R Type: Expression Integer +--E +@ +The numerator is +<<*>>= +--S 46 +nn:=(a*x+b)^2 +--R +--R 2 2 2 +--R (8) a x + 2a b x + b +--R Type: Polynomial Integer +--E +@ +<<*>>= +--S 47 +mm:=(a*x+b)^3 +--R +--R 3 3 2 2 2 3 +--R (9) a x + 3a b x + 3a b x + b +--R Type: Polynomial Integer +--E +@ +which expands to Spiegel's version. +<<*>>= +--S 48 14:89 Schaums and Axiom differ by a constant +result=nn/sqrt(mm) +--R +--R 2 2 2 +--R a x + 2a b x + b +--R (10) result= ------------------------------- +--R +----------------------------+ +--R | 3 3 2 2 2 3 +--R \|a x + 3a b x + 3a b x + b +--R Type: Equation Expression Integer +--E +@ +and this reduces to $\sqrt{ax+b}$ + +\section{\cite{1}:14.90~~~~~$\displaystyle +\int{x\sqrt{ax+b}~dx}$} +$$\int{x\sqrt{ax+b}}= +\frac{2(3ax-2b)}{15a^2}~\sqrt{(ax+b)^3} +$$ +<<*>>= +)clear all + +--S 49 +aa:=integrate(x*sqrt(a*x+b),x) +--R +--R +--R 2 2 2 +-------+ +--R (6a x + 2a b x - 4b )\|a x + b +--R (1) -------------------------------- +--R 2 +--R 15a +--R Type: Union(Expression Integer,...) +--E + +--S 50 +bb:=(2*(3*a*x-2*b))/(15*a^2)*sqrt((a*x+b)^3) +--R +--R +--R +----------------------------+ +--R | 3 3 2 2 2 3 +--R (6a x - 4b)\|a x + 3a b x + 3a b x + b +--R (2) ------------------------------------------ +--R 2 +--R 15a +--R Type: Expression Integer +--E + +--S 51 +cc:=aa-bb +--R +--R (3) +--R +----------------------------+ +--R | 3 3 2 2 2 3 +--R (- 6a x + 4b)\|a x + 3a b x + 3a b x + b +--R + +--R 2 2 2 +-------+ +--R (6a x + 2a b x - 4b )\|a x + b +--R / +--R 2 +--R 15a +--R Type: Expression Integer +--E + +--S 52 14:90 Schaums and Axiom agree +dd:=rootSimp cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.91~~~~~$\displaystyle +\int{x^2\sqrt{ax+b}~dx}$} +$$\int{x^2\sqrt{ax+b}}= +\frac{2(15a^2x^2-12abx+8b^2)}{105a^2}~\sqrt{(a+bx)^3} +$$ +Note: the sqrt term is almost certainly $\sqrt{(ax+b)}$ +<<*>>= +)clear all + +--S 53 +aa:=integrate(x^2*sqrt(a*x+b),x) +--R +--R +--R 3 3 2 2 2 3 +-------+ +--R (30a x + 6a b x - 8a b x + 16b )\|a x + b +--R (1) -------------------------------------------- +--R 3 +--R 105a +--R Type: Union(Expression Integer,...) +--E + +--S 54 +bb:=(2*(15*a^2*x^2-12*a*b*x+8*b^2))/(105*a^3)*sqrt((a+b*x)^3) +--R +--R +--R +----------------------------+ +--R 2 2 2 | 3 3 2 2 2 3 +--R (30a x - 24a b x + 16b )\|b x + 3a b x + 3a b x + a +--R (2) -------------------------------------------------------- +--R 3 +--R 105a +--R Type: Expression Integer +--E + +--S 55 14:91 Axiom cannot simplify this expression. Schaums typo? +cc:=aa-bb +--R +--R (3) +--R +----------------------------+ +--R 2 2 2 | 3 3 2 2 2 3 +--R (- 30a x + 24a b x - 16b )\|b x + 3a b x + 3a b x + a +--R + +--R 3 3 2 2 2 3 +-------+ +--R (30a x + 6a b x - 8a b x + 16b )\|a x + b +--R / +--R 3 +--R 105a +--R Type: Expression Integer +--E + +@ +Notice that if we factor the numerator of 'aa' we get an expression that +differs from schaums on by the order of the variables in the square root. +(We can square the term (a*x+b) and drag it under the square root to get +the cubic term). It appears that Schaums has a typo. +<<*>>= +--S 56 +factor numer aa +--R +--R 2 2 2 +-------+ +--R (4) 2(a x + b)(15a x - 12a b x + 8b )\|a x + b +--RType: Factored SparseMultivariatePolynomial(Integer,Kernel Expression Integer) +--E +@ + +\section{\cite{1}:14.92~~~~~$\displaystyle +\int{\frac{\sqrt{ax+b}}{x}~dx}$} +$$\int{\frac{\sqrt{ax+b}}{x}}= +2\sqrt{ax+b}+b~\int{\frac{1}{x\sqrt{ax+b}}} +$$ +<<*>>= +)clear all + +--S 57 +aa:=integrate(sqrt(a*x+b)/x,x) +--R +--R +--R (1) +--R +-+ +-------+ +--R +-+ - 2\|b \|a x + b + a x + 2b +-------+ +--R [\|b log(----------------------------) + 2\|a x + b , +--R x +--R +-------+ +--R +---+ \|a x + b +-------+ +--R - 2\|- b atan(----------) + 2\|a x + b ] +--R +---+ +--R \|- b +--R Type: Union(List Expression Integer,...) +--E + +--S 58 +dd:=integrate(1/(x*sqrt(a*x+b)),x) +--R +--R +--R +-------+ +-+ +---+ +-------+ +--R - 2b\|a x + b + (a x + 2b)\|b \|- b \|a x + b +--R log(-------------------------------) 2atan(----------------) +--R x b +--R (2) [------------------------------------,- -----------------------] +--R +-+ +---+ +--R \|b \|- b +--R Type: Union(List Expression Integer,...) +--E + +--S 59 +bb1:=2*sqrt(a*x+b)+b*dd.1 +--R +--R +--R +-------+ +-+ +--R - 2b\|a x + b + (a x + 2b)\|b +-+ +-------+ +--R b log(-------------------------------) + 2\|b \|a x + b +--R x +--R (3) -------------------------------------------------------- +--R +-+ +--R \|b +--R Type: Expression Integer +--E + +--S 60 +bb2:=2*sqrt(a*x+b)+b*dd.2 +--R +--R +--R +---+ +-------+ +--R \|- b \|a x + b +---+ +-------+ +--R - 2b atan(----------------) + 2\|- b \|a x + b +--R b +--R (4) ----------------------------------------------- +--R +---+ +--R \|- b +--R Type: Expression Integer +--E + +--S 61 +cc11:=bb1-aa.1 +--R +--R +--R (5) +--R +-------+ +-+ +-+ +-------+ +--R - 2b\|a x + b + (a x + 2b)\|b - 2\|b \|a x + b + a x + 2b +--R b log(-------------------------------) - b log(----------------------------) +--R x x +--R ---------------------------------------------------------------------------- +--R +-+ +--R \|b +--R Type: Expression Integer +--E + +--S 62 +cc12:=bb1-aa.2 +--R +--R +--R +-------+ +-+ +-------+ +--R - 2b\|a x + b + (a x + 2b)\|b +---+ +-+ \|a x + b +--R b log(-------------------------------) + 2\|- b \|b atan(----------) +--R x +---+ +--R \|- b +--R (6) -------------------------------------------------------------------- +--R +-+ +--R \|b +--R Type: Expression Integer +--E + +--S 63 +cc21:=bb2-aa.1 +--R +--R +--R (7) +--R +-+ +-------+ +---+ +-------+ +--R +---+ +-+ - 2\|b \|a x + b + a x + 2b \|- b \|a x + b +--R - \|- b \|b log(----------------------------) - 2b atan(----------------) +--R x b +--R ------------------------------------------------------------------------- +--R +---+ +--R \|- b +--R Type: Expression Integer +--E + +--S 64 +cc22:=bb2-aa.2 +--R +--R +--R +---+ +-------+ +-------+ +--R \|- b \|a x + b \|a x + b +--R - 2b atan(----------------) - 2b atan(----------) +--R b +---+ +--R \|- b +--R (8) ------------------------------------------------- +--R +---+ +--R \|- b +--R Type: Expression Integer +--E + +--S 65 14:92 Schaums and Axiom agree +dd22:=ratDenom cc22 +--R +--R (9) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.93~~~~~$\displaystyle +\int{\frac{\sqrt{ax+b}}{x^2}~dx}$} +$$\int{\frac{\sqrt{ax+b}}{x^2}}= +-\frac{\sqrt{ax+b}}{x}+\frac{a}{2}~\int{\frac{1}{x\sqrt{ax+b}}} +$$ +<<*>>= +)clear all + +--S 65 +aa:=integrate(sqrt(a*x+b)/x^2,x) +--R +--R +--R (1) +--R +-------+ +-+ +--R - 2b\|a x + b + (a x + 2b)\|b +-+ +-------+ +--R a x log(-------------------------------) - 2\|b \|a x + b +--R x +--R [----------------------------------------------------------, +--R +-+ +--R 2x\|b +--R +---+ +-------+ +--R \|- b \|a x + b +---+ +-------+ +--R - a x atan(----------------) - \|- b \|a x + b +--R b +--R -----------------------------------------------] +--R +---+ +--R x\|- b +--R Type: Union(List Expression Integer,...) +--E + +--S 66 +dd:=integrate(1/(x*sqrt(a*x+b)),x) +--R +--R +--R +-------+ +-+ +---+ +-------+ +--R - 2b\|a x + b + (a x + 2b)\|b \|- b \|a x + b +--R log(-------------------------------) 2atan(----------------) +--R x b +--R (2) [------------------------------------,- -----------------------] +--R +-+ +---+ +--R \|b \|- b +--R Type: Union(List Expression Integer,...) +--E + +--S 67 +bb1:=-sqrt(a*x+b)/x+a/2*dd.1 +--R +--R +--R +-------+ +-+ +--R - 2b\|a x + b + (a x + 2b)\|b +-+ +-------+ +--R a x log(-------------------------------) - 2\|b \|a x + b +--R x +--R (3) ---------------------------------------------------------- +--R +-+ +--R 2x\|b +--R Type: Expression Integer +--E + +--S 68 +bb2:=-sqrt(a*x+b)/x+a/2*dd.2 +--R +--R +--R +---+ +-------+ +--R \|- b \|a x + b +---+ +-------+ +--R - a x atan(----------------) - \|- b \|a x + b +--R b +--R (4) ----------------------------------------------- +--R +---+ +--R x\|- b +--R Type: Expression Integer +--E + +--S 69 +cc11:=bb1-aa.1 +--R +--R +--R (5) 0 +--R Type: Expression Integer +--E + +--S 70 +cc21:=bb-aa.1 +--R +--R +--R (6) +--R +-------+ +-+ +--R - 2b\|a x + b + (a x + 2b)\|b +-+ +-------+ +-+ +--R - a x log(-------------------------------) + 2\|b \|a x + b + 2bb x\|b +--R x +--R ------------------------------------------------------------------------ +--R +-+ +--R 2x\|b +--R Type: Expression Integer +--E + +--S 71 +cc12:=bb1-aa.2 +--R +--R +--R (7) +--R +-------+ +-+ +---+ +-------+ +--R +---+ - 2b\|a x + b + (a x + 2b)\|b +-+ \|- b \|a x + b +--R a\|- b log(-------------------------------) + 2a\|b atan(----------------) +--R x b +--R -------------------------------------------------------------------------- +--R +---+ +-+ +--R 2\|- b \|b +--R Type: Expression Integer +--E + +--S 72 14:93 Schaums and Axiom agree +cc22:=bb2-aa.2 +--R +--R +--R (8) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.94~~~~~$\displaystyle +\int{\frac{x^m}{\sqrt{ax+b}}~dx}$} +$$\int{\frac{x^m}{\sqrt{ax+b}}}= +\frac{2x^m\sqrt{ax+b}}{(2m+1)a}-\frac{2mb}{(2m+1)a} +~\int{\frac{x^{m-1}}{\sqrt{ax+b}}} +$$ +<<*>>= +)clear all + +--S 73 14:94 Axiom cannot do this integral +aa:=integrate(x^m/sqrt(a*x+b),x) +--R +--R +--R x m +--I ++ %L +--I (1) | ----------- d%L +--R ++ +--------+ +--I \|b + %L a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.95~~~~~$\displaystyle +\int{\frac{dx}{x^m\sqrt{ax+b}}}$} +$$\int{\frac{1}{x^m\sqrt{ax+b}}}= +-\frac{\sqrt{ax+b}}{(m-1)bx^{m-1}}-\frac{(2m-3)a}{(2m-2)b} +~\int{\frac{1}{x^{m-1}\sqrt{ax+b}}} +$$ +<<*>>= +)clear all + +--S 74 14:95 Axiom cannot do this integral +aa:=integrate(1/(x^m*sqrt(a*x+b)),x) +--R +--R +--R x +--R ++ 1 +--I (1) | -------------- d%L +--R ++ m +--------+ +--I %L \|b + %L a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.96~~~~~$\displaystyle +\int{x^m\sqrt{ax+b}~dx}$} +$$\int{x^m\sqrt{ax+b}}= +\frac{2x^m}{(2m+3)a}(ax+b)^{3/2} +-\frac{2mb}{(2m+3)a}~\int{x^{m-1}\sqrt{ax+b}} +$$ +<<*>>= +)clear all + +--S 75 14:96 Axiom cannot do this integral +aa:=integrate(x^m*sqrt(a*x+b),x) +--R +--R +--R x +--R ++ m +--------+ +--I (1) | %L \|b + %L a d%L +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.97~~~~~$\displaystyle +\int{\frac{\sqrt{ax+b}}{x^m}~dx}$} +$$\int{\frac{\sqrt{ax+b}}{x^m}}= +-\frac{\sqrt{ax+b}}{(m-1)x^{m-1}} ++\frac{a}{2(m-1)}~\int{\frac{1}{x^{m-1}\sqrt{ax+b}}} +$$ +<<*>>= +)clear all + +--S 76 14:97 Axiom cannot do this integral +aa:=integrate(sqrt(a*x+b)/x^m,x) +--R +--R +--R x +--------+ +--I ++ \|b + %L a +--I (1) | ----------- d%L +--R ++ m +--I %L +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.98~~~~~$\displaystyle +\int{\frac{\sqrt{ax+b}}{x^m}~dx}$} +$$\int{\frac{\sqrt{ax+b}}{x^m}}= +\frac{-(ax+b)^{3/2}}{(m-1)bx^{m-1}} +-\frac{(2m-5)a}{(2m-2)b}~\int{\frac{\sqrt{ax+b}}{x^{m-1}}} +$$ +Note: 14.98 is the same as 14.97 +<<*>>= +)clear all + +--S 77 14:98 Axiom cannot do this integral +aa:=integrate(sqrt(a*x+b)/x^m,x) +--R +--R +--R x +--------+ +--I ++ \|b + %L a +--I (1) | ----------- d%L +--R ++ m +--I %L +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.99~~~~~$\displaystyle +\int{(ax+b)^{m/2}~dx}$} +$$\int{(ax+b)^{m/2}}= +\frac{2(ax+b)^{(m+2)/2}}{a(m+2)} +$$ +<<*>>= +)clear all + +--S 78 +aa:=integrate((a*x+b)^(m/2),x) +--R +--R +--R m log(a x + b) +--R -------------- +--R 2 +--R (2a x + 2b)%e +--R (1) --------------------------- +--R a m + 2a +--R Type: Union(Expression Integer,...) +--E + +--S 79 +bb:=(2*(a*x+b)^((m+2)/2))/(a*(m+2)) +--R +--R +--R m + 2 +--R ----- +--R 2 +--R 2(a x + b) +--R (2) --------------- +--R a m + 2a +--R Type: Expression Integer +--E + +--S 80 +cc:=aa-bb +--R +--R +--R m log(a x + b) m + 2 +--R -------------- ----- +--R 2 2 +--R (2a x + 2b)%e - 2(a x + b) +--R (3) --------------------------------------------- +--R a m + 2a +--R Type: Expression Integer +--E + +--S 81 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 82 +dd:=explog cc +--R +--R m + 2 m +--R ----- - +--R 2 2 +--R - 2(a x + b) + (2a x + 2b)(a x + b) +--R (5) ----------------------------------------- +--R a m + 2a +--R Type: Expression Integer +--E + +--S 83 14:99 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.100~~~~~$\displaystyle +\int{x(ax+b)^{m/2}~dx}$} +$$\int{x(ax+b)^{m/2}}= +\frac{2(ax+b)^{(m+4)/2}}{a^2(m+4)} +-\frac{2b(ax+b)^{(m+2)/2}}{a^2(m+2)} +$$ +<<*>>= +)clear all + +--S 84 +aa:=integrate(x*(a*x+b)^(m/2),x) +--R +--R +--R m log(a x + b) +--R -------------- +--R 2 2 2 2 2 +--R ((2a m + 4a )x + 2a b m x - 4b )%e +--R (1) ------------------------------------------------- +--R 2 2 2 2 +--R a m + 6a m + 8a +--R Type: Union(Expression Integer,...) +--E + +--S 85 +bb:=(2*(a*x+b)^((m+4)/2))/(a^2*(m+4))-(2*b*(a*x+b)^((m+2)/2))/(a^2*(m+2)) +--R +--R +--R m + 4 m + 2 +--R ----- ----- +--R 2 2 +--R (2m + 4)(a x + b) + (- 2b m - 8b)(a x + b) +--R (2) ---------------------------------------------------- +--R 2 2 2 2 +--R a m + 6a m + 8a +--R Type: Expression Integer +--E + +--S 86 +cc:=aa-bb +--R +--R +--R (3) +--R m log(a x + b) +--R -------------- +--R 2 2 2 2 2 +--R ((2a m + 4a )x + 2a b m x - 4b )%e +--R + +--R m + 4 m + 2 +--R ----- ----- +--R 2 2 +--R (- 2m - 4)(a x + b) + (2b m + 8b)(a x + b) +--R / +--R 2 2 2 2 +--R a m + 6a m + 8a +--R Type: Expression Integer +--E + +--S 87 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 88 +dd:=explog cc +--R +--R (5) +--R m + 4 m + 2 +--R ----- ----- +--R 2 2 +--R (- 2m - 4)(a x + b) + (2b m + 8b)(a x + b) +--R + +--R m +--R - +--R 2 2 2 2 2 +--R ((2a m + 4a )x + 2a b m x - 4b )(a x + b) +--R / +--R 2 2 2 2 +--R a m + 6a m + 8a +--R Type: Expression Integer +--E + +--S 89 14:100 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.101~~~~~$\displaystyle +\int{x^2(ax+b)^{m/2}~dx}$} +$$\int{x^2(ax+b)^{m/2}}= +\frac{2(ax+b)^{(m+6)/2}}{a^3(m+6)} +-\frac{4b(ax+b)^{(m+4)/2}}{a^3(m+4)} ++\frac{2b^2(ax+b)^{(m+2)/2}}{a^3(m+2)} +$$ +<<*>>= +)clear all + +--S 90 +aa:=integrate(x^2*(a*x+b)^(m/2),x) +--R +--R +--R (1) +--R 3 2 3 3 3 2 2 2 2 2 3 +--R ((2a m + 12a m + 16a )x + (2a b m + 4a b m)x - 8a b m x + 16b ) +--R * +--R m log(a x + b) +--R -------------- +--R 2 +--R %e +--R / +--R 3 3 3 2 3 3 +--R a m + 12a m + 44a m + 48a +--R Type: Union(Expression Integer,...) +--E + +--S 91 +bb:=(2*(a*x+b)^((m+6)/2))/(a^3*(m+6))-_ + (4*b*(a*x+b)^((m+4)/2))/(a^3*(m+4))+_ + (2*b^2*(a*x+b)^((m+2)/2))/(a^3*(m+2)) +--R +--R +--R (2) +--R m + 6 m + 4 +--R ----- ----- +--R 2 2 2 2 +--R (2m + 12m + 16)(a x + b) + (- 4b m - 32b m - 48b)(a x + b) +--R + +--R m + 2 +--R ----- +--R 2 2 2 2 2 +--R (2b m + 20b m + 48b )(a x + b) +--R / +--R 3 3 3 2 3 3 +--R a m + 12a m + 44a m + 48a +--R Type: Expression Integer +--E + +--S 92 +cc:=aa-bb +--R +--R +--R (3) +--R 3 2 3 3 3 2 2 2 2 2 3 +--R ((2a m + 12a m + 16a )x + (2a b m + 4a b m)x - 8a b m x + 16b ) +--R * +--R m log(a x + b) +--R -------------- +--R 2 +--R %e +--R + +--R m + 6 m + 4 +--R ----- ----- +--R 2 2 2 2 +--R (- 2m - 12m - 16)(a x + b) + (4b m + 32b m + 48b)(a x + b) +--R + +--R m + 2 +--R ----- +--R 2 2 2 2 2 +--R (- 2b m - 20b m - 48b )(a x + b) +--R / +--R 3 3 3 2 3 3 +--R a m + 12a m + 44a m + 48a +--R Type: Expression Integer +--E + +--S 93 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 94 +dd:=explog cc +--R +--R (5) +--R m + 6 m + 4 +--R ----- ----- +--R 2 2 2 2 +--R (- 2m - 12m - 16)(a x + b) + (4b m + 32b m + 48b)(a x + b) +--R + +--R m + 2 +--R ----- +--R 2 2 2 2 2 +--R (- 2b m - 20b m - 48b )(a x + b) +--R + +--R 3 2 3 3 3 2 2 2 2 2 3 +--R ((2a m + 12a m + 16a )x + (2a b m + 4a b m)x - 8a b m x + 16b ) +--R * +--R m +--R - +--R 2 +--R (a x + b) +--R / +--R 3 3 3 2 3 3 +--R a m + 12a m + 44a m + 48a +--R Type: Expression Integer +--E + +--S 95 14:101 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.102~~~~~$\displaystyle +\int{\frac{(ax+b)^{m/2}}{x}~dx}$} +$$\int{\frac{(ax+b)^{m/2}}{x}}= +\frac{2(ax+b)^{m/2}}{m} ++b~\int{\frac{(ax+b)^{(m-2)/2}}{x}} +$$ +<<*>>= +)clear all + +--S 96 14:102 Axiom cannot do this integral +aa:=integrate((a*x+b)^(m/2)/x,x) +--R +--R +--R m +--R - +--R x 2 +--I ++ (b + %L a) +--I (1) | ----------- d%L +--I ++ %L +--R Type: Union(Expression Integer,...) +--E +@ +\section{\cite{1}:14.103~~~~~$\displaystyle +\int{\frac{(ax+b)^{m/2}}{x^2}~dx}$} +$$\int{\frac{(ax+b)^{m/2}}{x^2}}= +-\frac{(ax+b)^{(m+2)/2}}{bx} ++\frac{ma}{2b}~\int{\frac{(ax+b)^{m/2}}{x}} +$$ +<<*>>= +)clear all + +--S 97 14:103 Axiom cannot do this integral +aa:=integrate((a*x+b)^(m/2)/x^2,x) +--R +--R +--R m +--R - +--R x 2 +--I ++ (b + %L a) +--I (1) | ----------- d%L +--R ++ 2 +--I %L +--R Type: Union(Expression Integer,...) +--E +@ +\section{\cite{1}:14.104~~~~~$\displaystyle +\int{\frac{dx}{x(ax+b)^{m/2}}}$} +$$\int{\frac{1}{x(ax+b)^{m/2}}}= +\frac{2}{(m-2)b(ax+b)^{(m-2)/2}} ++\frac{1}{b}~\int{\frac{1}{x(ax+b)^{(m-2)/2}}} +$$ +<<*>>= +)clear all + +--S 98 14:104 Axiom cannot do this integral +aa:=integrate(1/(x*(a*x+b)^(m/2)),x) +--R +--R +--R x +--R ++ 1 +--I (1) | -------------- d%L +--R ++ m +--R - +--R 2 +--I %L (b + %L a) +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp61-62 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum2.input.pdf b/src/axiom-website/CATS/schaum2.input.pdf new file mode 100644 index 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+\int{\tan{ax}}~dx$} +$$\int{\tan{ax}}= +-\frac{1}{a}\ln~\cos{ax}= +\frac{1}{a}\ln~\sec{ax} +$$ +<<*>>= +)spool schaum20.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(tan(a*x),x) +--R +--R +--R 2 +--R log(tan(a x) + 1) +--R (1) ------------------ +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb1:=-1/a*log(cos(a*x)) +--R +--R log(cos(a x)) +--R (2) - ------------- +--R a +--R Type: Expression Integer +--E + +--S 3 +bb2:=1/a*log(sec(a*x)) +--R +--R log(sec(a x)) +--R (3) ------------- +--R a +--R Type: Expression Integer +--E + +--S 4 +cc1:=aa-bb1 +--R +--R 2 +--R log(tan(a x) + 1) + 2log(cos(a x)) +--R (4) ----------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 5 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (5) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 6 +dd1:=tanrule cc1 +--R +--R 2 2 +--R sin(a x) + cos(a x) +--R log(---------------------) + 2log(cos(a x)) +--R 2 +--R cos(a x) +--R (6) ------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 7 +ee1:=expandLog dd1 +--R +--R 2 2 +--R log(sin(a x) + cos(a x) ) +--R (7) -------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 8 +sincossqrrule:=rule(sin(a)^2+cos(a)^2 == 1) +--R +--R 2 2 +--I (8) sin(a) + cos(a) + %K == %K + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 9 14:429 Schaums and Axiom agree +ff1:=sincossqrrule ee1 +--R +--R (9) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.430~~~~~$\displaystyle +\int{\tan^2{ax}}~dx$} +$$\int{\tan^2{ax}}= +\frac{\tan{ax}}{x}-x +$$ +<<*>>= +)clear all + +--S 10 +aa:=integrate(tan(a*x)^2,x) +--R +--R +--R tan(a x) - a x +--R (1) -------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 11 +bb:=tan(a*x)/a-x +--R +--R tan(a x) - a x +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 12 14:430 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.431~~~~~$\displaystyle +\int{\tan^3{ax}}~dx$} +$$\int{\tan^3{ax}}= +\frac{\tan^2{ax}}{2a}+\frac{1}{a}\ln~\cos{ax} +$$ +<<*>>= +)clear all + +--S 13 +aa:=integrate(tan(a*x)^3,x) +--R +--R +--R 2 2 +--R - log(tan(a x) + 1) + tan(a x) +--R (1) -------------------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 14 +bb:=tan(a*x)^2/(2*a)+1/a*log(cos(a*x)) +--R +--R 2 +--R 2log(cos(a x)) + tan(a x) +--R (2) -------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 15 +cc:=aa-bb +--R +--R 2 +--R - log(tan(a x) + 1) - 2log(cos(a x)) +--R (3) ------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 16 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 17 +dd:=tanrule cc +--R +--R 2 2 +--R sin(a x) + cos(a x) +--R - log(---------------------) - 2log(cos(a x)) +--R 2 +--R cos(a x) +--R (5) --------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 18 +ee:=expandLog dd +--R +--R 2 2 +--R log(sin(a x) + cos(a x) ) +--R (6) - -------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 19 +sincossqrrule:=rule(sin(a)^2+cos(a)^2 == 1) +--R +--R 2 2 +--I (7) sin(a) + cos(a) + %L == %L + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 20 14:431 Schaums and Axiom agree +ff:=sincossqrrule ee +--R +--R (8) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.432~~~~~$\displaystyle +\int{\tan^n{ax}\sec^2{ax}}~dx$} +$$\int{\tan^n{ax}\sec^2{ax}}= +\frac{\tan^{n+1}{ax}}{(n+1)a} +$$ +<<*>>= +)clear all + +--S 21 +aa:=integrate(tan(a*x)^n*sec(a*x)^2,x) +--R +--R +--R sin(a x) +--R n log(--------) +--R cos(a x) +--R sin(a x)%e +--R (1) ------------------------- +--R (a n + a)cos(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 22 +bb:=tan(a*x)^(n+1)/((n+1)*a) +--R +--R n + 1 +--R tan(a x) +--R (2) ------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 23 +cc:=aa-bb +--R +--R sin(a x) +--R n log(--------) +--R cos(a x) n + 1 +--R sin(a x)%e - cos(a x)tan(a x) +--R (3) ------------------------------------------------- +--R (a n + a)cos(a x) +--R Type: Expression Integer +--E + +--S 24 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 25 +dd:=explog cc +--R +--R n + 1 sin(a x) n +--R - cos(a x)tan(a x) + sin(a x)(--------) +--R cos(a x) +--R (5) --------------------------------------------- +--R (a n + a)cos(a x) +--R Type: Expression Integer +--E + +--S 26 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (6) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 27 +ee:=tanrule dd +--R +--R sin(a x) n + 1 sin(a x) n +--R - cos(a x)(--------) + sin(a x)(--------) +--R cos(a x) cos(a x) +--R (7) ----------------------------------------------- +--R (a n + a)cos(a x) +--R Type: Expression Integer +--E + +--S 28 14:432 Schaums and Axiom agree +ff:=complexNormalize ee +--R +--R (8) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.433~~~~~$\displaystyle +\int{\frac{\sec^2{ax}}{\tan{ax}}}~dx$} +$$\int{\frac{\sec^2{ax}}{\tan{ax}}}= +\frac{1}{a}\ln~\tan{ax} +$$ +<<*>>= +)clear all + +--S 29 +aa:=integrate(sec(a*x)^2/tan(a*x),x) +--R +--R +--R sin(a x) 2cos(a x) +--R log(------------) - log(- ------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (1) --------------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 30 +bb:=1/a*log(tan(a*x)) +--R +--R log(tan(a x)) +--R (2) ------------- +--R a +--R Type: Expression Integer +--E + +--S 31 +cc:=aa-bb +--R +--R sin(a x) 2cos(a x) +--R - log(tan(a x)) + log(------------) - log(- ------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (3) --------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 32 +dd:=expandLog cc +--R +--R - log(tan(a x)) + log(sin(a x)) - log(cos(a x)) - log(- 2) +--R (4) ---------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 33 14:433 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R log(- 2) +--R (5) - -------- +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.434~~~~~$\displaystyle +\int{\frac{dx}{\tan{ax}}}~dx$} +$$\int{\frac{1}{\tan{ax}}}= +\frac{1}{a}\ln~\sin{ax} +$$ +<<*>>= +)clear all + +--S 34 +aa:=integrate(1/tan(a*x),x) +--R +--R +--R 2 +--R - log(tan(a x) + 1) + 2log(tan(a x)) +--R (1) ------------------------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 35 +bb:=1/a*log(sin(a*x)) +--R +--R log(sin(a x)) +--R (2) ------------- +--R a +--R Type: Expression Integer +--E + +--S 36 +cc:=aa-bb +--R +--R 2 +--R - log(tan(a x) + 1) + 2log(tan(a x)) - 2log(sin(a x)) +--R (3) ------------------------------------------------------ +--R 2a +--R Type: Expression Integer +--E + +--S 37 +complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.435~~~~~$\displaystyle +\int{x\tan{ax}}~dx$} +$$\int{x\tan{ax}}= +\frac{1}{a^2}\left\{\frac{(ax)^3}{3}+\frac{(ax)^5}{15}+\frac{2(ax)^7}{105} ++\cdots+\frac{2^{2n}(2^{2n}-1)B_n(ax)^{2n+1}}{(2n+1)!}+\cdots\right\} +$$ +<<*>>= +)clear all + +--S 38 14:435 Axiom cannot compute this integral +aa:=integrate(x*tan(a*x),x) +--R +--R +--R x +--R ++ +--I (1) | %I tan(%I a)d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.436~~~~~$\displaystyle +\int{\frac{\tan{ax}}{x}}~dx$} +$$\int{\frac{\tan{ax}}{x}}= +ax+\frac{(ax)^3}{9}+\frac{2(ax)^5}{75}+\cdots ++\frac{2^{2n}(2^{2n}-1)B_n(ax)^{2n-1}}{(2n-1)(2n)!}+\cdots +$$ +<<*>>= +)clear all + +--S 39 14:436 Axiom cannot compute this integral +aa:=integrate(tan(a*x)/x,x) +--R +--R +--R x +--I ++ tan(%I a) +--I (1) | --------- d%I +--I ++ %I +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.437~~~~~$\displaystyle +\int{x\tan^2{ax}}~dx$} +$$\int{x\tan^2{ax}}= +\frac{x\tan{ax}}{a}+\frac{1}{a^2}\ln~\cos{ax}-\frac{x^2}{2} +$$ +<<*>>= +)clear all + +--S 40 +aa:=integrate(x*tan(a*x)^2,x) +--R +--R +--R 2 2 2 +--R - log(tan(a x) + 1) + 2a x tan(a x) - a x +--R (1) ------------------------------------------- +--R 2 +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 41 +bb:=(x*tan(a*x))/a+1/a^2*log(cos(a*x))-x^2/2 +--R +--R 2 2 +--R 2log(cos(a x)) + 2a x tan(a x) - a x +--R (2) ------------------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 42 +cc:=aa-bb +--R +--R 2 +--R - log(tan(a x) + 1) - 2log(cos(a x)) +--R (3) ------------------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 43 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 44 +dd:=tanrule cc +--R +--R 2 2 +--R sin(a x) + cos(a x) +--R - log(---------------------) - 2log(cos(a x)) +--R 2 +--R cos(a x) +--R (5) --------------------------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 45 +ee:=expandLog dd +--R +--R 2 2 +--R log(sin(a x) + cos(a x) ) +--R (6) - -------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 46 +sincossqrrule:=rule(sin(a)^2+cos(a)^2 == 1) +--R +--R 2 2 +--I (7) sin(a) + cos(a) + %R == %R + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 47 14:437 Schaums and Axiom agree +ff:=sincossqrrule ee +--R +--R (8) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.438~~~~~$\displaystyle +\int{\frac{dx}{p+q\tan{ax}}}~dx$} +$$\int{\frac{1}{p+q\tan{ax}}}= +\frac{px}{p^2+q^2}+\frac{q}{a(p^2+q^2)}\ln(q\sin{ax}+p\cos{ax}) +$$ +<<*>>= +)clear all + +--S 48 +aa:=integrate(1/(p+q*tan(a*x)),x) +--R +--R +--R 2 +--R - q log(tan(a x) + 1) + 2q log(q tan(a x) + p) + 2a p x +--R (1) -------------------------------------------------------- +--R 2 2 +--R 2a q + 2a p +--R Type: Union(Expression Integer,...) +--E + +--S 49 +bb:=(p*x)/(p^2+q^2)+q/(a*(p^2+q^2))*log(q*sin(a*x)+p*cos(a*x)) +--R +--R q log(q sin(a x) + p cos(a x)) + a p x +--R (2) -------------------------------------- +--R 2 2 +--R a q + a p +--R Type: Expression Integer +--E + +--S 50 +cc:=aa-bb +--R +--R (3) +--R 2 +--R - q log(tan(a x) + 1) + 2q log(q tan(a x) + p) +--R + +--R - 2q log(q sin(a x) + p cos(a x)) +--R / +--R 2 2 +--R 2a q + 2a p +--R Type: Expression Integer +--E + +--S 51 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 52 +dd:=tanrule cc +--R +--R (5) +--R 2 2 +--R sin(a x) + cos(a x) +--R - q log(---------------------) - 2q log(q sin(a x) + p cos(a x)) +--R 2 +--R cos(a x) +--R + +--R q sin(a x) + p cos(a x) +--R 2q log(-----------------------) +--R cos(a x) +--R / +--R 2 2 +--R 2a q + 2a p +--R Type: Expression Integer +--E + +--S 53 +ee:=expandLog dd +--R +--R 2 2 +--R q log(sin(a x) + cos(a x) ) +--R (6) - ---------------------------- +--R 2 2 +--R 2a q + 2a p +--R Type: Expression Integer +--E + +--S 54 +sincossqrrule:=rule(sin(a)^2+cos(a)^2 == 1) +--R +--R 2 2 +--I (7) sin(a) + cos(a) + %S == %S + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 55 14:438 Schaums and Axiom agree +ff:=sincossqrrule ee +--R +--R (8) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.439~~~~~$\displaystyle +\int{\tan^n{ax}}~dx$} +$$\int{\tan^n{ax}}= +\frac{\tan^{n-1}{ax}}{(n-1)a}-\int{\tan^{n-2}{ax}} +$$ +<<*>>= +)clear all + +--S 56 14:439 Axiom cannot compute this integral +aa:=integrate(tan(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | tan(%I a) d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p80 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum20.input.pdf b/src/axiom-website/CATS/schaum20.input.pdf new file mode 100644 index 0000000..851a4ee --- /dev/null +++ b/src/axiom-website/CATS/schaum20.input.pdf @@ -0,0 +1,1939 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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+aa:=integrate(cot(a*x),x) +--R +--R +--R sin(2a x) 2 +--R 2log(-------------) - log(-------------) +--R cos(2a x) + 1 cos(2a x) + 1 +--R (1) ---------------------------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=1/a*log(sin(a*x)) +--R +--R log(sin(a x)) +--R (2) ------------- +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R sin(2a x) 2 +--R 2log(-------------) - 2log(sin(a x)) - log(-------------) +--R cos(2a x) + 1 cos(2a x) + 1 +--R (3) --------------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 4 +dd:=expandLog cc +--R +--R 2log(sin(2a x)) - 2log(sin(a x)) - log(cos(2a x) + 1) - log(2) +--R (4) -------------------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 5 14:440 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.441~~~~~$\displaystyle +\int{\cot^2{ax}}~dx$} +$$\int{\cot^2{ax}}= +-\frac{\cot{ax}}{a}-x +$$ +<<*>>= +)clear all + +--S 6 +aa:=integrate(cot(a*x)^2,x) +--R +--R +--R - a x sin(2a x) - cos(2a x) - 1 +--R (1) ------------------------------- +--R a sin(2a x) +--R Type: Union(Expression Integer,...) +--E + +--S 7 +bb:=-cot(a*x)/a-x +--R +--R - cot(a x) - a x +--R (2) ---------------- +--R a +--R Type: Expression Integer +--E + +--S 8 +cc:=aa-bb +--R +--R cot(a x)sin(2a x) - cos(2a x) - 1 +--R (3) --------------------------------- +--R a sin(2a x) +--R Type: Expression Integer +--E + +--S 9 +cotrule:=rule(cot(a) == cos(a)/sin(a)) +--R +--R cos(a) +--R (4) cot(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 10 +dd:=cotrule cc +--R +--R cos(a x)sin(2a x) + (- cos(2a x) - 1)sin(a x) +--R (5) --------------------------------------------- +--R a sin(a x)sin(2a x) +--R Type: Expression Integer +--E + +--S 11 14:441 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.442~~~~~$\displaystyle +\int{\cot^3{ax}}~dx$} +$$\int{\cot^3{ax}}= +-\frac{\cot^2{ax}}{2a}-\frac{1}{a}\ln\sin{ax} +$$ +<<*>>= +)clear all + +--S 12 +aa:=integrate(cot(a*x)^3,x) +--R +--R +--R (1) +--R sin(2a x) 2 +--R (- 2cos(2a x) + 2)log(-------------) + (cos(2a x) - 1)log(-------------) +--R cos(2a x) + 1 cos(2a x) + 1 +--R + +--R cos(2a x) + 1 +--R / +--R 2a cos(2a x) - 2a +--R Type: Union(Expression Integer,...) +--E + +--S 13 +bb:=-cot(a*x)^2/(2*a)-1/a*log(sin(a*x)) +--R +--R 2 +--R - 2log(sin(a x)) - cot(a x) +--R (2) ---------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 14 +cc:=aa-bb +--R +--R (3) +--R sin(2a x) +--R (- 2cos(2a x) + 2)log(-------------) + (2cos(2a x) - 2)log(sin(a x)) +--R cos(2a x) + 1 +--R + +--R 2 2 +--R (cos(2a x) - 1)log(-------------) + (cos(2a x) - 1)cot(a x) + cos(2a x) +--R cos(2a x) + 1 +--R + +--R 1 +--R / +--R 2a cos(2a x) - 2a +--R Type: Expression Integer +--E + +--S 15 +cotrule:=rule(cot(a) == cos(a)/sin(a)) +--R +--R cos(a) +--R (4) cot(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 16 +dd:=cotrule cc +--R +--R (5) +--R 2 sin(2a x) +--R (- 2cos(2a x) + 2)sin(a x) log(-------------) +--R cos(2a x) + 1 +--R + +--R 2 +--R (2cos(2a x) - 2)sin(a x) log(sin(a x)) +--R + +--R 2 2 2 +--R (cos(2a x) - 1)sin(a x) log(-------------) + (cos(2a x) + 1)sin(a x) +--R cos(2a x) + 1 +--R + +--R 2 2 +--R cos(a x) cos(2a x) - cos(a x) +--R / +--R 2 +--R (2a cos(2a x) - 2a)sin(a x) +--R Type: Expression Integer +--E + +--S 17 +ee:=expandLog dd +--R +--R (6) +--R 2 +--R (- 2cos(2a x) + 2)sin(a x) log(sin(2a x)) +--R + +--R 2 +--R (2cos(2a x) - 2)sin(a x) log(sin(a x)) +--R + +--R 2 +--R (cos(2a x) - 1)sin(a x) log(cos(2a x) + 1) +--R + +--R 2 2 +--R ((log(2) + 1)cos(2a x) - log(2) + 1)sin(a x) + cos(a x) cos(2a x) +--R + +--R 2 +--R - cos(a x) +--R / +--R 2 +--R (2a cos(2a x) - 2a)sin(a x) +--R Type: Expression Integer +--E + +--S 18 14:442 Schaums and Axiom agree +ff:=complexNormalize ee +--R +--R (7) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.443~~~~~$\displaystyle +\int{\cot^n{ax}\csc^2{ax}}~dx$} +$$\int{\cot^n{ax}\csc^2{ax}}= +-\frac{\cot^{n+1}{ax}}{(n+1)a} +$$ +<<*>>= +)clear all + +--S 19 +aa:=integrate(cot(a*x)^n*csc(a*x)^2,x) +--R +--R +--R cos(a x) +--R n log(--------) +--R sin(a x) +--R cos(a x)%e +--R (1) - ------------------------- +--R (a n + a)sin(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 20 +bb:=-cot(a*x)^(n+1)/((n+1)*a) +--R +--R n + 1 +--R cot(a x) +--R (2) - ------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 21 +cc:=aa-bb +--R +--R cos(a x) +--R n log(--------) +--R sin(a x) n + 1 +--R - cos(a x)%e + sin(a x)cot(a x) +--R (3) --------------------------------------------------- +--R (a n + a)sin(a x) +--R Type: Expression Integer +--E + +--S 22 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 23 +dd:=explog cc +--R +--R n + 1 cos(a x) n +--R sin(a x)cot(a x) - cos(a x)(--------) +--R sin(a x) +--R (5) ------------------------------------------- +--R (a n + a)sin(a x) +--R Type: Expression Integer +--E + +--S 24 +cotrule:=rule(cot(a) == cos(a)/sin(a)) +--R +--R cos(a) +--R (6) cot(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 25 +ee:=cotrule dd +--R +--R cos(a x) n + 1 cos(a x) n +--R sin(a x)(--------) - cos(a x)(--------) +--R sin(a x) sin(a x) +--R (7) --------------------------------------------- +--R (a n + a)sin(a x) +--R Type: Expression Integer +--E + +--S 26 14:443 Schaums and Axiom agree +ff:=complexNormalize ee +--R +--R (8) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.444~~~~~$\displaystyle +\int{\frac{\csc^2{ax}}{\cot{ax}}}~dx$} +$$\int{\frac{\csc^2{ax}}{\cot{ax}}}= +-\frac{1}{a}\ln\cot{ax} +$$ +<<*>>= +)clear all + +--S 27 +aa:=integrate(csc(a*x)^2/cot(a*x),x) +--R +--R +--R sin(a x) 2cos(a x) +--R log(------------) - log(- ------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (1) --------------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 28 +bb:=-1/a*log(cot(a*x)) +--R +--R log(cot(a x)) +--R (2) - ------------- +--R a +--R Type: Expression Integer +--E + +--S 29 +cc:=aa-bb +--R +--R sin(a x) 2cos(a x) +--R log(------------) + log(cot(a x)) - log(- ------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (3) ------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 30 +cotrule:=rule(cot(a) == cos(a)/sin(a)) +--R +--R cos(a) +--R (4) cot(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 31 +dd:=cotrule cc +--R +--R sin(a x) cos(a x) 2cos(a x) +--R log(------------) + log(--------) - log(- ------------) +--R cos(a x) + 1 sin(a x) cos(a x) + 1 +--R (5) ------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 32 14:444 Schaums and Axiom differ by a constant +ee:=expandLog dd +--R +--R log(- 2) +--R (6) - -------- +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.445~~~~~$\displaystyle +\int{\frac{dx}{\cot{ax}}}~dx$} +$$\int{\frac{1}{\cot{ax}}}= +-\frac{1}{a}\ln\cos{ax} +$$ +<<*>>= +)clear all + +--S 33 +aa:=integrate(1/cot(a*x),x) +--R +--R +--R 2 +--R log(-------------) +--R cos(2a x) + 1 +--R (1) ------------------ +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 34 +bb:=-1/a*log(cos(a*x)) +--R +--R log(cos(a x)) +--R (2) - ------------- +--R a +--R Type: Expression Integer +--E + +--S 35 +cc:=aa-bb +--R +--R 2 +--R 2log(cos(a x)) + log(-------------) +--R cos(2a x) + 1 +--R (3) ----------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 36 +dd:=expandLog cc +--R +--R - log(cos(2a x) + 1) + 2log(cos(a x)) + log(2) +--R (4) ---------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 37 14:445 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.446~~~~~$\displaystyle +\int{x\cot{ax}}~dx$} +$$\int{x\cot{ax}}= +\frac{1}{a^2}\left\{ax +-\frac{(ax)^3}{9}-\frac{(ax)^5}{225} +-\cdots-\frac{2^{2n}B_n(ax)^{2n+1}}{(2n+1)!}-\cdots\right\} +$$ +<<*>>= +)clear all + +--S 38 14:446 Axiom cannot compute this integral +aa:=integrate(x*cot(a*x),x) +--R +--R +--R x +--R ++ +--I (1) | %I cot(%I a)d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.447~~~~~$\displaystyle +\int{\frac{\cot{ax}}{x}}~dx$} +$$\int{\frac{\cot{ax}}{x}}= +-\frac{1}{ax}-\frac{ax}{3}-\frac{(ax)^3}{135}-\cdots +-\frac{2^{2n}B_n(ax)^{2n-1}}{(2n-1)(2n)!}-\cdots +$$ +<<*>>= +)clear all + +--S 39 14:447 Axiom cannot compute this integral +aa:=integrate(cot(a*x)/x,x) +--R +--R +--R x +--I ++ cot(%I a) +--I (1) | --------- d%I +--I ++ %I +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.448~~~~~$\displaystyle +\int{x\cot^2{ax}}~dx$} +$$\int{x\cot^2{ax}}= +-\frac{x\cot{ax}}{a}+\frac{1}{a^2}\ln\sin{ax}-\frac{x^2}{2} +$$ +<<*>>= +)clear all + +--S 40 +aa:=integrate(x*cot(a*x)^2,x) +--R +--R +--R (1) +--R sin(2a x) 2 +--R 2sin(2a x)log(-------------) - sin(2a x)log(-------------) +--R cos(2a x) + 1 cos(2a x) + 1 +--R + +--R 2 2 +--R - a x sin(2a x) - 2a x cos(2a x) - 2a x +--R / +--R 2 +--R 2a sin(2a x) +--R Type: Union(Expression Integer,...) +--E + +--S 41 +bb:=-(x*cot(a*x))/a+1/a^2*log(sin(a*x))-x^2/2 +--R +--R 2 2 +--R 2log(sin(a x)) - 2a x cot(a x) - a x +--R (2) ------------------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 42 +cc:=aa-bb +--R +--R (3) +--R sin(2a x) +--R 2sin(2a x)log(-------------) - 2sin(2a x)log(sin(a x)) +--R cos(2a x) + 1 +--R + +--R 2 +--R - sin(2a x)log(-------------) + 2a x cot(a x)sin(2a x) - 2a x cos(2a x) +--R cos(2a x) + 1 +--R + +--R - 2a x +--R / +--R 2 +--R 2a sin(2a x) +--R Type: Expression Integer +--E + +--S 43 +dd:=expandLog cc +--R +--R (4) +--R 2sin(2a x)log(sin(2a x)) - 2sin(2a x)log(sin(a x)) +--R + +--R - sin(2a x)log(cos(2a x) + 1) + (2a x cot(a x) - log(2))sin(2a x) +--R + +--R - 2a x cos(2a x) - 2a x +--R / +--R 2 +--R 2a sin(2a x) +--R Type: Expression Integer +--E + +--S 44 14:448 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.449~~~~~$\displaystyle +\int{\frac{dx}{p+q\cot{ax}}}~dx$} +$$\int{\frac{1}{p+q\cot{ax}}}= +\frac{px}{p^2+q^2}-\frac{q}{a(p^2+q^2)}\ln(p\sin{ax}+q\cos{ax}) +$$ +<<*>>= +)clear all + +--S 45 +aa:=integrate(1/(p+q*cot(a*x)),x) +--R +--R +--R (1) +--R p sin(2a x) + q cos(2a x) + q 2 +--R - 2q log(-----------------------------) + q log(-------------) + 2a p x +--R cos(2a x) + 1 cos(2a x) + 1 +--R ----------------------------------------------------------------------- +--R 2 2 +--R 2a q + 2a p +--R Type: Union(Expression Integer,...) +--E + +--S 46 +bb:=(p*x)/(p^2+q^2)-q/(a*(p^2+q^2))*log(p*sin(a*x)+q*cos(a*x)) +--R +--R - q log(p sin(a x) + q cos(a x)) + a p x +--R (2) ---------------------------------------- +--R 2 2 +--R a q + a p +--R Type: Expression Integer +--E + +--S 47 +cc:=aa-bb +--R +--R (3) +--R p sin(2a x) + q cos(2a x) + q +--R - 2q log(-----------------------------) + 2q log(p sin(a x) + q cos(a x)) +--R cos(2a x) + 1 +--R + +--R 2 +--R q log(-------------) +--R cos(2a x) + 1 +--R / +--R 2 2 +--R 2a q + 2a p +--R Type: Expression Integer +--E + +--S 48 +sindblrule:=rule(sin(2*a) == 2*sin(a)*cos(a)) +--R +--R (4) sin(2a) == 2cos(a)sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 49 +dd:=sindblrule cc +--R +--R (5) +--R 2q log(p sin(a x) + q cos(a x)) +--R + +--R 2p cos(a x)sin(a x) + q cos(2a x) + q 2 +--R - 2q log(-------------------------------------) + q log(-------------) +--R cos(2a x) + 1 cos(2a x) + 1 +--R / +--R 2 2 +--R 2a q + 2a p +--R Type: Expression Integer +--E + +--S 50 +cosdblrule:=rule(cos(2*a) == 2*cos(a)^2-1) +--R +--R 2 +--R (6) cos(2a) == 2cos(a) - 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 51 +ee:=cosdblrule dd +--R +--R (7) +--R p sin(a x) + q cos(a x) +--R 2q log(p sin(a x) + q cos(a x)) - 2q log(-----------------------) +--R cos(a x) +--R + +--R 1 +--R q log(---------) +--R 2 +--R cos(a x) +--R / +--R 2 2 +--R 2a q + 2a p +--R Type: Expression Integer +--E + +--S 52 14:449 Schaums and Axiom agree +ff:=expandLog % +--R +--R (8) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.450~~~~~$\displaystyle +\int{\cot^n{ax}}~dx$} +$$\int{\cot^n{ax}}= +-\frac{\cot^{n-1}{ax}}{(n-1)a}-\int{\cos^{n-2}{ax}} +$$ +<<*>>= +)clear all + +--S 53 14:450 Axiom cannot compute this integral +aa:=integrate(cot(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | cot(%I a) d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p81 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum21.input.pdf b/src/axiom-website/CATS/schaum21.input.pdf new file mode 100644 index 0000000..63365fe --- /dev/null +++ b/src/axiom-website/CATS/schaum21.input.pdf @@ -0,0 +1,2068 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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+aa:=integrate(sec(a*x),x) +--R +--R +--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1 +--R log(-----------------------) - log(-----------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (1) ----------------------------------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb1:=1/a*log(sec(a*x)+tan(a*x)) +--R +--R log(tan(a x) + sec(a x)) +--R (2) ------------------------ +--R a +--R Type: Expression Integer +--E + +--S 3 +bb2:=1/a*log(tan((a*x)/2+%pi/4)) +--R +--R 2a x + %pi +--R log(tan(----------)) +--R 4 +--R (3) -------------------- +--R a +--R Type: Expression Integer +--E + +--S 4 +cc1:=aa-bb1 +--R +--R (4) +--R sin(a x) + cos(a x) + 1 +--R - log(tan(a x) + sec(a x)) + log(-----------------------) +--R cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R a +--R Type: Expression Integer +--E + +--S 5 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (5) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 6 +dd1:=tanrule cc1 +--R +--R (6) +--R sin(a x) + cos(a x)sec(a x) sin(a x) + cos(a x) + 1 +--R - log(---------------------------) + log(-----------------------) +--R cos(a x) cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R a +--R Type: Expression Integer +--E + +--S 7 +secrule:=rule(sec(a) == 1/cos(a)) +--R +--R 1 +--R (7) sec(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 8 +ee1:=secrule dd1 +--R +--R (8) +--R sin(a x) + 1 sin(a x) + cos(a x) + 1 +--R - log(------------) + log(-----------------------) +--R cos(a x) cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R a +--R Type: Expression Integer +--E + +--S 9 +ff1:=expandLog ee1 +--R +--R (9) +--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) + 1) +--R + +--R - log(sin(a x) - cos(a x) - 1) + log(cos(a x)) +--R / +--R a +--R Type: Expression Integer +--E + +--S 10 +gg1:=complexNormalize ff1 +--R +--R log(- 1) +--R (10) -------- +--R a +--R Type: Expression Integer +--E + +--S 11 +cc2:=aa-bb2 +--R +--R (11) +--R 2a x + %pi sin(a x) + cos(a x) + 1 +--R - log(tan(----------)) + log(-----------------------) +--R 4 cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R a +--R Type: Expression Integer +--E + +--S 12 +dd2:=tanrule cc2 +--R +--R (12) +--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1 +--R log(-----------------------) - log(-----------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R + +--R 2a x + %pi +--R sin(----------) +--R 4 +--R - log(---------------) +--R 2a x + %pi +--R cos(----------) +--R 4 +--R / +--R a +--R Type: Expression Integer +--E + +--S 13 +ee2:=expandLog dd2 +--R +--R (13) +--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) - cos(a x) - 1) +--R + +--R 2a x + %pi 2a x + %pi +--R - log(sin(----------)) + log(cos(----------)) +--R 4 4 +--R / +--R a +--R Type: Expression Integer +--E + +--S 14 14:451 Schaums and Axiom differ by a constant +ff2:=complexNormalize ee2 +--R +--R log(- 1) +--R (14) -------- +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.452~~~~~$\displaystyle +\int{\sec^2{ax}}~dx$} +$$\int{\sec^2{ax}}= +\frac{\tan{ax}}{a} +$$ +<<*>>= +)clear all + +--S 15 +aa:=integrate(sec(a*x)^2,x) +--R +--R +--R sin(a x) +--R (1) ---------- +--R a cos(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 16 +bb:=tan(a*x)/a +--R +--R tan(a x) +--R (2) -------- +--R a +--R Type: Expression Integer +--E + +--S 17 +cc:=aa-bb +--R +--R - cos(a x)tan(a x) + sin(a x) +--R (3) ----------------------------- +--R a cos(a x) +--R Type: Expression Integer +--E + +--S 18 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 19 14:452 Schaums and Axiom agree +dd:=tanrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.453~~~~~$\displaystyle +\int{\sec^3{ax}}~dx$} +$$\int{\sec^3{ax}}= +\frac{\sec{ax}\tan{ax}}{2a}+\frac{1}{2a}\ln(\sec{ax}+\tan{ax}) +$$ +<<*>>= +)clear all + +--S 20 +aa:=integrate(sec(a*x)^3,x) +--R +--R +--R (1) +--R 2 sin(a x) + cos(a x) + 1 +--R cos(a x) log(-----------------------) +--R cos(a x) + 1 +--R + +--R 2 sin(a x) - cos(a x) - 1 +--R - cos(a x) log(-----------------------) + sin(a x) +--R cos(a x) + 1 +--R / +--R 2 +--R 2a cos(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 21 +bb:=(sec(a*x)*tan(a*x))/(2*a)+1/(2*a)*log(sec(a*x)+tan(a*x)) +--R +--R log(tan(a x) + sec(a x)) + sec(a x)tan(a x) +--R (2) ------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 22 +cc:=aa-bb +--R +--R (3) +--R 2 +--R - cos(a x) log(tan(a x) + sec(a x)) +--R + +--R 2 sin(a x) + cos(a x) + 1 +--R cos(a x) log(-----------------------) +--R cos(a x) + 1 +--R + +--R 2 sin(a x) - cos(a x) - 1 2 +--R - cos(a x) log(-----------------------) - cos(a x) sec(a x)tan(a x) +--R cos(a x) + 1 +--R + +--R sin(a x) +--R / +--R 2 +--R 2a cos(a x) +--R Type: Expression Integer +--E + +--S 23 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 24 +dd:=tanrule cc +--R +--R (5) +--R 2 sin(a x) + cos(a x)sec(a x) +--R - cos(a x) log(---------------------------) +--R cos(a x) +--R + +--R 2 sin(a x) + cos(a x) + 1 +--R cos(a x) log(-----------------------) +--R cos(a x) + 1 +--R + +--R 2 sin(a x) - cos(a x) - 1 +--R - cos(a x) log(-----------------------) + (- cos(a x)sec(a x) + 1)sin(a x) +--R cos(a x) + 1 +--R / +--R 2 +--R 2a cos(a x) +--R Type: Expression Integer +--E + +--S 25 +secrule:=rule(sec(a) == 1/cos(a)) +--R +--R 1 +--R (6) sec(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 26 +ee:=secrule dd +--R +--R (7) +--R sin(a x) + 1 sin(a x) + cos(a x) + 1 +--R - log(------------) + log(-----------------------) +--R cos(a x) cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 27 +ff:=expandLog ee +--R +--R (8) +--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) + 1) +--R + +--R - log(sin(a x) - cos(a x) - 1) + log(cos(a x)) +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 28 14:453 Schaums and Axiom differ by a constant +gg:=complexNormalize ff +--R +--R log(- 1) +--R (9) -------- +--R 2a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.454~~~~~$\displaystyle +\int{\sec^n{ax}\tan{ax}}~dx$} +$$\int{\sec^n{ax}\tan{ax}}= +\frac{\sec^n{ax}}{na} +$$ +<<*>>= +)clear all + +--S 29 +aa:=integrate(sec(a*x)^n*tan(a*x),x) +--R +--R 1 +--R n log(---------) +--R 2 +--R cos(a x) +--R ---------------- +--R 2 +--R %e +--R (1) ------------------ +--R a n +--R Type: Union(Expression Integer,...) +--E + +--S 30 +bb:=sec(a*x)^n/(n*a) +--R +--R n +--R sec(a x) +--R (2) --------- +--R a n +--R Type: Expression Integer +--E + +--S 31 +cc:=aa-bb +--R +--R 1 +--R n log(---------) +--R 2 +--R cos(a x) +--R ---------------- +--R 2 n +--R %e - sec(a x) +--R (3) ------------------------------ +--R a n +--R Type: Expression Integer +--E + +--S 32 14:454 Schaums and Axiom agree +normalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.455~~~~~$\displaystyle +\int{\frac{dx}{\sec{ax}}}~dx$} +$$\int{\frac{1}{\sec{ax}}}= +\frac{\sin{ax}}{a} +$$ +<<*>>= +)clear all + +--S 33 +aa:=integrate(1/sec(a*x),x) +--R +--R +--R sin(a x) +--R (1) -------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 34 +bb:=sin(a*x)/a +--R +--R sin(a x) +--R (2) -------- +--R a +--R Type: Expression Integer +--E + +--S 35 14:455 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.456~~~~~$\displaystyle +\int{x\sec{ax}}~dx$} +$$\int{x\sec{ax}}= +\frac{1}{a^2}\left\{\frac{(ax)^2}{2}+\frac{(ax)^4}{8}+\frac{5(ax)^6}{144} ++\cdots+\frac{E_n(ax)^{2n+2}}{(2n+2)(2n)!}+\cdots\right\} +$$ +<<*>>= +)clear all + +--S 36 14:456 Axiom cannot compute this integral +aa:=integrate(x*sec(a*x),x) +--R +--R +--R x +--R ++ +--I (1) | %N sec(%N a)d%N +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.457~~~~~$\displaystyle +\int{\frac{\sec{ax}}{x}}~dx$} +$$\int{\frac{\sec{ax}}{x}}= +\ln{x}+\frac{(ax)^2}{4}+\frac{5(ax)^4}{96}+\frac{61(ax)^6}{4320} ++\cdots+\frac{E_n(ax)^{2n}}{(2n)(2n)!}+\cdots +$$ +<<*>>= +)clear all + +--S 37 14:457 Axiom cannot compute this integral +aa:=integrate(sec(a*x)/x,x) +--R +--R +--R x +--I ++ sec(%N a) +--I (1) | --------- d%N +--I ++ %N +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.458~~~~~$\displaystyle +\int{x\sec^2{ax}}~dx$} +$$\int{x\sec^2{ax}}= +\frac{x}{a}\tan{ax}+\frac{1}{a^2}\ln\cos{ax} +$$ +<<*>>= +)clear all + +--S 38 +aa:=integrate(x*sec(a*x)^2,x) +--R +--R +--R (1) +--R 2 2cos(a x) +--R - cos(a x)log(------------) + cos(a x)log(- ------------) + a x sin(a x) +--R cos(a x) + 1 cos(a x) + 1 +--R ------------------------------------------------------------------------ +--R 2 +--R a cos(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 39 +bb:=x/a*tan(a*x)+1/a^2*log(cos(a*x)) +--R +--R log(cos(a x)) + a x tan(a x) +--R (2) ---------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 40 +cc:=aa-bb +--R +--R (3) +--R 2 +--R - cos(a x)log(cos(a x)) - cos(a x)log(------------) +--R cos(a x) + 1 +--R + +--R 2cos(a x) +--R cos(a x)log(- ------------) - a x cos(a x)tan(a x) + a x sin(a x) +--R cos(a x) + 1 +--R / +--R 2 +--R a cos(a x) +--R Type: Expression Integer +--E + +--S 41 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 42 +dd:=tanrule cc +--R +--R 2 2cos(a x) +--R - log(cos(a x)) - log(------------) + log(- ------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (5) --------------------------------------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 43 14:458 Schaums and Axiom differ by a constant +ee:=expandLog dd +--R +--R - log(2) + log(- 2) +--R (6) ------------------- +--R 2 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.459~~~~~$\displaystyle +\int{\frac{dx}{q+p\sec{ax}}}~dx$} +$$\int{\frac{1}{q+p\sec{ax}}}= +\frac{x}{q}-\frac{p}{q}\int{\frac{dx}{p+q\cos{ax}}} +$$ +<<*>>= +)clear all + +--S 44 +aa:=integrate(1/(q+p*sec(a*x)),x) +--R +--R +--R (1) +--R +-------+ +--R | 2 2 2 2 +-------+ +--R (- p cos(a x) - q)\|q - p + (q - p )sin(a x) | 2 2 +--R p log(------------------------------------------------) + a x\|q - p +--R q cos(a x) + p +--R [-----------------------------------------------------------------------, +--R +-------+ +--R | 2 2 +--R a q\|q - p +--R +---------+ +--R | 2 2 +---------+ +--R sin(a x)\|- q + p | 2 2 +--R - 2p atan(-----------------------) + a x\|- q + p +--R (q + p)cos(a x) + q + p +--R ----------------------------------------------------] +--R +---------+ +--R | 2 2 +--R a q\|- q + p +--R Type: Union(List Expression Integer,...) +--E + +--S 45 +t1:=integrate(1/(p+q*cos(a*x)),x) +--R +--R (2) +--R +-------+ +--R | 2 2 2 2 +--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) +--R log(--------------------------------------------------) +--R q cos(a x) + p +--R [-------------------------------------------------------, +--R +-------+ +--R | 2 2 +--R a\|q - p +--R +---------+ +--R | 2 2 +--R sin(a x)\|- q + p +--R 2atan(-----------------------) +--R (q + p)cos(a x) + q + p +--R ------------------------------] +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Union(List Expression Integer,...) +--E + +--S 46 +bb1:=x/q-p/q*t1.1 +--R +--R (3) +--R +-------+ +--R | 2 2 2 2 +-------+ +--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) | 2 2 +--R - p log(--------------------------------------------------) + a x\|q - p +--R q cos(a x) + p +--R --------------------------------------------------------------------------- +--R +-------+ +--R | 2 2 +--R a q\|q - p +--R Type: Expression Integer +--E + +--S 47 +bb2:=x/q-p/q*t1.2 +--R +--R +---------+ +--R | 2 2 +---------+ +--R sin(a x)\|- q + p | 2 2 +--R - 2p atan(-----------------------) + a x\|- q + p +--R (q + p)cos(a x) + q + p +--R (4) ---------------------------------------------------- +--R +---------+ +--R | 2 2 +--R a q\|- q + p +--R Type: Expression Integer +--E + +--S 48 +cc1:=aa.1-bb1 +--R +--R (5) +--R +-------+ +--R | 2 2 2 2 +--R (- p cos(a x) - q)\|q - p + (q - p )sin(a x) +--R p log(------------------------------------------------) +--R q cos(a x) + p +--R + +--R +-------+ +--R | 2 2 2 2 +--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) +--R p log(--------------------------------------------------) +--R q cos(a x) + p +--R / +--R +-------+ +--R | 2 2 +--R a q\|q - p +--R Type: Expression Integer +--E + +--S 49 +cc2:=aa.1-bb2 +--R +--R (6) +--R +-------+ +--R +---------+ | 2 2 2 2 +--R | 2 2 (- p cos(a x) - q)\|q - p + (q - p )sin(a x) +--R p\|- q + p log(------------------------------------------------) +--R q cos(a x) + p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 sin(a x)\|- q + p +--R 2p\|q - p atan(-----------------------) +--R (q + p)cos(a x) + q + p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a q\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 50 +cc3:=aa.2-bb1 +--R +--R (7) +--R +-------+ +--R +---------+ | 2 2 2 2 +--R | 2 2 (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) +--R p\|- q + p log(--------------------------------------------------) +--R q cos(a x) + p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 sin(a x)\|- q + p +--R - 2p\|q - p atan(-----------------------) +--R (q + p)cos(a x) + q + p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a q\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 51 14:459 Schaums and Axiom agree +cc4:=aa.2-bb2 +--R +--R (8) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.460~~~~~$\displaystyle +\int{\sec^n{ax}}~dx$} +$$\int{\sec^n{ax}}= +\frac{\sec^{n-2}{ax}\tan{ax}}{a(n-1)} ++\frac{n-2}{n-1}\int{\sec^{n-2}{ax}} +$$ +<<*>>= +)clear all + +--S 52 14:460 Axiom cannot compute this integral +aa:=integrate(sec(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | sec(%N a) d%N +--R ++ +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp81-82 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum22.input.pdf b/src/axiom-website/CATS/schaum22.input.pdf 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+\eject +\section{\cite{1}:14.461~~~~~$\displaystyle +\int{\csc{ax}}~dx$} +$$\int{\csc{ax}}= +\frac{1}{a}\ln(\csc{ax}-\cot{ax})= +\frac{1}{a}\ln\tan{\frac{ax}{2}} +$$ +<<*>>= +)spool schaum23.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(csc(a*x),x) +--R +--R +--R sin(a x) +--R log(------------) +--R cos(a x) + 1 +--R (1) ----------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb1:=1/a*log(csc(a*x)-cot(a*x)) +--R +--R log(csc(a x) - cot(a x)) +--R (2) ------------------------ +--R a +--R Type: Expression Integer +--E + +--S 3 +bb2:=1/a*log(tan((a*x)/2)) +--R +--R a x +--R log(tan(---)) +--R 2 +--R (3) ------------- +--R a +--R Type: Expression Integer +--E + +--S 4 +cc1:=aa-bb1 +--R +--R sin(a x) +--R log(------------) - log(csc(a x) - cot(a x)) +--R cos(a x) + 1 +--R (4) -------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 5 +cotrule:=rule(cot(a) == cos(a)/sin(a)) +--R +--R cos(a) +--R (5) cot(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 6 +dd1:=cotrule cc1 +--R +--R sin(a x) csc(a x)sin(a x) - cos(a x) +--R log(------------) - log(---------------------------) +--R cos(a x) + 1 sin(a x) +--R (6) ---------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 7 +cscrule:=rule(csc(a) == 1/sin(a)) +--R +--R 1 +--R (7) csc(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 8 +ee1:=cscrule dd1 +--R +--R sin(a x) - cos(a x) + 1 +--R log(------------) - log(--------------) +--R cos(a x) + 1 sin(a x) +--R (8) --------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 9 +ff1:=expandLog ee1 +--R +--R 2log(sin(a x)) - log(cos(a x) + 1) - log(cos(a x) - 1) - log(- 1) +--R (9) ----------------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 10 +gg1:=complexNormalize ff1 +--R +--R 2log(- 1) +--R (10) - --------- +--R a +--R Type: Expression Integer +--E + +--S 11 +cc2:=aa-bb2 +--R +--R a x sin(a x) +--R - log(tan(---)) + log(------------) +--R 2 cos(a x) + 1 +--R (11) ----------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 12 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (12) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 13 +dd2:=tanrule cc2 +--R +--R a x +--R sin(---) +--R sin(a x) 2 +--R log(------------) - log(--------) +--R cos(a x) + 1 a x +--R cos(---) +--R 2 +--R (13) --------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 14 +ee2:=expandLog dd2 +--R +--R a x a x +--R log(sin(a x)) - log(sin(---)) - log(cos(a x) + 1) + log(cos(---)) +--R 2 2 +--R (14) ----------------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 15 14:461 Schaums and Axiom agree +ff2:=complexNormalize ee2 +--R +--R (15) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.462~~~~~$\displaystyle +\int{\csc^2{ax}}~dx$} +$$\int{\csc^2{ax}}= +-\frac{\cot{ax}}{a} +$$ +<<*>>= +)clear all + +--S 16 +aa:=integrate(csc(a*x)^2,x) +--R +--R +--R cos(a x) +--R (1) - ---------- +--R a sin(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 17 +bb:=-cot(a*x)/a +--R +--R cot(a x) +--R (2) - -------- +--R a +--R Type: Expression Integer +--E + +--S 18 +cc:=aa-bb +--R +--R cot(a x)sin(a x) - cos(a x) +--R (3) --------------------------- +--R a sin(a x) +--R Type: Expression Integer +--E + +--S 19 +cotrule:=rule(cot(a) == cos(a)/sin(a)) +--R +--R cos(a) +--R (4) cot(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 20 14:462 Schaums and Axiom agree +dd:=cotrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.463~~~~~$\displaystyle +\int{\csc^3{ax}}~dx$} +$$\int{\csc^3{ax}}= +-\frac{\csc{ax}\cot{ax}}{2a}+\frac{1}{2a}\ln\tan{\frac{ax}{2}} +$$ +<<*>>= +)clear all + +--S 21 +aa:=integrate(csc(a*x)^3,x) +--R +--R +--R 2 sin(a x) +--R (cos(a x) - 1)log(------------) + cos(a x) +--R cos(a x) + 1 +--R (1) ------------------------------------------- +--R 2 +--R 2a cos(a x) - 2a +--R Type: Union(Expression Integer,...) +--E + +--S 22 +bb:=-(csc(a*x)*cot(a*x))/(2*a)+1/(2*a)*log(tan((a*x)/2)) +--R +--R a x +--R log(tan(---)) - cot(a x)csc(a x) +--R 2 +--R (2) -------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 23 +cc:=aa-bb +--R +--R (3) +--R 2 a x 2 sin(a x) +--R (- cos(a x) + 1)log(tan(---)) + (cos(a x) - 1)log(------------) +--R 2 cos(a x) + 1 +--R + +--R 2 +--R (cos(a x) - 1)cot(a x)csc(a x) + cos(a x) +--R / +--R 2 +--R 2a cos(a x) - 2a +--R Type: Expression Integer +--E + +--S 24 +cotrule:=rule(cot(a) == cos(a)/sin(a)) +--R +--R cos(a) +--R (4) cot(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 25 +dd:=cotrule cc +--R +--R (5) +--R 2 a x +--R (- cos(a x) + 1)sin(a x)log(tan(---)) +--R 2 +--R + +--R 2 sin(a x) +--R (cos(a x) - 1)sin(a x)log(------------) + cos(a x)sin(a x) +--R cos(a x) + 1 +--R + +--R 3 +--R (cos(a x) - cos(a x))csc(a x) +--R / +--R 2 +--R (2a cos(a x) - 2a)sin(a x) +--R Type: Expression Integer +--E + +--S 26 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (6) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 27 +ee:=tanrule dd +--R +--R (7) +--R 2 sin(a x) +--R (cos(a x) - 1)sin(a x)log(------------) +--R cos(a x) + 1 +--R + +--R a x +--R sin(---) +--R 2 2 +--R (- cos(a x) + 1)sin(a x)log(--------) + cos(a x)sin(a x) +--R a x +--R cos(---) +--R 2 +--R + +--R 3 +--R (cos(a x) - cos(a x))csc(a x) +--R / +--R 2 +--R (2a cos(a x) - 2a)sin(a x) +--R Type: Expression Integer +--E + +--S 28 +cscrule:=rule(csc(a) == 1/sin(a)) +--R +--R 1 +--R (8) csc(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 29 +ff:=cscrule ee +--R +--R (9) +--R 2 2 sin(a x) +--R (cos(a x) - 1)sin(a x) log(------------) +--R cos(a x) + 1 +--R + +--R a x +--R sin(---) +--R 2 2 2 2 3 +--R (- cos(a x) + 1)sin(a x) log(--------) + cos(a x)sin(a x) + cos(a x) +--R a x +--R cos(---) +--R 2 +--R + +--R - cos(a x) +--R / +--R 2 2 +--R (2a cos(a x) - 2a)sin(a x) +--R Type: Expression Integer +--E + +--S 30 +gg:=expandLog ff +--R +--R (10) +--R 2 2 +--R (cos(a x) - 1)sin(a x) log(sin(a x)) +--R + +--R 2 2 a x +--R (- cos(a x) + 1)sin(a x) log(sin(---)) +--R 2 +--R + +--R 2 2 +--R (- cos(a x) + 1)sin(a x) log(cos(a x) + 1) +--R + +--R 2 2 a x 2 3 +--R (cos(a x) - 1)sin(a x) log(cos(---)) + cos(a x)sin(a x) + cos(a x) +--R 2 +--R + +--R - cos(a x) +--R / +--R 2 2 +--R (2a cos(a x) - 2a)sin(a x) +--R Type: Expression Integer +--E + +--S 31 14:463 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (11) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.464~~~~~$\displaystyle +\int{\csc^n{ax}\cot{ax}}~dx$} +$$\int{\csc^n{ax}\cot{ax}}= +-\frac{csc^n{ax}}{na} +$$ +<<*>>= +)clear all + +--S 32 +aa:=integrate(csc(a*x)^n*cot(a*x),x) +--R +--R +--R 1 +--R n log(- -------------) +--R 2 +--R cos(a x) - 1 +--R ---------------------- +--R 2 +--R %e +--R (1) - ------------------------ +--R a n +--R Type: Union(Expression Integer,...) +--E + +--S 33 +bb:=-csc(a*x)^n/(n*a) +--R +--R n +--R csc(a x) +--R (2) - --------- +--R a n +--R Type: Expression Integer +--E + +--S 34 +cc:=aa-bb +--R +--R 1 +--R n log(- -------------) +--R 2 +--R cos(a x) - 1 +--R ---------------------- +--R 2 n +--R - %e + csc(a x) +--R (3) -------------------------------------- +--R a n +--R Type: Expression Integer +--E + +--S 35 14:464 Schaums and Axiom agree +normalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.465~~~~~$\displaystyle +\int{\frac{dx}{\csc{ax}}}~dx$} +$$\int{\frac{1}{\csc{ax}}}= +-\frac{\cos{ax}}{a} +$$ +<<*>>= +)clear all + +--S 36 +aa:=integrate(1/csc(a*x),x) +--R +--R +--R cos(a x) +--R (1) - -------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 37 +bb:=-cos(a*x)/a +--R +--R cos(a x) +--R (2) - -------- +--R a +--R Type: Expression Integer +--E + +--S 38 14:465 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.466~~~~~$\displaystyle +\int{x\csc{ax}}~dx$} +$$\int{x\csc{ax}}= +\frac{1}{a^2}\left\{ax+\frac{(ax)^3}{18}+\frac{7(ax)^5}{1800} ++\cdots+\frac{2(2^{2n-1}-1)B_n(ax)^{2n+1}}{(2n+1)!}+\cdots\right\} +$$ +<<*>>= +)clear all + +--S 39 14:466 Axiom cannot compute this integral +aa:=integrate(x*csc(a*x),x) +--R +--R +--R x +--R ++ +--I (1) | %H csc(%H a)d%H +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.467~~~~~$\displaystyle +\int{\frac{\csc{ax}}{x}}~dx$} +$$\int{\frac{\csc{ax}}{x}}= +-\frac{1}{ax}+\frac{(ax)}{6}+\frac{7(ax)^3}{1800} ++\cdots+\frac{2(2^{2n-1}-1)B_n(ax)^{2n-1}}{(2n-1)(2n)!}+\cdots +$$ +<<*>>= +)clear all + +--S 40 14:467 Axiom cannot compute this integral +aa:=integrate(csc(a*x)/x,x) +--R +--R +--R x +--I ++ csc(%H a) +--I (1) | --------- d%H +--I ++ %H +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.468~~~~~$\displaystyle +\int{x\csc^2{ax}}~dx$} +$$\int{x\csc^2{ax}}= +-\frac{x\cot{ax}}{a}+\frac{1}{a^2}\ln\sin{ax} +$$ +<<*>>= +)clear all + +--S 41 +aa:=integrate(x*csc(a*x)^2,x) +--R +--R +--R sin(a x) 2 +--R sin(a x)log(------------) - sin(a x)log(------------) - a x cos(a x) +--R cos(a x) + 1 cos(a x) + 1 +--R (1) -------------------------------------------------------------------- +--R 2 +--R a sin(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 42 +bb:=-(x*cot(a*x))/a+1/a^2*log(sin(a*x)) +--R +--R log(sin(a x)) - a x cot(a x) +--R (2) ---------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 43 +cc:=aa-bb +--R +--R (3) +--R sin(a x) +--R - sin(a x)log(sin(a x)) + sin(a x)log(------------) +--R cos(a x) + 1 +--R + +--R 2 +--R - sin(a x)log(------------) + a x cot(a x)sin(a x) - a x cos(a x) +--R cos(a x) + 1 +--R / +--R 2 +--R a sin(a x) +--R Type: Expression Integer +--E + +--S 44 +cotrule:=rule(cot(a) == cos(a)/sin(a)) +--R +--R cos(a) +--R (4) cot(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 45 +dd:=cotrule cc +--R +--R sin(a x) 2 +--R - log(sin(a x)) + log(------------) - log(------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (5) ------------------------------------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 46 14:468 Schaums and Axiom differ by a constant +ee:=expandLog dd +--R +--R log(2) +--R (6) - ------ +--R 2 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.469~~~~~$\displaystyle +\int{\frac{dx}{q+p\csc{ax}}}~dx$} +$$\int{\frac{1}{q+p\csc{ax}}}= +\frac{x}{q}-\frac{p}{q}\int{\frac{1}{p+q\sin{ax}}} +$$ +<<*>>= +)clear all + +--S 47 +aa:=integrate(1/(q+p*csc(a*x)),x) +--R +--R +--R (1) +--R [ +--R p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (p q - p )sin(a x) + (q - p q)cos(a x) + q - p q +--R / +--R q sin(a x) + p +--R + +--R +-------+ +--R | 2 2 +--R a x\|q - p +--R / +--R +-------+ +--R | 2 2 +--R a q\|q - p +--R , +--R +---------+ +--R | 2 2 +---------+ +--R (p sin(a x) + q cos(a x) + q)\|- q + p | 2 2 +--R 2p atan(-----------------------------------------) + a x\|- q + p +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R --------------------------------------------------------------------] +--R +---------+ +--R | 2 2 +--R a q\|- q + p +--R Type: Union(List Expression Integer,...) +--E + +--S 48 +t1:=integrate(1/(p+q*sin(a*x)),x) +--R +--R (2) +--R [ +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R / +--R +-------+ +--R | 2 2 +--R a\|q - p +--R , +--R +---------+ +--R | 2 2 +--R (p sin(a x) + q cos(a x) + q)\|- q + p +--R 2atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R - ------------------------------------------------] +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Union(List Expression Integer,...) +--E + +--S 49 +bb1:=x/q-p/q*t1.1 +--R +--R (3) +--R - +--R p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R + +--R +-------+ +--R | 2 2 +--R a x\|q - p +--R / +--R +-------+ +--R | 2 2 +--R a q\|q - p +--R Type: Expression Integer +--E + +--S 50 +bb2:=x/q-p/q*t1.2 +--R +--R +---------+ +--R | 2 2 +---------+ +--R (p sin(a x) + q cos(a x) + q)\|- q + p | 2 2 +--R 2p atan(-----------------------------------------) + a x\|- q + p +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R (4) -------------------------------------------------------------------- +--R +---------+ +--R | 2 2 +--R a q\|- q + p +--R Type: Expression Integer +--E + +--S 51 +cc1:=aa.1-bb1 +--R +--R (5) +--R p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (p q - p )sin(a x) + (q - p q)cos(a x) + q - p q +--R / +--R q sin(a x) + p +--R + +--R p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R / +--R +-------+ +--R | 2 2 +--R a q\|q - p +--R Type: Expression Integer +--E + +--S 52 +cc2:=aa.2-bb1 +--R +--R (6) +--R +---------+ +--R | 2 2 +--R p\|- q + p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p +--R 2p\|q - p atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a q\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 53 +cc3:=aa.1-bb2 +--R +--R (7) +--R +---------+ +--R | 2 2 +--R p\|- q + p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (p q - p )sin(a x) + (q - p q)cos(a x) + q - p q +--R / +--R q sin(a x) + p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p +--R - 2p\|q - p atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a q\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 54 14:469 Schaums and Axiom agree +cc4:=aa.2-bb2 +--R +--R (8) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.470~~~~~$\displaystyle +\int{\csc^n{ax}}~dx$} +$$\int{\csc^n{ax}}= +-\frac{\csc^{n-2}{ax}\cot{ax}}{a(n-1)} ++\frac{n-2}{n-1}\int{\csc^{n-2}{ax}} +$$ +<<*>>= +)clear all + +--S 55 14:470 Axiom cannot compute this integral +aa:=integrate(csc(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | csc(%H a) d%H +--R ++ +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p82 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum23.input.pdf b/src/axiom-website/CATS/schaum23.input.pdf new file mode 100644 index 0000000..a88073d --- /dev/null +++ b/src/axiom-website/CATS/schaum23.input.pdf @@ -0,0 +1,2155 @@ +%PDF-1.2 +7 0 obj 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b/src/axiom-website/CATS/schaum24.input.pamphlet new file mode 100644 index 0000000..71f28d5 --- /dev/null +++ b/src/axiom-website/CATS/schaum24.input.pamphlet @@ -0,0 +1,2555 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum24.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.471~~~~~$\displaystyle +\int{\sin^{-1}{\frac{x}{a}}}~dx$} +$$\int{\sin^{-1}{\frac{x}{a}}}= +x\sin^{-1}{\frac{x}{a}}+\sqrt{a^2-x^2} +$$ +<<*>>= +)spool schaum24.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(asin(x/a),x) +--R +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2x\|- x + a | 2 2 +--R - x atan(--------------) + 2\|- x + a +--R 2 2 +--R 2x - a +--R (1) ---------------------------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=s+asin(x/a)+sqrt(a^2-x^2) +--R +--R +---------+ +--R | 2 2 x +--R (2) \|- x + a + asin(-) + s +--R a +--R Type: Expression Integer +--E + +--S 3 14:471 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 2x\|- x + a x +--R - x atan(--------------) - 2asin(-) - 2s +--R 2 2 a +--R 2x - a +--R (3) ---------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.472~~~~~$\displaystyle +\int{x\sin^{-1}{\frac{x}{a}}}~dx$} +$$\int{x\sin^{-1}{\frac{x}{a}}}= +\left(\frac{x^2}{2}-\frac{a^2}{4}\right)\sin^{-1}{\frac{x}{a}} ++\frac{x\sqrt{a^2-x^2}}{4} +$$ +<<*>>= +)clear all + +--S 4 +aa:=integrate(x*asin(x/a),x) +--R +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 2 2x\|- x + a | 2 2 +--R (- 2x + a )atan(--------------) + 2x\|- x + a +--R 2 2 +--R 2x - a +--R (1) ------------------------------------------------- +--R 8 +--R Type: Union(Expression Integer,...) +--E + +--S 5 +bb:=(x^2/2-a^2/4)*asin(x/a)+(x*sqrt(a^2-x^2))/4 +--R +--R +---------+ +--R | 2 2 2 2 x +--R x\|- x + a + (2x - a )asin(-) +--R a +--R (2) --------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 6 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 2 2 2x\|- x + a 2 2 x +--R (- 2x + a )atan(--------------) + (- 4x + 2a )asin(-) +--R 2 2 a +--R 2x - a +--R (3) ------------------------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +@ +Here we try to understand why we cannot find a simplification +that makes these two expressions equal. If the expressions were +equal then we could use them as functions, substitute floating +point values and expect the same numeric results. So we try that here. +<<*>>= +)clear all +@ +This is the initial integrand. +<<*>>= +--S 7 +t1:=x*asin(x/a) +--R +--R x +--R (1) x asin(-) +--R a +--R Type: Expression Integer +--E +@ +This is the integral result provided by Axiom. +<<*>>= +--S 8 +t2:=integrate(t1,x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 2 2x\|- x + a | 2 2 +--R (- 2x + a )atan(--------------) + 2x\|- x + a +--R 2 2 +--R 2x - a +--R (2) ------------------------------------------------- +--R 8 +--R Type: Union(Expression Integer,...) +--E +@ +This is the derivative of the integral computed by Axiom +<<*>>= +--S 9 +t3:=D(t2,x) +--R +--R +---------+ +--R | 2 2 +--R 2x\|- x + a +--R x atan(--------------) +--R 2 2 +--R 2x - a +--R (3) - ---------------------- +--R 2 +--R Type: Expression Integer +--E +@ +This is the integral result provided by Schaums +<<*>>= +--S 10 +t4:=(x^2/2-a^2/4)*asin(x/a)+(x*sqrt(a^2-x^2))/4 +--R +--R +---------+ +--R | 2 2 2 2 x +--R x\|- x + a + (2x - a )asin(-) +--R a +--R (4) --------------------------------- +--R 4 +--R Type: Expression Integer +--E +@ +This is the derivative of the integral of the original function +according to Schaums. +<<*>>= +--S 11 +t5:=D(t4,x) +--R +--R (5) +--R +---------+ +--R +---------+ | 2 2 +---------+ +--R x | 2 2 2 3 |- x + a 2 2 | 2 2 +--R (4a x asin(-)\|- x + a - 2a x + a ) |--------- + (2x - a )\|- x + a +--R a | 2 +--R \| a +--R ---------------------------------------------------------------------------- +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 |- x + a +--R 4a\|- x + a |--------- +--R | 2 +--R \| a +--R Type: Expression Integer +--E +@ +Now we create a function for computing the integrand's values. +<<*>>= +--S 12 +f:=makeFloatFunction(t1,x,a) +--I Compiling function %BF with type (DoubleFloat,DoubleFloat) -> +--R DoubleFloat +--R +--I (6) theMap(MKBCFUNC;binaryFunction;SM;2!0,120) +--R Type: ((DoubleFloat,DoubleFloat) -> DoubleFloat) +--E +@ +Now we create a function for computing Axiom's values for its integrand. +<<*>>= +--S 13 +axiom:=makeFloatFunction(t3,x,a) +--I Compiling function %BJ with type (DoubleFloat,DoubleFloat) -> +--R DoubleFloat +--R +--I (7) theMap(MKBCFUNC;binaryFunction;SM;2!0,996) +--R Type: ((DoubleFloat,DoubleFloat) -> DoubleFloat) +--E +@ +Now we create a function for computing Schams values for its integrand. +<<*>>= +--S 14 +schaums:=makeFloatFunction(t5,x,a) +--I Compiling function %BK with type (DoubleFloat,DoubleFloat) -> +--R DoubleFloat +--R +--I (8) theMap(MKBCFUNC;binaryFunction;SM;2!0,62) +--R Type: ((DoubleFloat,DoubleFloat) -> DoubleFloat) +--E +@ +And now we compute the floating point values for each function +and compare the results. As can be clearly seen, the Axiom result +lies on a different branch cut from the Schaums result and the +functions are only equal within the branch cut range. This is a +generic problem with all of the inverse functions that are +multi-valued. +<<*>>= +--S 15 14:472 Schaums and Axiom agree (modulo branch cuts) +[ [f(i::Float,i::Float+1.0::Float)::Float,axiom(i::Float,i::Float+1.0::Float)::Float,schaums(i::Float,i::Float+1.0::Float)::Float] for i in 1..4] +--R +--R (9) +--R [[0.5235987755 9829892668,0.5235987755 9829892668,0.5235987755 9829881566], +--R [1.4594553124 539326738,1.4594553124 539326738,1.4594553124 539324518], +--R [2.5441862369 444430136,- 2.1682027434 402466604,2.5441862369 444430136], +--R [3.7091808720 064496363,- 2.5740044351 731374839,3.7091808720 064500804]] +--R Type: List List Float +--E +@ + +\section{\cite{1}:14.473~~~~~$\displaystyle +\int{x^2\sin^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\sin^{-1}\frac{x}{a}}= +\frac{x^3}{3}\sin^{-1}\frac{x}{a}+\frac{(x^2+2a^2)\sqrt{a^2-x^2}}{9} +$$ +<<*>>= +)clear all + +--S 16 +aa:=integrate(x^2*asin(x/a),x) +--R +--R +--R +---------+ +--R | 2 2 +---------+ +--R 3 2x\|- x + a 2 2 | 2 2 +--R - 3x atan(--------------) + (2x + 4a )\|- x + a +--R 2 2 +--R 2x - a +--R (1) --------------------------------------------------- +--R 18 +--R Type: Union(Expression Integer,...) +--E + +--S 17 +bb:=x^3/3*asin(x/a)+((x^2+2*a^2)*sqrt(a^2-x^2))/9 +--R +--R +---------+ +--R 2 2 | 2 2 3 x +--R (x + 2a )\|- x + a + 3x asin(-) +--R a +--R (2) ----------------------------------- +--R 9 +--R Type: Expression Integer +--E + +--S 18 14:473 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 3 2x\|- x + a 3 x +--R - x atan(--------------) - 2x asin(-) +--R 2 2 a +--R 2x - a +--R (3) ------------------------------------- +--R 6 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.474~~~~~$\displaystyle +\int{\frac{\sin^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{\sin^{-1}(x/a)}{x}}= +\frac{x}{a}+\frac{(x/a)^3}{2\cdot 3\cdot 3} ++\frac{1\cdot 3(x/a)^5}{2\cdot 4\cdot 5\cdot 5} ++\frac{1\cdot 3\cdot 5(x/a)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7}+\cdots +$$ +<<*>>= +)clear all + +--S 19 14:474 Axiom cannot compute this integral +aa:=integrate(asin(x/a)/x,x) +--R +--R +--I %H +--R x asin(--) +--R ++ a +--I (1) | -------- d%H +--I ++ %H +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.475~~~~~$\displaystyle +\int{\frac{\sin^{-1}{(x/a)}}{x^2}}~dx$} +$$\int{\frac{\sin^{-1}{(x/a)}}{x^2}}= +-\frac{\sin^{-1}(x/a)}{x} +-\frac{1}{a}\ln\left(\frac{a+\sqrt{a^2-x^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 20 +aa:=integrate(asin(x/a)/x^2,x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ +---------+ | 2 2 +--R | 2 2 | 2 2 2x\|- x + a +--R - x log(\|- x + a + a) + x log(\|- x + a - a) + a atan(--------------) +--R 2 2 +--R 2x - a +--R ---------------------------------------------------------------------------- +--R 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 21 +bb:=-asin(x/a)/x-1/a*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +--R \|- x + a + a x +--R - x log(----------------) - a asin(-) +--R x a +--R (2) ------------------------------------- +--R a x +--R Type: Expression Integer +--E + +--S 22 14:475 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R - x log(\|- x + a + a) + x log(\|- x + a - a) +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a 2x\|- x + a x +--R 2x log(----------------) + a atan(--------------) + 2a asin(-) +--R x 2 2 a +--R 2x - a +--R / +--R 2a x +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.476~~~~~$\displaystyle +\int{\left(sin^{-1}\frac{x}{a}\right)^2}~dx$} +$$\int{\left(sin^{-1}\frac{x}{a}\right)^2}= +x\left(\sin^{-1}\frac{x}{a}\right)^2-2x+2\sqrt{a^2-x^2}\sin^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 23 +aa:=integrate(asin(x/a)^2,x) +--R +--R +--R +---------+ 2 +---------+ +--R | 2 2 +---------+ | 2 2 +--R 2x\|- x + a | 2 2 2x\|- x + a +--R x atan(--------------) - 4\|- x + a atan(--------------) - 8x +--R 2 2 2 2 +--R 2x - a 2x - a +--R (1) ---------------------------------------------------------------- +--R 4 +--R Type: Union(Expression Integer,...) +--E + +--S 24 +bb:=x*asin(x/a)^2-2*x+2*sqrt(a^2-x^2)*asin(x/a) +--R +--R +---------+ +--R x | 2 2 x 2 +--R (2) 2asin(-)\|- x + a + x asin(-) - 2x +--R a a +--R Type: Expression Integer +--E + +--S 25 14:476 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R +---------+ 2 +---------+ +--R | 2 2 +---------+ | 2 2 +--R 2x\|- x + a | 2 2 2x\|- x + a +--R x atan(--------------) - 4\|- x + a atan(--------------) +--R 2 2 2 2 +--R 2x - a 2x - a +--R + +--R +---------+ +--R x | 2 2 x 2 +--R - 8asin(-)\|- x + a - 4x asin(-) +--R a a +--R / +--R 4 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.477~~~~~$\displaystyle +\int{\cos^{-1}\frac{x}{a}}~dx$} +$$\int{\cos^{-1}\frac{x}{a}}= +x\cos^{-1}\frac{x}{a}-\sqrt{a^2-x^2} +$$ +<<*>>= +)clear all + +--S 26 +aa:=integrate(acos(x/a),x) +--R +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2x\|- x + a | 2 2 +--R x atan(--------------) - 2\|- x + a +--R 2 2 +--R 2x - a +--R (1) -------------------------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E + +--S 27 +bb:=x*acos(x/a)-sqrt(a^2-x^2) +--R +--R +---------+ +--R | 2 2 x +--R (2) - \|- x + a + x acos(-) +--R a +--R Type: Expression Integer +--E + +--S 28 14:477 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 2x\|- x + a x +--R x atan(--------------) - 2x acos(-) +--R 2 2 a +--R 2x - a +--R (3) ----------------------------------- +--R 2 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.478~~~~~$\displaystyle +\int{x\cos^{-1}\frac{x}{a}}~dx$} +$$\int{x\cos^{-1}\frac{x}{a}}= +\left(\frac{x^2}{2}-\frac{a^2}{4}\right)\cos^{-1}\frac{x}{a} +-\frac{x\sqrt{a^2-x^2}}{4} +$$ +<<*>>= +)clear all + +--S 29 +aa:=integrate(x*acos(x/a),x) +--R +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 2 2x\|- x + a | 2 2 +--R (2x - a )atan(--------------) - 2x\|- x + a +--R 2 2 +--R 2x - a +--R (1) ----------------------------------------------- +--R 8 +--R Type: Union(Expression Integer,...) +--E + +--S 30 +bb:=(x^2/2-a^2/4)*acos(x/a)-(x*sqrt(a^2-x^2))/4 +--R +--R +---------+ +--R | 2 2 2 2 x +--R - x\|- x + a + (2x - a )acos(-) +--R a +--R (2) ----------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 31 14:478 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 2 2 2x\|- x + a 2 2 x +--R (2x - a )atan(--------------) + (- 4x + 2a )acos(-) +--R 2 2 a +--R 2x - a +--R (3) ----------------------------------------------------- +--R 8 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.479~~~~~$\displaystyle +\int{x^2\cos^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\cos^{-1}\frac{x}{a}}= +\frac{x^3}{3}\cos^{-1}\frac{x}{a}-\frac{(x^2+2a^2)\sqrt{a^2-x^2}}{9} +$$ +<<*>>= +)clear all + +--S 32 +aa:=integrate(x^2*acos(x/a),x) +--R +--R +--R +---------+ +--R | 2 2 +---------+ +--R 3 2x\|- x + a 2 2 | 2 2 +--R 3x atan(--------------) + (- 2x - 4a )\|- x + a +--R 2 2 +--R 2x - a +--R (1) --------------------------------------------------- +--R 18 +--R Type: Union(Expression Integer,...) +--E + +--S 33 +bb:=x^3/3*acos(x/a)-((x^2+2*a^2)*sqrt(a^2-x^2))/9 +--R +--R +---------+ +--R 2 2 | 2 2 3 x +--R (- x - 2a )\|- x + a + 3x acos(-) +--R a +--R (2) ------------------------------------- +--R 9 +--R Type: Expression Integer +--E + +--S 34 14:479 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 3 2x\|- x + a 3 x +--R x atan(--------------) - 2x acos(-) +--R 2 2 a +--R 2x - a +--R (3) ----------------------------------- +--R 6 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.480~~~~~$\displaystyle +\int{\frac{\cos^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{\cos^{-1}(x/a)}{x}}= +\frac{x}{2}\ln{x}-\int{\frac{\sin^{-1}(x/a)}{x}} +$$ +<<*>>= +)clear all + +--S 35 14:480 Axiom cannot compute this integral +aa:=integrate(acos(x/a)/x,x) +--R +--R +--I %H +--R x acos(--) +--R ++ a +--I (1) | -------- d%H +--I ++ %H +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.481~~~~~$\displaystyle +\int{\frac{\cos^{-1}(x/a)}{x^2}}~dx$} +$$\int{\frac{\cos^{-1}(x/a)}{x^2}}= +-\frac{\cos^{-1}(x/a)}{x}+\frac{1}{a}\ln\left(\frac{a+\sqrt{a^2-x^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 36 +aa:=integrate(acos(x/a)/x^2,x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ +---------+ | 2 2 +--R | 2 2 | 2 2 2x\|- x + a +--R x log(\|- x + a + a) - x log(\|- x + a - a) - a atan(--------------) +--R 2 2 +--R 2x - a +--R -------------------------------------------------------------------------- +--R 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 37 +bb:=-acos(x/a)/x+1/a*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +--R \|- x + a + a x +--R x log(----------------) - a acos(-) +--R x a +--R (2) ----------------------------------- +--R a x +--R Type: Expression Integer +--E + +--S 38 14:481 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R x log(\|- x + a + a) - x log(\|- x + a - a) +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a 2x\|- x + a x +--R - 2x log(----------------) - a atan(--------------) + 2a acos(-) +--R x 2 2 a +--R 2x - a +--R / +--R 2a x +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.482~~~~~$\displaystyle +\int{\left(\cos^{-1}\frac{x}{a}\right)^2}~dx$} +$$\int{\left(\cos^{-1}\frac{x}{a}\right)^2}= +x\left(\cos^{-1}\frac{x}{a}\right)^2-2x-2\sqrt{a^2-x^2}\cos^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 39 +aa:=integrate(acos(x/a)^2,x) +--R +--R +--R +---------+ 2 +---------+ +--R | 2 2 +---------+ | 2 2 +--R 2x\|- x + a | 2 2 2x\|- x + a +--R x atan(--------------) - 4\|- x + a atan(--------------) - 8x +--R 2 2 2 2 +--R 2x - a 2x - a +--R (1) ---------------------------------------------------------------- +--R 4 +--R Type: Union(Expression Integer,...) +--E + +--S 40 +bb:=x*acos(x/a)^2-2*x-2*sqrt(a^2-x^2)*acos(x/a) +--R +--R +---------+ +--R x | 2 2 x 2 +--R (2) - 2acos(-)\|- x + a + x acos(-) - 2x +--R a a +--R Type: Expression Integer +--E + +--S 41 14:482 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R +---------+ 2 +---------+ +--R | 2 2 +---------+ | 2 2 +--R 2x\|- x + a | 2 2 2x\|- x + a +--R x atan(--------------) - 4\|- x + a atan(--------------) +--R 2 2 2 2 +--R 2x - a 2x - a +--R + +--R +---------+ +--R x | 2 2 x 2 +--R 8acos(-)\|- x + a - 4x acos(-) +--R a a +--R / +--R 4 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.483~~~~~$\displaystyle +\int{\tan^{-1}\frac{x}{a}}~dx$} +$$\int{\tan^{-1}\frac{x}{a}}= +x\tan^{-1}\frac{x}{a}-\frac{a}{2}\ln(x^2+a^2) +$$ +<<*>>= +)clear all + +--S 42 +aa:=integrate(atan(x/a),x) +--R +--R +--R 2 2 2a x +--R - a log(x + a ) - x atan(-------) +--R 2 2 +--R x - a +--R (1) ---------------------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E + +--S 43 +bb:=x*atan(x/a)-a/2*log(x^2+a^2) +--R +--R 2 2 x +--R - a log(x + a ) + 2x atan(-) +--R a +--R (2) ----------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 44 +cc:=aa-bb +--R +--R x 2a x +--R - 2x atan(-) - x atan(-------) +--R a 2 2 +--R x - a +--R (3) ------------------------------ +--R 2 +--R Type: Expression Integer +--E + +--S 45 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 46 +dd:=atanrule cc +--R +--R 2 2 +--R x + 2%i a x - a - x + %i a +--R %i x log(-----------------) + 2%i x log(----------) +--R 2 2 x + %i a +--R x - 2%i a x - a +--R (5) --------------------------------------------------- +--R 4 +--R Type: Expression Complex Integer +--E + +--S 47 14:483 SCHAUMS AND AXIOM DIFFER? (BRANCH CUTS?) +ee:=expandLog dd +--R +--R %i x log(- 1) +--R (6) ------------- +--R 2 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.484~~~~~$\displaystyle +\int{x\tan^{-1}\frac{x}{a}}~dx$} +$$\int{x\tan^{-1}\frac{x}{a}}= +\frac{1}{2}(x^2+a^2)\tan^{-1}\frac{x}{a}-\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 48 14:484 Axiom cannot compute this integral +aa:=integrate(x*tan(x/a),x) +--R +--R +--R x +--I ++ %H +--I (1) | %H tan(--)d%H +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.485~~~~~$\displaystyle +\int{x^2\tan^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\tan^{-1}\frac{x}{a}}= +\frac{x^3}{3}\tan^{-1}\frac{x}{a}-\frac{ax^2}{6}+\frac{a^3}{6}\ln(x^2+a^2) +$$ +<<*>>= +)clear all + +--S 49 +aa:=integrate(x^2*atan(x/a),x) +--R +--R +--R 3 2 2 3 2a x 2 +--R a log(x + a ) - x atan(-------) - a x +--R 2 2 +--R x - a +--R (1) --------------------------------------- +--R 6 +--R Type: Union(Expression Integer,...) +--E + +--S 50 +bb:=x^3/2*atan(x/a)-(a*x^2)/6+a^3/6*log(x^2+a^2) +--R +--R 3 2 2 3 x 2 +--R a log(x + a ) + 3x atan(-) - a x +--R a +--R (2) ---------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 51 14:485 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R 3 x 3 2a x +--R - 3x atan(-) - x atan(-------) +--R a 2 2 +--R x - a +--R (3) ------------------------------ +--R 6 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.486~~~~~$\displaystyle +\int{\frac{\tan^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{\tan^{-1}(x/a)}{x}}= +\frac{x}{a}-\frac{(x/a)^3}{3^2}+\frac{(x/a)^5}{5^2}-\frac{(x/a)^7}{7^2}+\cdots +$$ +<<*>>= +)clear all + +--S 52 14:486 Axiom cannot compute this integral +aa:=integrate(atan(x/a)/x,x) +--R +--R +--I %H +--R x atan(--) +--R ++ a +--I (1) | -------- d%H +--I ++ %H +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.487~~~~~$\displaystyle +\int{\frac{\tan^{-1}(x/a)}{x^2}}~dx$} +$$\int{\frac{\tan^{-1}(x/a)}{x^2}}= +-\frac{1}{x}\tan^{-1}\frac{x}{a} +-\frac{1}{2a}\ln\left(\frac{x^2+a^2}{x^2}\right) +$$ +<<*>>= +)clear all + +--S 53 +aa:=integrate(atan(x/a)/x^2,x) +--R +--R +--R 2 2 2a x +--R - x log(x + a ) + 2x log(x) + a atan(-------) +--R 2 2 +--R x - a +--R (1) ---------------------------------------------- +--R 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 54 +bb:=-1/x*atan(x/a)-1/(2*a)*log((x^2+a^2)/x^2) +--R +--R 2 2 +--R x + a x +--R - x log(-------) - 2a atan(-) +--R 2 a +--R x +--R (2) ----------------------------- +--R 2a x +--R Type: Expression Integer +--E + +--S 55 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R 2 2 x + a x 2a x +--R - x log(x + a ) + 2x log(x) + x log(-------) + 2a atan(-) + a atan(-------) +--R 2 a 2 2 +--R x x - a +--R ---------------------------------------------------------------------------- +--R 2a x +--R Type: Expression Integer +--E + +--S 56 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 57 +dd:=atanrule cc +--R +--R (5) +--R 2 2 +--R 2 2 x + 2%i a x - a +--R - 2x log(x + a ) + 4x log(x) - %i a log(-----------------) +--R 2 2 +--R x - 2%i a x - a +--R + +--R 2 2 +--R x + a - x + %i a +--R 2x log(-------) - 2%i a log(----------) +--R 2 x + %i a +--R x +--R / +--R 4a x +--R Type: Expression Complex Integer +--E + +--S 58 14:487 SCHAUMS AND AXIOM DIFFER? (branch cuts?) +ee:=expandLog dd +--R +--R %i log(- 1) +--R (6) - ----------- +--R 2x +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.488~~~~~$\displaystyle +\int{\cot^{-1}\frac{x}{a}}~dx$} +$$\int{\cot^{-1}\frac{x}{a}}= +x\cot^{-1}\frac{x}{a}+\frac{a}{2}\ln(x^2+a^2) +$$ +<<*>>= +)clear all + +--S 59 +aa:=integrate(acot(x/a),x) +--R +--R +--R 2 2 2a x +--R a log(x + a ) + x atan(-------) +--R 2 2 +--R x - a +--R (1) -------------------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E + +--S 60 +bb:=x*acot(x/a)+a/2*log(x^2+a^2) +--R +--R 2 2 x +--R a log(x + a ) + 2x acot(-) +--R a +--R (2) --------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 61 +cc:=aa-bb +--R +--R 2a x x +--R x atan(-------) - 2x acot(-) +--R 2 2 a +--R x - a +--R (3) ---------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 62 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 63 +dd:=atanrule cc +--R +--R 2 2 +--R x + 2%i a x - a x +--R - %i x log(-----------------) - 4x acot(-) +--R 2 2 a +--R x - 2%i a x - a +--R (5) ------------------------------------------ +--R 4 +--R Type: Expression Complex Integer +--E + +--S 64 +acotrule:=rule(acot(x) == -%i/2*log((%i*x-1)/(%i*x+1))) +--R +--R x + %i +--R %i log(------) +--R x - %i +--R (6) acot(x) == - -------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 65 +ee:=acotrule dd +--R +--R 2 2 +--R x + 2%i a x - a x + %i a +--R - %i x log(-----------------) + 2%i x log(--------) +--R 2 2 x - %i a +--R x - 2%i a x - a +--R (7) --------------------------------------------------- +--R 4 +--R Type: Expression Complex Integer +--E + +--S 66 14:488 Axiom and Schaums agree +ff:=expandLog % +--R +--R (8) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.489~~~~~$\displaystyle +\int{x\cot^{-1}\frac{x}{a}}~dx$} +$$\int{x\cot^{-1}\frac{x}{a}}= +\frac{1}{2}(x^2+a^2)\cot^{-1}\frac{x}{a}+\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 67 +aa:=integrate(x*acot(x/a),x) +--R +--R +--R 2 2 2a x +--R (x + a )atan(-------) + 2a x +--R 2 2 +--R x - a +--R (1) ----------------------------- +--R 4 +--R Type: Union(Expression Integer,...) +--E + +--S 68 +bb:=1/2*(x^2+a^2)*acot(x/a)+(a*x)/2 +--R +--R 2 2 x +--R (x + a )acot(-) + a x +--R a +--R (2) ---------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 69 +cc:=aa-bb +--R +--R 2 2 2a x 2 2 x +--R (x + a )atan(-------) + (- 2x - 2a )acot(-) +--R 2 2 a +--R x - a +--R (3) --------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 70 +acotrule:=rule(acot(x) == -%i/2*log((%i*x-1)/(%i*x+1))) +--R +--R x + %i +--R %i log(------) +--R x - %i +--R (4) acot(x) == - -------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 71 +dd:=acotrule cc +--R +--R 2 2 x + %i a 2 2 2a x +--R (%i x + %i a )log(--------) + (x + a )atan(-------) +--R x - %i a 2 2 +--R x - a +--R (5) ----------------------------------------------------- +--R 4 +--R Type: Expression Complex Integer +--E + +--S 72 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 73 +ee:=atanrule dd +--R +--R (7) +--R 2 2 +--R 2 2 x + 2%i a x - a 2 2 x + %i a +--R (- %i x - %i a )log(-----------------) + (2%i x + 2%i a )log(--------) +--R 2 2 x - %i a +--R x - 2%i a x - a +--R ------------------------------------------------------------------------ +--R 8 +--R Type: Expression Complex Integer +--E + +--S 74 14:489 Axiom and Schaums agree +ff:=expandLog ee +--R +--R (8) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.490~~~~~$\displaystyle +\int{x^2\cot^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\cot^{-1}\frac{x}{a}}= +\frac{x^3}{3}\cot^{-1}\frac{x}{a}+\frac{ax^2}{6}-\frac{a^3}{6}\ln(x^2+a^2) +$$ +<<*>>= +)clear all + +--S 75 +aa:=integrate(x^2*acot(x/a),x) +--R +--R +--R 3 2 2 3 2a x 2 +--R - a log(x + a ) + x atan(-------) + a x +--R 2 2 +--R x - a +--R (1) ----------------------------------------- +--R 6 +--R Type: Union(Expression Integer,...) +--E + +--S 76 +bb:=x^3/3*acot(x/a)+(a*x^2)/6-a^3/6*log(x^2+a^2) +--R +--R 3 2 2 3 x 2 +--R - a log(x + a ) + 2x acot(-) + a x +--R a +--R (2) ------------------------------------ +--R 6 +--R Type: Expression Integer +--E + +--S 77 +cc:=aa-bb +--R +--R 3 2a x 3 x +--R x atan(-------) - 2x acot(-) +--R 2 2 a +--R x - a +--R (3) ---------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 78 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 79 +dd:=atanrule cc +--R +--R 2 2 +--R 3 x + 2%i a x - a 3 x +--R - %i x log(-----------------) - 4x acot(-) +--R 2 2 a +--R x - 2%i a x - a +--R (5) ------------------------------------------ +--R 12 +--R Type: Expression Complex Integer +--E + +--S 80 +acotrule:=rule(acot(x) == -%i/2*log((%i*x-1)/(%i*x+1))) +--R +--R x + %i +--R %i log(------) +--R x - %i +--R (6) acot(x) == - -------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 81 +ee:=acotrule dd +--R +--R 2 2 +--R 3 x + 2%i a x - a 3 x + %i a +--R - %i x log(-----------------) + 2%i x log(--------) +--R 2 2 x - %i a +--R x - 2%i a x - a +--R (7) --------------------------------------------------- +--R 12 +--R Type: Expression Complex Integer +--E + +--S 82 14:490 Axiom and Schaums agree +ff:=expandLog ee +--R +--R (8) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.491~~~~~$\displaystyle +\int{\frac{\cot^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{\cot^{-1}(x/a)}{x}}= +\frac{\pi}{2}\ln{x}-\int{\frac{\tan^{-1}(x/a)}{x}} +$$ +<<*>>= +)clear all + +--S 83 14:491 Axiom cannot compute this integral +aa:=integrate(acot(x/a)/x,x) +--R +--R +--I %H +--R x acot(--) +--R ++ a +--I (1) | -------- d%H +--I ++ %H +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.492~~~~~$\displaystyle +\int{\frac{\cot^{-1}(x/a)}{x^2}}~dx$} +$$\int{\frac{\cot^{-1}(x/a)}{x^2}}= +-\frac{cot^{-1}(x/a)}{x}+\frac{1}{2a}\ln\left(\frac{x^2+a^2}{x^2}\right) +$$ +<<*>>= +)clear all + +--S 84 +aa:=integrate(acot(x/a)/x^2,x) +--R +--R +--R 2 2 2a x +--R x log(x + a ) - 2x log(x) - a atan(-------) +--R 2 2 +--R x - a +--R (1) -------------------------------------------- +--R 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 85 +bb:=-acot(x/a)/x+1/(2*a)*log((x^2+a^2)/x^2) +--R +--R 2 2 +--R x + a x +--R x log(-------) - 2a acot(-) +--R 2 a +--R x +--R (2) --------------------------- +--R 2a x +--R Type: Expression Integer +--E + +--S 86 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R 2 2 x + a 2a x x +--R x log(x + a ) - 2x log(x) - x log(-------) - a atan(-------) + 2a acot(-) +--R 2 2 2 a +--R x x - a +--R -------------------------------------------------------------------------- +--R 2a x +--R Type: Expression Integer +--E + +--S 87 +acotrule:=rule(acot(x) == -%i/2*log((%i*x-1)/(%i*x+1))) +--R +--R x + %i +--R %i log(------) +--R x - %i +--R (4) acot(x) == - -------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 88 +dd:=acotrule cc +--R +--R (5) +--R 2 2 +--R 2 2 x + %i a x + a +--R x log(x + a ) - 2x log(x) - %i a log(--------) - x log(-------) +--R x - %i a 2 +--R x +--R + +--R 2a x +--R - a atan(-------) +--R 2 2 +--R x - a +--R / +--R 2a x +--R Type: Expression Complex Integer +--E + +--S 89 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 90 +ee:=atanrule dd +--R +--R (7) +--R 2 2 +--R 2 2 x + 2%i a x - a +--R 2x log(x + a ) - 4x log(x) + %i a log(-----------------) +--R 2 2 +--R x - 2%i a x - a +--R + +--R 2 2 +--R x + %i a x + a +--R - 2%i a log(--------) - 2x log(-------) +--R x - %i a 2 +--R x +--R / +--R 4a x +--R Type: Expression Complex Integer +--E + +--S 91 14:492 Schaums and Axiom agree +ff:=expandLog ee +--R +--R (8) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.493~~~~~$\displaystyle +\int{\sec^{-1}\frac{x}{a}}~dx$} +$$\int{\sec^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +x\sec^{-1}\frac{x}{a}-a\ln(x+\sqrt{x^2-a^2}) +{\rm \ if\ }0 < \sec^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +x\sec^{-1}\frac{x}{a}+a\ln(x+\sqrt{x^2-a^2}) +{\rm \ if\ }\frac{\pi}{2} < \sec^{-1}\frac{x}{a} < \pi +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 92 +aa:=integrate(asec(x/a),x) +--R +--R +--R (1) +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +--R +-+ 2x\|2 \|- x + a 2a\|- x + a +--R - a\|2 atan(------------------) + x atan(--------------) +--R 2 2 2 +--R 3x - 2a x +--R + +--R x +--R - 2a atan(------------) +--R +---------+ +--R | 2 2 +--R \|- x + a +--R / +--R 2 +--R Type: Union(Expression Integer,...) +--E + +--S 93 +bb1:=x*asec(x/a)-a*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +--R | 2 2 x +--R (2) - a log(\|x - a + x) + x asec(-) +--R a +--R Type: Expression Integer +--E + +--S 94 +bb2:=x*asec(x/a)+a*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +--R | 2 2 x +--R (3) a log(\|x - a + x) + x asec(-) +--R a +--R Type: Expression Integer +--E + +--S 95 +cc1:=aa-bb1 +--R +--R (4) +--R +---------+ +--R +-------+ +-+ | 2 2 +--R | 2 2 +-+ 2x\|2 \|- x + a +--R 2a log(\|x - a + x) - a\|2 atan(------------------) +--R 2 2 +--R 3x - 2a +--R + +--R +---------+ +--R | 2 2 +--R 2a\|- x + a x x +--R x atan(--------------) - 2a atan(------------) - 2x asec(-) +--R 2 +---------+ a +--R x | 2 2 +--R \|- x + a +--R / +--R 2 +--R Type: Expression Integer +--E + +--S 96 14:493 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +--R +-------+ +-+ | 2 2 +--R | 2 2 +-+ 2x\|2 \|- x + a +--R - 2a log(\|x - a + x) - a\|2 atan(------------------) +--R 2 2 +--R 3x - 2a +--R + +--R +---------+ +--R | 2 2 +--R 2a\|- x + a x x +--R x atan(--------------) - 2a atan(------------) - 2x asec(-) +--R 2 +---------+ a +--R x | 2 2 +--R \|- x + a +--R / +--R 2 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.494~~~~~$\displaystyle +\int{x\sec^{-1}\frac{x}{a}}~dx$} +$$\int{x\sec^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{x^2}{2}\sec^{-1}\frac{x}{a}-\frac{a\sqrt{x^2-a^2}}{2} +{\rm \ if\ }0 < \sec^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +\frac{x^2}{2}\sec^{-1}\frac{x}{a}+\frac{a\sqrt{x^2-a^2}}{2} +{\rm \ if\ }\frac{\pi}{2} < \sec^{-1}\frac{x}{a} < \pi +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 97 +aa:=integrate(x*asec(x/a),x) +--R +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 2 2a\|- x + a | 2 2 +--R (x - 2a )atan(--------------) + 2a\|- x + a +--R 2 +--R x +--R (1) ----------------------------------------------- +--R 4 +--R Type: Union(Expression Integer,...) +--E + +--S 98 +bb1:=x^2/2*asec(x/a)-(a*sqrt(x^2-a^2))/2 +--R +--R +-------+ +--R | 2 2 2 x +--R - a\|x - a + x asec(-) +--R a +--R (2) ------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 99 +bb2:=x^2/2*asec(x/a)+(a*sqrt(x^2-a^2))/2 +--R +--R +-------+ +--R | 2 2 2 x +--R a\|x - a + x asec(-) +--R a +--R (3) ----------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 100 +cc1:=aa-bb1 +--R +--R (4) +--R +---------+ +--R | 2 2 +-------+ +---------+ +--R 2 2 2a\|- x + a | 2 2 | 2 2 2 x +--R (x - 2a )atan(--------------) + 2a\|x - a + 2a\|- x + a - 2x asec(-) +--R 2 a +--R x +--R --------------------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 101 14:494 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +--R | 2 2 +-------+ +---------+ +--R 2 2 2a\|- x + a | 2 2 | 2 2 2 x +--R (x - 2a )atan(--------------) - 2a\|x - a + 2a\|- x + a - 2x asec(-) +--R 2 a +--R x +--R --------------------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.495~~~~~$\displaystyle +\int{x^2\sec^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\sec^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{x^3}{3}\sec^{-1}\frac{x}{a}-\frac{ax\sqrt{x^2-a^2}}{6} +-\frac{a^3}{6}\ln(x+\sqrt{x^2-a^2})\\ +\\ +\displaystyle +\hbox{\hskip 3cm}{\rm \ if\ }0 < \sec^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +\frac{x^3}{3}\sec^{-1}\frac{x}{a}+\frac{ax\sqrt{x^2-a^2}}{6} ++\frac{a^3}{6}\ln(x+\sqrt{x^2-a^2})\\ +\\ +\displaystyle +\hbox{\hskip 3cm} +{\rm \ if\ }\frac{\pi}{2} < \sec^{-1}\frac{x}{a} < \pi\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 102 +aa:=integrate(x^2*asec(x/a),x) +--R +--R +--R (1) +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +--R 3 +-+ 2x\|2 \|- x + a 3 2a\|- x + a +--R - 2a \|2 atan(------------------) + x atan(--------------) +--R 2 2 2 +--R 3x - 2a x +--R + +--R +---------+ +--R 3 x | 2 2 +--R - 5a atan(------------) + a x\|- x + a +--R +---------+ +--R | 2 2 +--R \|- x + a +--R / +--R 6 +--R Type: Union(Expression Integer,...) +--E + +--S 103 +bb1:=x^3/3*asec(x/a)-(a*x*sqrt(x^2-a^2))/6-a^3/6*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 3 | 2 2 | 2 2 3 x +--R - a log(\|x - a + x) - a x\|x - a + 2x asec(-) +--R a +--R (2) ---------------------------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 104 +bb2:=x^3/3*asec(x/a)+(a*x*sqrt(x^2-a^2))/6+a^3/6*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 3 | 2 2 | 2 2 3 x +--R a log(\|x - a + x) + a x\|x - a + 2x asec(-) +--R a +--R (3) -------------------------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 105 +cc1:=aa-bb1 +--R +--R (4) +--R +---------+ +--R +-------+ +-+ | 2 2 +--R 3 | 2 2 3 +-+ 2x\|2 \|- x + a +--R a log(\|x - a + x) - 2a \|2 atan(------------------) +--R 2 2 +--R 3x - 2a +--R + +--R +---------+ +--R | 2 2 +-------+ +--R 3 2a\|- x + a 3 x | 2 2 +--R x atan(--------------) - 5a atan(------------) + a x\|x - a +--R 2 +---------+ +--R x | 2 2 +--R \|- x + a +--R + +--R +---------+ +--R | 2 2 3 x +--R a x\|- x + a - 2x asec(-) +--R a +--R / +--R 6 +--R Type: Expression Integer +--E + +--S 106 14:495 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +--R +-------+ +-+ | 2 2 +--R 3 | 2 2 3 +-+ 2x\|2 \|- x + a +--R - a log(\|x - a + x) - 2a \|2 atan(------------------) +--R 2 2 +--R 3x - 2a +--R + +--R +---------+ +--R | 2 2 +-------+ +--R 3 2a\|- x + a 3 x | 2 2 +--R x atan(--------------) - 5a atan(------------) - a x\|x - a +--R 2 +---------+ +--R x | 2 2 +--R \|- x + a +--R + +--R +---------+ +--R | 2 2 3 x +--R a x\|- x + a - 2x asec(-) +--R a +--R / +--R 6 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.496~~~~~$\displaystyle +\int{\frac{\sec^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{\sec^{-1}(x/a)}{x}}= +\frac{\pi}{2}\ln{x}+\frac{a}{x}+\frac{(a/x)^3}{2\cdot 3\cdot 3} ++\frac{1\cdot 3(a/x)^5}{2\cdot 4\cdot 5\cdot 5} ++\frac{1\cdot 3\cdot 5(a/x)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7}+\cdots +$$ +<<*>>= +)clear all + +--S 107 14:496 Axiom cannot compute this integral +aa:=integrate(asec(x/a)/x,x) +--R +--R +--I %H +--R x asec(--) +--R ++ a +--I (1) | -------- d%H +--I ++ %H +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.497~~~~~$\displaystyle +\int{\frac{\sec^{-1}(x/a)}{x^2}}~dx$} +$$\int{\frac{\sec^{-1}(x/a)}{x^2}}= +\left\{ +\begin{array}{l} +\displaystyle +-\frac{\sec^{-1}(x/a)}{x}+\frac{\sqrt{x^2-a^2}}{ax} +{\rm \ if\ }0 < \sec^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +-\frac{\sec^{-1}(x/a)}{x}-\frac{\sqrt{x^2-a^2}}{ax} +{\rm \ if\ }\frac{\pi}{2} < \sec^{-1}\frac{x}{a} < \pi\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 108 +aa:=integrate(asec(x/a)/x^2,x) +--R +--R +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +--R 2x\|2 \|- x + a +-+ 2a\|- x + a +--R x atan(------------------) - a\|2 atan(--------------) +--R 2 2 2 +--R 3x - 2a x +--R (1) ------------------------------------------------------ +--R +-+ +--R 2a x\|2 +--R Type: Union(Expression Integer,...) +--E + +--S 109 +bb1:=-asec(x/a)/x+sqrt(x^2-a^2)/(a*x) +--R +--R +-------+ +--R | 2 2 x +--R \|x - a - a asec(-) +--R a +--R (2) ---------------------- +--R a x +--R Type: Expression Integer +--E + +--S 110 +bb2:=-asec(x/a)/x-sqrt(x^2-a^2)/(a*x) +--R +--R +-------+ +--R | 2 2 x +--R - \|x - a - a asec(-) +--R a +--R (3) ------------------------ +--R a x +--R Type: Expression Integer +--E + +--S 111 +cc1:=aa-bb1 +--R +--R (4) +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +-------+ +--R 2x\|2 \|- x + a +-+ 2a\|- x + a +-+ | 2 2 +--R x atan(------------------) - a\|2 atan(--------------) - 2\|2 \|x - a +--R 2 2 2 +--R 3x - 2a x +--R + +--R +-+ x +--R 2a\|2 asec(-) +--R a +--R / +--R +-+ +--R 2a x\|2 +--R Type: Expression Integer +--E + +--S 112 14:497 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +-------+ +--R 2x\|2 \|- x + a +-+ 2a\|- x + a +-+ | 2 2 +--R x atan(------------------) - a\|2 atan(--------------) + 2\|2 \|x - a +--R 2 2 2 +--R 3x - 2a x +--R + +--R +-+ x +--R 2a\|2 asec(-) +--R a +--R / +--R +-+ +--R 2a x\|2 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.498~~~~~$\displaystyle +\int{\csc^{-1}\frac{x}{a}}~dx$} +$$\int{\csc^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +x\csc^{-1}\frac{x}{a}+a\ln(x+\sqrt{x^2-a^2}) +{\rm \ if\ }0 < \csc^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +x\csc^{-1}\frac{x}{a}-a\ln(x+\sqrt{x^2-a^2}) +{\rm \ if\ }-\frac{\pi}{2} < \csc^{-1}\frac{x}{a} < 0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 113 +aa:=integrate(acsc(x/a),x) +--R +--R +--R (1) +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +--R +-+ 2x\|2 \|- x + a 2a\|- x + a +--R a\|2 atan(------------------) - x atan(--------------) +--R 2 2 2 +--R 3x - 2a x +--R + +--R x +--R 2a atan(------------) +--R +---------+ +--R | 2 2 +--R \|- x + a +--R / +--R 2 +--R Type: Union(Expression Integer,...) +--E + +--S 114 +bb1:=x*acsc(x/a)+a*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +--R | 2 2 x +--R (2) a log(\|x - a + x) + x acsc(-) +--R a +--R Type: Expression Integer +--E + +--S 115 +bb2:=x*acsc(x/a)-a*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +--R | 2 2 x +--R (3) - a log(\|x - a + x) + x acsc(-) +--R a +--R Type: Expression Integer +--E + +--S 116 +cc1:=aa-bb1 +--R +--R (4) +--R +---------+ +--R +-------+ +-+ | 2 2 +--R | 2 2 +-+ 2x\|2 \|- x + a +--R - 2a log(\|x - a + x) + a\|2 atan(------------------) +--R 2 2 +--R 3x - 2a +--R + +--R +---------+ +--R | 2 2 +--R 2a\|- x + a x x +--R - x atan(--------------) + 2a atan(------------) - 2x acsc(-) +--R 2 +---------+ a +--R x | 2 2 +--R \|- x + a +--R / +--R 2 +--R Type: Expression Integer +--E + +--S 117 14:498 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +--R +-------+ +-+ | 2 2 +--R | 2 2 +-+ 2x\|2 \|- x + a +--R 2a log(\|x - a + x) + a\|2 atan(------------------) +--R 2 2 +--R 3x - 2a +--R + +--R +---------+ +--R | 2 2 +--R 2a\|- x + a x x +--R - x atan(--------------) + 2a atan(------------) - 2x acsc(-) +--R 2 +---------+ a +--R x | 2 2 +--R \|- x + a +--R / +--R 2 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.499~~~~~$\displaystyle +\int{x\csc^{-1}\frac{x}{a}}~dx$} +$$\int{x\csc^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{x^2}{2}\csc^{-1}\frac{x}{a}+\frac{a\sqrt{x^2-a^2}}{2} +{\rm \ if\ }0 < \csc^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +\frac{x^2}{2}\csc^{-1}\frac{x}{a}-\frac{a\sqrt{x^2-a^2}}{2} +{\rm \ if\ }-\frac{\pi}{2} < \csc^{-1}\frac{x}{a} < 0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 118 +aa:=integrate(x*acsc(x/a),x) +--R +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 2 2a\|- x + a | 2 2 +--R (- x + 2a )atan(--------------) - 2a\|- x + a +--R 2 +--R x +--R (1) ------------------------------------------------- +--R 4 +--R Type: Union(Expression Integer,...) +--E + +--S 119 +bb1:=x^2/2*acsc(x/a)+(a*sqrt(x^2-a^2))/2 +--R +--R +-------+ +--R | 2 2 2 x +--R a\|x - a + x acsc(-) +--R a +--R (2) ----------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 120 +bb2:=x^2/2*acsc(x/a)-(a*sqrt(x^2-a^2))/2 +--R +--R +-------+ +--R | 2 2 2 x +--R - a\|x - a + x acsc(-) +--R a +--R (3) ------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 121 +cc1:=aa-bb1 +--R +--R (4) +--R +---------+ +--R | 2 2 +-------+ +---------+ +--R 2 2 2a\|- x + a | 2 2 | 2 2 2 x +--R (- x + 2a )atan(--------------) - 2a\|x - a - 2a\|- x + a - 2x acsc(-) +--R 2 a +--R x +--R ----------------------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 122 14:499 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +--R | 2 2 +-------+ +---------+ +--R 2 2 2a\|- x + a | 2 2 | 2 2 2 x +--R (- x + 2a )atan(--------------) + 2a\|x - a - 2a\|- x + a - 2x acsc(-) +--R 2 a +--R x +--R ----------------------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.500~~~~~$\displaystyle +\int{x^2\csc^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\csc^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{x^3}{3}\csc^{-1}\frac{x}{a}+\frac{ax\sqrt{x^2-a^2}}{6} ++\frac{a^3}{6}\ln(x+\sqrt{x^2-a^2})\\ +\\ +\displaystyle +\hbox{\hskip 3cm}{\rm \ if\ }0 < \csc^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +\frac{x^3}{3}\sec^{-1}\frac{x}{a}-\frac{ax\sqrt{x^2-a^2}}{6} +-\frac{a^3}{6}\ln(x+\sqrt{x^2-a^2})\\ +\\ +\displaystyle +\hbox{\hskip 3cm} +{\rm \ if\ }-\frac{\pi}{2} < \csc^{-1}\frac{x}{a} < 0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 123 +aa:=integrate(x^2*acsc(x/a),x) +--R +--R +--R (1) +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +--R 3 +-+ 2x\|2 \|- x + a 3 2a\|- x + a +--R 2a \|2 atan(------------------) - x atan(--------------) +--R 2 2 2 +--R 3x - 2a x +--R + +--R +---------+ +--R 3 x | 2 2 +--R 5a atan(------------) - a x\|- x + a +--R +---------+ +--R | 2 2 +--R \|- x + a +--R / +--R 6 +--R Type: Union(Expression Integer,...) +--E + +--S 124 +bb1:=x^3/3*acsc(x/a)+(a*x*sqrt(x^2-a^2))/6+a^3/6*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 3 | 2 2 | 2 2 3 x +--R a log(\|x - a + x) + a x\|x - a + 2x acsc(-) +--R a +--R (2) -------------------------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 125 +bb2:=x^3/3*acsc(x/a)-(a*x*sqrt(x^2-a^2))/6-a^3/6*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 3 | 2 2 | 2 2 3 x +--R - a log(\|x - a + x) - a x\|x - a + 2x acsc(-) +--R a +--R (3) ---------------------------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 126 +cc1:=aa-bb1 +--R +--R (4) +--R +---------+ +--R +-------+ +-+ | 2 2 +--R 3 | 2 2 3 +-+ 2x\|2 \|- x + a +--R - a log(\|x - a + x) + 2a \|2 atan(------------------) +--R 2 2 +--R 3x - 2a +--R + +--R +---------+ +--R | 2 2 +-------+ +--R 3 2a\|- x + a 3 x | 2 2 +--R - x atan(--------------) + 5a atan(------------) - a x\|x - a +--R 2 +---------+ +--R x | 2 2 +--R \|- x + a +--R + +--R +---------+ +--R | 2 2 3 x +--R - a x\|- x + a - 2x acsc(-) +--R a +--R / +--R 6 +--R Type: Expression Integer +--E + +--S 127 14:500 Axiom cannot simplify this expression +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +--R +-------+ +-+ | 2 2 +--R 3 | 2 2 3 +-+ 2x\|2 \|- x + a +--R a log(\|x - a + x) + 2a \|2 atan(------------------) +--R 2 2 +--R 3x - 2a +--R + +--R +---------+ +--R | 2 2 +-------+ +--R 3 2a\|- x + a 3 x | 2 2 +--R - x atan(--------------) + 5a atan(------------) + a x\|x - a +--R 2 +---------+ +--R x | 2 2 +--R \|- x + a +--R + +--R +---------+ +--R | 2 2 3 x +--R - a x\|- x + a - 2x acsc(-) +--R a +--R / +--R 6 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.501~~~~~$\displaystyle +\int{\frac{\csc^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{\csc^{-1}(x/a)}{x}}= +-\left(\frac{a}{x}+\frac{(a/x)^3}{2\cdot 3\cdot 3} ++\frac{1\cdot 3(a/x)^5}{2\cdot 4\cdot 5\cdot 5} ++\frac{1\cdot 3\cdot 5(a/x)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7}+\cdots\right) +$$ +<<*>>= +)clear all + +--S 128 14:501 Axiom cannot compute this integral +aa:=integrate(acsc(x/a)/x,x) +--R +--R +--I %H +--R x acsc(--) +--R ++ a +--I (1) | -------- d%H +--I ++ %H +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.502~~~~~$\displaystyle +\int{\frac{\csc^{-1}(x/a)}{x^2}}~dx$} +$$\int{\frac{\csc^{-1}(x/a)}{x^2}}= +\left\{ +\begin{array}{l} +\displaystyle +-\frac{csc^{-1}(x/a)}{x}-\frac{\sqrt{x^2-a^2}}{ax} +{\rm \ if\ }0 < \csc^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +-\frac{csc^{-1}(x/a)}{x}+\frac{\sqrt{x^2-a^2}}{ax} +{\rm \ if\ }-\frac{\pi}{2} < \csc^{-1}\frac{x}{a} < 0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 129 +aa:=integrate(acsc(x/a)/x^2,x) +--R +--R +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +--R 2x\|2 \|- x + a +-+ 2a\|- x + a +--R - x atan(------------------) + a\|2 atan(--------------) +--R 2 2 2 +--R 3x - 2a x +--R (1) -------------------------------------------------------- +--R +-+ +--R 2a x\|2 +--R Type: Union(Expression Integer,...) +--E + +--S 130 +bb1:=-acsc(x/a)/x-sqrt(x^2-a^2)/(a*x) +--R +--R +-------+ +--R | 2 2 x +--R - \|x - a - a acsc(-) +--R a +--R (2) ------------------------ +--R a x +--R Type: Expression Integer +--E + +--S 131 +bb2:=-acsc(x/a)/x+sqrt(x^2-a^2)/(a*x) +--R +--R +-------+ +--R | 2 2 x +--R \|x - a - a acsc(-) +--R a +--R (3) ---------------------- +--R a x +--R Type: Expression Integer +--E + +--S 132 +cc1:=aa-bb1 +--R +--R (4) +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +--R 2x\|2 \|- x + a +-+ 2a\|- x + a +--R - x atan(------------------) + a\|2 atan(--------------) +--R 2 2 2 +--R 3x - 2a x +--R + +--R +-------+ +--R +-+ | 2 2 +-+ x +--R 2\|2 \|x - a + 2a\|2 acsc(-) +--R a +--R / +--R +-+ +--R 2a x\|2 +--R Type: Expression Integer +--E + +--S 133 14:502 Axiom cannot simplify this expression +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +---------+ +--R +-+ | 2 2 | 2 2 +--R 2x\|2 \|- x + a +-+ 2a\|- x + a +--R - x atan(------------------) + a\|2 atan(--------------) +--R 2 2 2 +--R 3x - 2a x +--R + +--R +-------+ +--R +-+ | 2 2 +-+ x +--R - 2\|2 \|x - a + 2a\|2 acsc(-) +--R a +--R / +--R +-+ +--R 2a x\|2 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.503~~~~~$\displaystyle +\int{x^m\sin^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\sin^{-1}\frac{x}{a}}= +\frac{x^{m+1}}{m+1}\sin^{-1}\frac{x}{a}-\frac{1}{m+1}\int{\frac{x^{m+1}}{\sqrt{a^2-x^2}}} +$$ +<<*>>= +)clear all + +--S 134 14:503 Axiom cannot compute this integral +aa:=integrate(x^m*asin(x/a),x) +--R +--R +--R x +--I ++ %H m +--I (1) | asin(--)%H d%H +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.504~~~~~$\displaystyle +\int{x^m\cos^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\cos^{-1}\frac{x}{a}}= +\frac{x^{m+1}}{m+1}\cos^{-1}\frac{x}{a}+\frac{1}{m+1}\int{\frac{x^{m+1}}{\sqrt{a^2-x^2}}} +$$ +<<*>>= +)clear all + +--S 135 14:504 Axiom cannot compute this integral +aa:=integrate(x^m*acos(x/a),x) +--R +--R +--R x +--I ++ %H m +--I (1) | acos(--)%H d%H +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.505~~~~~$\displaystyle +\int{x^m\tan^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\tan^{-1}\frac{x}{a}}= +\frac{x^{m_1}}{m+1}\tan^{-1}\frac{x}{a} +-\frac{a}{m+1}\int{\frac{x^{m+1}}{x^2+a^2}} +$$ +This appears to be an interesting integral. Axiom found a closed +form solution to the problem. However, the t1 integral below does +not have a closed form solution. Note that we did not return a +result for the prior two integrals, nor for the next integral. They +have the same form but are expressed in terms of asin, acos, and acot. +<<*>>= +)clear all + +--S 136 +aa:=integrate(x*m*atan(x/a),x) +--R +--R +--R 2 2 2a x +--R (- m x - a m)atan(-------) - 2a m x +--R 2 2 +--R x - a +--R (1) ------------------------------------ +--R 4 +--R Type: Union(Expression Integer,...) +--E + +--S 137 +t1:=integrate(x^(m+1)/(x^2+a^2),x) +--E +@ +Since we cannot get a closed form version of the prior integral +we proceed to try to prove that Axiom got a correct answer. We +do this by computing the derivate of 'aa' above and finding the +difference from the original formula. + +So first we generate the derivative: +<<*>>= + +--S 138 +bb:=D(aa,x) +--R +--R 2a x +--R m x atan(-------) +--R 2 2 +--R x - a +--R (3) - ----------------- +--R 2 +--R Type: Expression Integer +--E +@ +Then we input the original expression +<<*>>= +--S 139 +aa1:=x*m*atan(x/a) +--R +--R x +--R (4) m x atan(-) +--R a +--R Type: Expression Integer +--E +@ +Now we take their difference +<<*>>= +--S 140 +dd:=aa1-bb +--R +--R x 2a x +--R 2m x atan(-) + m x atan(-------) +--R a 2 2 +--R x - a +--R (5) -------------------------------- +--R 2 +--R Type: Expression Integer +--E +@ +Now we input the atan transformation +<<*>>= +--S 141 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E +@ +And apply the transformation to the difference +<<*>>= +--S 142 +ee:=atanrule dd +--R +--R 2 2 +--R x + 2%i a x - a - x + %i a +--R - %i m x log(-----------------) - 2%i m x log(----------) +--R 2 2 x + %i a +--R x - 2%i a x - a +--R (7) --------------------------------------------------------- +--R 4 +--R Type: Expression Complex Integer +--E +@ +And now we simplify +<<*>>= +--S 143 14:505 SCHAUMS AND AXIOM DISAGREE? (branch cuts?) +ff:=expandLog ee +--R +--R %i m x log(- 1) +--R (8) - --------------- +--R 2 +--R Type: Expression Complex Integer +--E +@ +And we get the surprising result that they are not equal. +In fact, they differ by a complex value depending on x. +Likely there is a branch-cut issue lurking somewhere. + +\section{\cite{1}:14.506~~~~~$\displaystyle +\int{x^m\cot^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\cot^{-1}\frac{x}{a}}= +\frac{x^{m+1}}{m+1}\cot^{-1}\frac{x}{a} ++\frac{a}{m+1}\int{\frac{x^{m+1}}{x^2+a^2}} +$$ +<<*>>= +)clear all + +--S 144 14:506 Axiom cannot compute this integral +aa:=integrate(x^m*acot(x/a),x) +--R +--R +--R x +--I ++ %H m +--I (1) | acot(--)%H d%H +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.507~~~~~$\displaystyle +\int{x^m\sec^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\sec^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{x^{m+1}\sec^{-1}(x/a)}{m+1}-\frac{a}{m+1}\int{\frac{x^m}{\sqrt{x^2-a^2}}} +{\rm \ if\ }0 < \sec^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +\frac{x^{m+1}\sec^{-1}(x/a)}{m+1}+\frac{a}{m+1}\int{\frac{x^m}{\sqrt{x^2-a^2}}} +{\rm \ if\ }\frac{\pi}{2} < \sec^{-1}\frac{x}{a} < \pi\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 145 14:507 Axiom cannot compute this integral +aa:=integrate(x^m*asec(x/a),x) +--R +--R +--R x +--I ++ %H m +--I (1) | asec(--)%H d%H +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.508~~~~~$\displaystyle +\int{x^m\csc^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\csc^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{x^{m+1}\csc^{-1}(x/a)}{m+1}+\frac{a}{m+1}\int{\frac{x^m}{\sqrt{x^2-a^2}}} +{\rm \ if\ }0 < \csc^{-1}\frac{x}{a} < \frac{\pi}{2}\\ +\\ +\displaystyle +\frac{x^{m+1}\csc^{-1}(x/a)}{m+1}-\frac{a}{m+1}\int{\frac{x^m}{\sqrt{x^2-a^2}}} +{\rm \ if\ }-\frac{\pi}{2} < \csc^{-1}\frac{x}{a} < 0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 146 14:508 Axiom cannot compute this integral +aa:=integrate(x^m*acsc(x/a),x) +--R +--R +--R x +--I ++ %H m +--I (1) | acsc(--)%H d%H +--R ++ a +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp82-84 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum24.input.pdf b/src/axiom-website/CATS/schaum24.input.pdf new file mode 100644 index 0000000..7b5f0e3 --- /dev/null +++ b/src/axiom-website/CATS/schaum24.input.pdf @@ -0,0 +1,4546 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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+aa:=integrate(%e^(a*x),x) +--R +--R a x +--R %e +--R (1) ----- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=%e^(a*x)/a +--R +--R a x +--R %e +--R (2) ----- +--R a +--R Type: Expression Integer +--E + +--S 3 14:509 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.510~~~~~$\displaystyle +\int{xe^{ax}}~dx$} +$$\int{xe^{ax}}= +\frac{e^{ax}}{x}\left(x-\frac{1}{a}\right) +$$ +<<*>>= +)clear all + +--S 4 +aa:=integrate(x*%e^(a*x),x) +--R +--R a x +--R (a x - 1)%e +--R (1) -------------- +--R 2 +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 5 +bb:=%e^(a*x)/a*(x-1/a) +--R +--R a x +--R (a x - 1)%e +--R (2) -------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 6 14:510 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.511~~~~~$\displaystyle +\int{x^2e^{ax}}~dx$} +$$\int{x^2e^{ax}}= +\frac{e^{ax}}{x}\left(x^2-\frac{2x}{a}+\frac{2}{a^2}\right) +$$ +<<*>>= +)clear all + +--S 7 +aa:=integrate(x^2*%e^(a*x),x) +--R +--R 2 2 a x +--R (a x - 2a x + 2)%e +--R (1) ---------------------- +--R 3 +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 8 +bb:=%e^(a*x)/a*(x^2-(2*x)/a+2/a^2) +--R +--R 2 2 a x +--R (a x - 2a x + 2)%e +--R (2) ---------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 9 14:511 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.512~~~~~$\displaystyle +\int{x^ne^{ax}}~dx$} +$$\begin{array}{rl} +\displaystyle\int{x^ne^{ax}}=& +\displaystyle +\frac{x^ne^{ax}}{a}-\frac{n}{a}\int{x^{n-1}e^{ax}}\\ +\\ +&\displaystyle +=\frac{e^{ax}}{x}\left(x^n-\frac{nx^{n-1}}{a}+\frac{n(n-1)x^{n-2}}{a^2} +-\cdots \frac{(-1)^nn!}{a^n}\right) +\\ +&\hbox{\hskip 5cm}{\rm\ if\ }n={\rm positive integer} +\end{array} +$$ +<<*>>= +)clear all + +--S 10 14:512 Axiom cannot compute this integral +aa:=integrate(x^n*%e^(a*x),x) +--R +--R x +--I ++ %I a n +--I (1) | %e %I d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E + +@ + +\section{\cite{1}:14.513~~~~~$\displaystyle +\int{\frac{e^{ax}}{x}}~dx$} +$$\int{\frac{e^{ax}}{x}}= +\ln{x}+\frac{ax}{1\cdot 1!}+\frac{(ax)^2}{2\cdot 2!} ++\frac{(ax)^3}{3\cdot 3!}+\cdots +$$ +<<*>>= +)clear all + +--S 11 14:513 Schaums and Axiom agree by definition +aa:=integrate(%e^(a*x)/x,x) +--R +--R (1) Ei(a x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.514~~~~~$\displaystyle +\int{\frac{e^{ax}}{x^n}}~dx$} +$$\int{\frac{e^{ax}}{x^n}}= +\frac{-e^{ax}}{(n-1)x^{n-1}}+\frac{a}{n-1}\int{\frac{e^{ax}}{x^{n-1}}} +$$ +<<*>>= +)clear all + +--S 12 14:514 Axiom cannot compute this integral +aa:=integrate(%e^(a*x)/x^n,x) +--R +--I x %I a +--R ++ %e +--I (1) | ------ d%I +--R ++ n +--I %I +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.515~~~~~$\displaystyle +\int{\frac{dx}{p+qe^{ax}}}~dx$} +$$\int{\frac{1}{p+qe^{ax}}}= +\frac{x}{p}-\frac{1}{ap}\ln(p+qe^{ax}) +$$ +<<*>>= +)clear all + +--S 13 +aa:=integrate(1/(p+q*%e^(a*x)),x) +--R +--R a x +--R - log(q %e + p) + a x +--R (1) ------------------------ +--R a p +--R Type: Union(Expression Integer,...) +--E + +--S 14 +bb:=x/p-1/(a*p)*log(p+q*%e^(a*x)) +--R +--R a x +--R - log(q %e + p) + a x +--R (2) ------------------------ +--R a p +--R Type: Expression Integer +--E + +--S 15 14:515 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.516~~~~~$\displaystyle +\int{\frac{dx}{(p+qe^{ax})^2}}~dx$} +$$\int{\frac{dx}{(p+qe^{ax})^2}}= +\frac{x}{p^2}+\frac{1}{ap(p+qe^{ax})}-\frac{1}{ap^2}\ln(p+qe^{ax}) +$$ +<<*>>= +)clear all + +--S 16 +aa:=integrate(1/(p+q*%e^(a*x))^2,x) +--R +--R a x a x a x +--R (- q %e - p)log(q %e + p) + a q x %e + a p x + p +--R (1) --------------------------------------------------------- +--R 2 a x 3 +--R a p q %e + a p +--R Type: Union(Expression Integer,...) +--E + +--S 17 +bb:=x/p^2+1/(a*p*(p+q*%e^(a*x)))-1/(a*p^2)*log(p+q*%e^(a*x)) +--R +--R a x a x a x +--R (- q %e - p)log(q %e + p) + a q x %e + a p x + p +--R (2) --------------------------------------------------------- +--R 2 a x 3 +--R a p q %e + a p +--R Type: Expression Integer +--E + +--S 18 14:516 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.517~~~~~$\displaystyle +\int{\frac{dx}{pe^{ax}+qe^{ax}}}~dx$} +$$\int{\frac{dx}{pe^{ax}+qe^{ax}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{a\sqrt{pq}}\tan^{-1}\left(\sqrt{\frac{p}{q}}e^{ax}\right)\\ +\\ +\displaystyle +\frac{1}{2a\sqrt{-pq}} +\ln\left(\frac{e^{ax}-\sqrt{-q/p}}{e^{ax}+\sqrt{-q/p}}\right) +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 19 +aa:=integrate(1/(p*%e^(a*x)+q*%e^-(a*x)),x) +--R +--R a x 2 +-----+ a x +--R (p (%e ) - q)\|- p q + 2p q %e a x +---+ +--R log(-------------------------------------) %e \|p q +--R a x 2 atan(-----------) +--R p (%e ) + q q +--R (1) [------------------------------------------,-----------------] +--R +-----+ +---+ +--R 2a\|- p q a\|p q +--R Type: Union(List Expression Integer,...) +--E + +--S 20 +bb1:=1/(a*sqrt(p*q))*atan(sqrt(p/q)*%e^(a*x)) +--R +--R +-+ +--R a x |p +--R atan(%e |- ) +--R \|q +--R (2) --------------- +--R +---+ +--R a\|p q +--R Type: Expression Integer +--E + +--S 21 +bb2:=1/(2*a*sqrt(-p*q))*log((%e^(a*x)-sqrt(-q/p))/(%e^(a*x)+sqrt(-q/p))) +--R +--R +---+ +--R | q a x +--R - |- - + %e +--R \| p +--R log(----------------) +--R +---+ +--R | q a x +--R |- - + %e +--R \| p +--R (3) --------------------- +--R +-----+ +--R 2a\|- p q +--R Type: Expression Integer +--E + +--S 22 +cc1:=aa.1-bb1 +--R +--R (4) +--R a x 2 +-----+ a x +-+ +--R +---+ (p (%e ) - q)\|- p q + 2p q %e +-----+ a x |p +--R \|p q log(-------------------------------------) - 2\|- p q atan(%e |- ) +--R a x 2 \|q +--R p (%e ) + q +--R --------------------------------------------------------------------------- +--R +-----+ +---+ +--R 2a\|- p q \|p q +--R Type: Expression Integer +--E + +--S 23 +cc2:=aa.2-bb1 +--R +--R a x +---+ +-+ +--R %e \|p q a x |p +--R atan(-----------) - atan(%e |- ) +--R q \|q +--R (5) ----------------------------------- +--R +---+ +--R a\|p q +--R Type: Expression Integer +--E + +--S 24 +cc3:=aa.1-bb2 +--R +--R +---+ +--R | q a x +--R a x 2 +-----+ a x - |- - + %e +--R (p (%e ) - q)\|- p q + 2p q %e \| p +--R log(-------------------------------------) - log(----------------) +--R a x 2 +---+ +--R p (%e ) + q | q a x +--R |- - + %e +--R \| p +--R (6) ------------------------------------------------------------------ +--R +-----+ +--R 2a\|- p q +--R Type: Expression Integer +--E + +--S 25 14:517 Axiom cannot simplify these expressions +cc4:=aa.2-bb2 +--R +--R +---+ +--R | q a x +--R - |- - + %e a x +---+ +--R +---+ \| p +-----+ %e \|p q +--R - \|p q log(----------------) + 2\|- p q atan(-----------) +--R +---+ q +--R | q a x +--R |- - + %e +--R \| p +--R (7) ---------------------------------------------------------- +--R +-----+ +---+ +--R 2a\|- p q \|p q +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.518~~~~~$\displaystyle +\int{e^{ax}\sin{bx}}~dx$} +$$\int{e^{ax}\sin{bx}}= +\frac{e^{ax}(a\sin{bx}-b\cos{bx})}{a^2+b^2} +$$ +<<*>>= +)clear all + +--S 26 +aa:=integrate(%e^(a*x)*sin(b*x),x) +--R +--R a x a x +--R a %e sin(b x) - b cos(b x)%e +--R (1) --------------------------------- +--R 2 2 +--R b + a +--R Type: Union(Expression Integer,...) +--E + +--S 27 +bb:=((%e^(a*x))*(a*sin(b*x)-b*cos(b*x)))/(a^2+b^2) +--R +--R a x a x +--R a %e sin(b x) - b cos(b x)%e +--R (2) --------------------------------- +--R 2 2 +--R b + a +--R Type: Expression Integer +--E + +--S 28 14:518 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.519~~~~~$\displaystyle +\int{e^{ax}\cos{bx}}~dx$} +$$\int{e^{ax}\cos{bx}}= +\frac{e^{ax}(a\cos{bx}-b\sin{bx})}{a^2+b^2} +$$ +<<*>>= +)clear all + +--S 29 +aa:=integrate(%e^(a*x)*cos(b*x),x) +--R +--R a x a x +--R b %e sin(b x) + a cos(b x)%e +--R (1) --------------------------------- +--R 2 2 +--R b + a +--R Type: Union(Expression Integer,...) +--E + +--S 30 +bb:=((%e^(a*x))*(a*cos(b*x)+b*sin(b*x)))/(a^2+b^2) +--R +--R a x a x +--R b %e sin(b x) + a cos(b x)%e +--R (2) --------------------------------- +--R 2 2 +--R b + a +--R Type: Expression Integer +--E + +--S 31 14:519 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.520~~~~~$\displaystyle +\int{xe^{ax}\sin{bx}}~dx$} +$$\int{xe^{ax}\sin{bx}}= +\frac{xe^{ax}(a\sin{bx}-b\cos{bx})}{a^2+b^2} +-\frac{e^{ax}\left((a^2-b^2)\sin{bx}-2ab\cos{bx}\right)}{(a^2+b^2)^2} +$$ +<<*>>= +)clear all + +--S 32 +aa:=integrate(x*%e^(a*x)*sin(b*x),x) +--R +--R (1) +--R 2 3 2 2 a x 3 2 a x +--R ((a b + a )x + b - a )%e sin(b x) + ((- b - a b)x + 2a b)cos(b x)%e +--R --------------------------------------------------------------------------- +--R 4 2 2 4 +--R b + 2a b + a +--R Type: Union(Expression Integer,...) +--E + +--S 33 +bb:=(x*%e^(a*x)*(a*sin(b*x)-b*cos(b*x)))/(a^2+b^2)-(%e^(a*x)*((a^2-b^2)*sin(b*x)-2*a*b*cos(b*x)))/(a^2+b^2)^2 +--R +--R (2) +--R 2 3 2 2 a x 3 2 a x +--R ((a b + a )x + b - a )%e sin(b x) + ((- b - a b)x + 2a b)cos(b x)%e +--R --------------------------------------------------------------------------- +--R 4 2 2 4 +--R b + 2a b + a +--R Type: Expression Integer +--E + +--S 34 14:520 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.521~~~~~$\displaystyle +\int{xe^{ax}\cos{bx}}~dx$} +$$\int{xe^{ax}\cos{bx}}= +\frac{xe^{ax}(a\cos{bx}-b\sin{bx})}{a^2+b^2} +-\frac{e^{ax}\left((a^2-b^2)\cos{bx}-2ab\sin{bx}\right)}{(a^2+b^2)^2} +$$ +<<*>>= +)clear all + +--S 35 +aa:=integrate(x*%e^(a*x)*cos(b*x),x) +--R +--R (1) +--R 3 2 a x 2 3 2 2 a x +--R ((b + a b)x - 2a b)%e sin(b x) + ((a b + a )x + b - a )cos(b x)%e +--R ------------------------------------------------------------------------- +--R 4 2 2 4 +--R b + 2a b + a +--R Type: Union(Expression Integer,...) +--E + +--S 36 +bb:=(x*%e^(a*x)*(a*cos(b*x)+b*sin(b*x)))/(a^2+b^2)-(%e^(a*x)*((a^2-b^2)*cos(b*x)+2*a*b*sin(b*x)))/(a^2+b^2)^2 +--R +--R (2) +--R 3 2 a x 2 3 2 2 a x +--R ((b + a b)x - 2a b)%e sin(b x) + ((a b + a )x + b - a )cos(b x)%e +--R ------------------------------------------------------------------------- +--R 4 2 2 4 +--R b + 2a b + a +--R Type: Expression Integer +--E + +--S 37 14:521 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.522~~~~~$\displaystyle +\int{e^{ax}\ln{x}}~dx$} +$$\int{e^{ax}\ln{x}}= +\frac{e^{ax}\ln{x}}{a}-\frac{1}{a}\int{\frac{e^{ax}}{x}} +$$ +<<*>>= +)clear all + +--S 38 14:522 Schaums and Axiom agree by definition +aa:=integrate(%e^(a*x)*log(x),x) +--R +--R a x +--R %e log(x) - Ei(a x) +--R (1) --------------------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.523~~~~~$\displaystyle +\int{e^{ax}\sin^n{bx}}~dx$} +$$\int{e^{ax}\sin^n{bx}}= +\frac{e^{ax}\sin^{n-1}{bx}}{a^2+n^2b^2}(a\sin{bx}-nb\cos{bx}) ++\frac{n(n-1)b^2}{a^2+n^2b^2}\int{e^{ax}\sin^{n-2}{bx}} +$$ +<<*>>= +)clear all + +--S 39 14:523 Axiom cannot compute this integral +aa:=integrate(%e^(a*x)*sin(b*x)^n,x) +--R +--R x +--I ++ %I a n +--I (1) | %e sin(%I b) d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.524~~~~~$\displaystyle +\int{e^{ax}\cos^n{bx}}~dx$} +$$\int{e^{ax}\cos^n{bx}}= +\frac{e^{ax}\cos^{n-1}{bx}}{a^2+n^2b^2}(a\cos{bx}-nb\sin{bx}) ++\frac{n(n-1)b^2}{a^2+n^2b^2}\int{e^{ax}\cos^{n-2}{bx}} +$$ +<<*>>= +)clear all + +--S 40 14:524 Axiom cannot compute this integral +aa:=integrate(%e^(a*x)*cos(b*x)^n,x) +--R +--R x +--I ++ %I a n +--I (1) | %e cos(%I b) d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p85 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum25.input.pdf b/src/axiom-website/CATS/schaum25.input.pdf new file mode 100644 index 0000000..3ceaeb9 --- /dev/null +++ b/src/axiom-website/CATS/schaum25.input.pdf @@ -0,0 +1,2023 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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+/Info 1 0 R +>> +startxref +112625 +%%EOF diff --git a/src/axiom-website/CATS/schaum26.input.pamphlet b/src/axiom-website/CATS/schaum26.input.pamphlet new file mode 100644 index 0000000..9bdeec5 --- /dev/null +++ b/src/axiom-website/CATS/schaum26.input.pamphlet @@ -0,0 +1,552 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum26.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.525~~~~~$\displaystyle +\int{ln x}~dx$} +$$\int{ln x}= +x\ln{x}-x +$$ +<<*>>= +)spool schaum26.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(log(x),x) +--R +--R +--R (1) x log(x) - x +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=x*log(x)-x +--R +--R (2) x log(x) - x +--R Type: Expression Integer +--E + +--S 3 14:525 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.526~~~~~$\displaystyle +\int{x\ln{x}}~dx$} +$$\int{x\ln{x}}= +\frac{x^2}{2}\left(\ln{x}-\frac{1}{2}\right) +$$ +<<*>>= +)clear all + +--S 4 +aa:=integrate(x*log(x),x) +--R +--R +--R 2 2 +--R 2x log(x) - x +--R (1) -------------- +--R 4 +--R Type: Union(Expression Integer,...) +--E + +--S 5 +bb:=x^2/2*(log(x)-1/2) +--R +--R 2 2 +--R 2x log(x) - x +--R (2) -------------- +--R 4 +--R Type: Expression Integer +--E + +--S 6 14:526 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.527~~~~~$\displaystyle +\int{x^m\ln{x}}~dx$} +$$\int{x^m\ln{x}}= +\frac{x^{m+1}}{m+1}\left(\ln{x}-\frac{1}{m+1}\right) +$$ +<<*>>= +)clear all + +--S 7 +aa:=integrate(x^m*log(x),x) +--R +--R +--R m log(x) +--R ((m + 1)x log(x) - x)%e +--R (1) ------------------------------- +--R 2 +--R m + 2m + 1 +--R Type: Union(Expression Integer,...) +--E + +--S 8 +bb:=x^(m+1)/(m+1)*(log(x)-1/(m+1)) +--R +--R m + 1 +--R ((m + 1)log(x) - 1)x +--R (2) ------------------------- +--R 2 +--R m + 2m + 1 +--R Type: Expression Integer +--E + +--S 9 +cc:=aa-bb +--R +--R m log(x) m + 1 +--R ((m + 1)x log(x) - x)%e + ((- m - 1)log(x) + 1)x +--R (3) ------------------------------------------------------------- +--R 2 +--R m + 2m + 1 +--R Type: Expression Integer +--E + +--S 10 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 11 +dd:=explog cc +--R +--R m + 1 m +--R ((- m - 1)log(x) + 1)x + ((m + 1)x log(x) - x)x +--R (5) ----------------------------------------------------- +--R 2 +--R m + 2m + 1 +--R Type: Expression Integer +--E + +--S 12 14:527 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.528~~~~~$\displaystyle +\int{\frac{\ln{x}}{x}}~dx$} +$$\int{\frac{\ln{x}}{x}}= +\frac{1}{2}\ln^2{x} +$$ +<<*>>= +)clear all + +--S 13 +aa:=integrate(log(x)/x,x) +--R +--R +--R 2 +--R log(x) +--R (1) ------- +--R 2 +--R Type: Union(Expression Integer,...) +--E + +--S 14 +bb:=1/2*log(x)^2 +--R +--R 2 +--R log(x) +--R (2) ------- +--R 2 +--R Type: Expression Integer +--E + +--S 15 14:528 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.529~~~~~$\displaystyle +\int{\frac{\ln{x}}{x^2}}~dx$} +$$\int{\frac{\ln{x}}{x^2}}= +-\frac{\ln{x}}{x}-\frac{1}{x} +$$ +<<*>>= +)clear all + +--S 16 +aa:=integrate(log(x)/x^2,x) +--R +--R +--R - log(x) - 1 +--R (1) ------------ +--R x +--R Type: Union(Expression Integer,...) +--E + +--S 17 +bb:=-log(x)/x-1/x +--R +--R - log(x) - 1 +--R (2) ------------ +--R x +--R Type: Expression Integer +--E + +--S 18 14:529 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.530~~~~~$\displaystyle +\int{\ln^2{x}}~dx$} +$$\int{\ln^2{x}}= +x\ln^2{x}-2x\ln{x}+2x +$$ +<<*>>= +)clear all + +--S 19 +aa:=integrate(log(x)^2,x) +--R +--R +--R 2 +--R (1) x log(x) - 2x log(x) + 2x +--R Type: Union(Expression Integer,...) +--E + +--S 20 +bb:=x*log(x)^2-2*x*log(x)+2*x +--R +--R 2 +--R (2) x log(x) - 2x log(x) + 2x +--R Type: Expression Integer +--E + +--S 21 14:530 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.531~~~~~$\displaystyle +\int{\frac{\ln^n{x}}{x}}~dx$} +$$\int{\frac{\ln^n{x}}{x}}= +\frac{ln^{n+1}{x}}{n+1} +$$ +<<*>>= +)clear all + +--S 22 +aa:=integrate(log(x)^n/x,x) +--R +--R +--R n log(log(x)) +--R log(x)%e +--R (1) --------------------- +--R n + 1 +--R Type: Union(Expression Integer,...) +--E + +--S 23 +bb:=log(x)^(n+1)/(n+1) +--R +--R n + 1 +--R log(x) +--R (2) ----------- +--R n + 1 +--R Type: Expression Integer +--E + +--S 24 +cc:=aa-bb +--R +--R n log(log(x)) n + 1 +--R log(x)%e - log(x) +--R (3) ----------------------------------- +--R n + 1 +--R Type: Expression Integer +--E + +--S 25 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 26 +dd:=explog cc +--R +--R n + 1 n +--R - log(x) + log(x)log(x) +--R (5) ----------------------------- +--R n + 1 +--R Type: Expression Integer +--E + +--S 27 14:531 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.532~~~~~$\displaystyle +\int{\frac{dx}{x\ln{x}}}$} +$$\int{\frac{1}{x\ln{x}}}= +\ln(\ln{x}) +$$ +<<*>>= +)clear all + +--S 28 +aa:=integrate(1/(x*log(x)),x) +--R +--R +--R (1) log(log(x)) +--R Type: Union(Expression Integer,...) +--E + +--S 29 +bb:=log(log(x)) +--R +--R (2) log(log(x)) +--R Type: Expression Integer +--E + +--S 30 14:532 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.533~~~~~$\displaystyle +\int{\frac{dx}{\ln{x}}}$} +$$\int{\frac{1}{\ln{x}}}= +\ln(\ln{x})+\ln{x}+\frac{\ln^2{x}}{2\cdot 2!} ++\frac{\ln^3{x}}{3\cdot 3!}+\cdots +$$ +<<*>>= +)clear all + +--S 31 14:533 Schaums and Axiom agree by definition +aa:=integrate(1/log(x),x) +--R +--R +--R (1) li(x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.534~~~~~$\displaystyle +\int{\frac{x^m}{\ln{x}}}~dx$} +$$\int{\frac{x^m}{\ln{x}}}= +\ln(\ln{x})+(m+1)\ln{x}+\frac{(m+1)^2\ln^2{x}}{2\cdot 2!} ++\frac{(m+1)^3\ln^3{x}}{3\cdot 3!}+\cdots +$$ +<<*>>= +)clear all + +--S 32 14:534 Axiom cannot compute this integral +aa:=integrate(x^m/log(x),x) +--R +--R +--R x m +--I ++ %I +--I (1) | ------- d%I +--I ++ log(%I) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.535~~~~~$\displaystyle +\int{\ln^n{x}}~dx$} +$$\int{\ln^n{x}}= +x\ln^n{x}-n\int{\ln^{n-1}{x}} +$$ +<<*>>= +)clear all + +--S 33 14:535 Axiom cannot compute this integral +aa:=integrate(log(x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | log(%I) d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.536~~~~~$\displaystyle +\int{x^m\ln^n{x}}~dx$} +$$\int{x^m\ln^n{x}}= +\frac{x^{m+1}\ln^n{x}}{m+1}-\frac{n}{m+1}\int{x^m\ln^{n-1}{x}} +$$ +<<*>>= +)clear all + +--S 34 14:536 Axiom cannot compute this integral +aa:=integrate(x^m*log(x)^n,x) +--R +--R +--R x +--R ++ m n +--I (1) | %I log(%I) d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.537~~~~~$\displaystyle +\int{\ln{(x^2+a^2)}}~dx$} +$$\int{\ln{(x^2+a^2)}}= +x\ln(x^2+a^2)-2x+2a\tan^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 35 +aa:=integrate(log(x^2+a^2),x) +--R +--R +--R 2 2 x +--R (1) x log(x + a ) + 2a atan(-) - 2x +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 36 +bb:=x*log(x^2+a^2)-2*x+2*a*atan(x/a) +--R +--R 2 2 x +--R (2) x log(x + a ) + 2a atan(-) - 2x +--R a +--R Type: Expression Integer +--E + +--S 37 14:537 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.538~~~~~$\displaystyle +\int{\ln(x^2-a^2)}~dx$} +$$\int{\ln(x^2-a^2)}= +x\ln(x^2-a^2)-2x+a\ln\left(\frac{x+a}{x-a}\right) +$$ +<<*>>= +)clear all + +--S 38 +aa:=integrate(log(x^2-a^2),x) +--R +--R +--R 2 2 +--R (1) x log(x - a ) + a log(x + a) - a log(x - a) - 2x +--R Type: Union(Expression Integer,...) +--E + +--S 39 +bb:=x*log(x^2-a^2)-2*x+a*log((x+a)/(x-a)) +--R +--R 2 2 x + a +--R (2) x log(x - a ) + a log(-----) - 2x +--R x - a +--R Type: Expression Integer +--E + +--S 40 +cc:=aa-bb +--R +--R x + a +--R (3) a log(x + a) - a log(x - a) - a log(-----) +--R x - a +--R Type: Expression Integer +--E + +--S 41 14:538 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.539~~~~~$\displaystyle +\int{x^m\ln(x^2\pm a^2)}~dx$} +$$\int{x^m\ln(x^2\pm a^2)}= +\frac{x^{m-1}\ln(x^2\pm a^2)}{m+1} +-\frac{2}{m+1}\int{\frac{x^{m+2}}{x^2\pm a^2}} +$$ +<<*>>= +)clear all + +--S 42 +aa:=integrate(x^m*log(x^2+a^2),x) +--R +--R +--R x +--R ++ 2 2 m +--I (1) | log(a + %I )%I d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E + +)clear all + +--S 43 14:539 Axiom cannot compute this integral +aa:=integrate(x^m*log(x^2-a^2),x) +--R +--R +--R x +--R ++ 2 2 m +--I (1) | log(- a + %I )%I d%I +--R ++ +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p86 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum26.input.pdf b/src/axiom-website/CATS/schaum26.input.pdf new file mode 100644 index 0000000..3e64ec6 --- /dev/null +++ b/src/axiom-website/CATS/schaum26.input.pdf @@ -0,0 +1,1950 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.540~~~~~$\displaystyle +\int{\sinh{ax}}~dx$} +$$\int{\sinh{ax}}= +\frac{\cosh{ax}}{a} +$$ +<<*>>= +)spool schaum27.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(sinh(a*x),x) +--R +--R cosh(a x) +--R (1) --------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=cosh(a*x)/a +--R +--R cosh(a x) +--R (2) --------- +--R a +--R Type: Expression Integer +--E + +--S 3 14:540 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.541~~~~~$\displaystyle +\int{x\sinh{ax}}~dx$} +$$\int{x\sinh{ax}}= +\frac{x*\cosh{ax}}{a}-\frac{\sinh{ax}}{a^2} +$$ +<<*>>= +)clear all + +--S 4 +aa:=integrate(x*sinh(a*x),x) +--R +--R +--R - sinh(a x) + a x cosh(a x) +--R (1) --------------------------- +--R 2 +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 5 +bb:=(x*cosh(a*x))/a-sinh(a*x)/a^2 +--R +--R - sinh(a x) + a x cosh(a x) +--R (2) --------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 6 14:541 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.542~~~~~$\displaystyle +\int{x^2\sinh{ax}}~dx$} +$$\int{x^2\sinh{ax}}= +\left(\frac{x^2}{a}+\frac{2}{a^3}\right)\cosh{ax}-\frac{2x}{a^2}\sinh{ax} +$$ +<<*>>= +)clear all + +--S 7 +aa:=integrate(x^2*sinh(a*x),x) +--R +--R +--R 2 2 +--R - 2a x sinh(a x) + (a x + 2)cosh(a x) +--R (1) -------------------------------------- +--R 3 +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 8 +bb:=(x^2/a+2/a^3)*cosh(a*x)-(2*x)/a^2*sinh(a*x) +--R +--R 2 2 +--R - 2a x sinh(a x) + (a x + 2)cosh(a x) +--R (2) -------------------------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 9 14:542 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.543~~~~~$\displaystyle +\int{\frac{\sinh{ax}}{x}}~dx$} +$$\int{\frac{\sinh{ax}}{x}}= +ax+\frac{(ax)^3}{3\cdot 3!}+\frac{(ax)^5}{5\cdot 5!}+\cdots +$$ +<<*>>= +)clear all + +--S 10 14:543 Axiom cannot compute this integral +aa:=integrate(sinh(a*x)/x,x) +--R +--R +--R x +--I ++ sinh(%N a) +--I (1) | ---------- d%N +--I ++ %N +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.544~~~~~$\displaystyle +\int{\frac{\sinh{ax}}{x^2}}~dx$} +$$\int{\frac{\sinh{ax}}{x^2}}= +-\frac{\sinh{ax}}{x}+\int{\frac{\cosh{ax}}{x}} +$$ +<<*>>= +)clear all + +--S 11 14:544 Axiom cannot compute this integral +aa:=integrate(sinh(a*x)/x^2,x) +--R +--R +--R x +--I ++ sinh(%N a) +--I (1) | ---------- d%N +--R ++ 2 +--I %N +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.545~~~~~$\displaystyle +\int{\frac{dx}{\sinh{ax}}}~dx$} +$$\int{\frac{1}{\sinh{ax}}}= +\frac{1}{a}\ln\tanh{\frac{ax}{2}} +$$ +<<*>>= +)clear all + +--S 12 +aa:=integrate(1/sinh(a*x),x) +--R +--R +--R - log(sinh(a x) + cosh(a x) + 1) + log(sinh(a x) + cosh(a x) - 1) +--R (1) ----------------------------------------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 13 +bb:=1/a*log(tanh(a*x)/2) +--R +--R tanh(a x) +--R log(---------) +--R 2 +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 14 14:545 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R tanh(a x) +--R - log(---------) - log(sinh(a x) + cosh(a x) + 1) +--R 2 +--R + +--R log(sinh(a x) + cosh(a x) - 1) +--R / +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.546~~~~~$\displaystyle +\int{\frac{x~dx}{\sinh{ax}}}~dx$} +$$\int{\frac{x}{\sinh{ax}}}= +\frac{1}{a^2}\left\{ax-\frac{(ax)^3}{18}+\frac{7(ax)^5}{1800}-\cdots ++\frac{2(-1)^n(2^{2n-1})B_n(ax)^{2n+1}}{(2n+1)!}+\cdots\right\} +$$ +<<*>>= +)clear all + +--S 15 14:546 Axiom cannot compute this integral +aa:=integrate(x/sinh(a*x),x) +--R +--R +--R x +--I ++ %N +--I (1) | ---------- d%N +--I ++ sinh(%N a) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.547~~~~~$\displaystyle +\int{\sinh^2{ax}}~dx$} +$$\int{\sinh^2{ax}}= +\frac{\sinh{ax}\cosh{ax}}{2a}-\frac{x}{2} +$$ +<<*>>= +)clear all + +--S 16 +aa:=integrate(sinh(a*x)^2,x) +--R +--R +--R cosh(a x)sinh(a x) - a x +--R (1) ------------------------ +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 17 +bb:=(sinh(a*x)*cosh(a*x))/(2*a)-x/2 +--R +--R cosh(a x)sinh(a x) - a x +--R (2) ------------------------ +--R 2a +--R Type: Expression Integer +--E + +--S 18 14:547 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.548~~~~~$\displaystyle +\int{x\sinh^2{ax}}~dx$} +$$\int{x\sinh^2{ax}}= +\frac{x*\sinh{2ax}}{4a}-\frac{\cosh{2ax}}{8a^2}-\frac{x^2}{4} +$$ +<<*>>= +)clear all + +--S 19 +aa:=integrate(x*sinh(a*x)^2,x) +--R +--R +--R 2 2 2 2 +--R - sinh(a x) + 4a x cosh(a x)sinh(a x) - cosh(a x) - 2a x +--R (1) ----------------------------------------------------------- +--R 2 +--R 8a +--R Type: Union(Expression Integer,...) +--E + +--S 20 +bb:=(x*sinh(2*a*x))/(4*a)-cosh(2*a*x)/(8*a^2)-x^2/4 +--R +--R 2 2 +--R 2a x sinh(2a x) - cosh(2a x) - 2a x +--R (2) ------------------------------------ +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 21 +cc:=aa-bb +--R +--R (3) +--R 2 +--R - 2a x sinh(2a x) - sinh(a x) + 4a x cosh(a x)sinh(a x) + cosh(2a x) +--R + +--R 2 +--R - cosh(a x) +--R / +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 22 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 23 +dd:=sinhsqrrule cc +--R +--R (5) +--R 2 +--R - 4a x sinh(2a x) + 8a x cosh(a x)sinh(a x) + cosh(2a x) - 2cosh(a x) + 1 +--R -------------------------------------------------------------------------- +--R 2 +--R 16a +--R Type: Expression Integer +--E + +--S 24 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 25 +ee:=coshsqrrule dd +--R +--R - x sinh(2a x) + 2x cosh(a x)sinh(a x) +--R (7) -------------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 26 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %K sinh(y + x) - %K sinh(y - x) +--I (8) %K cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 27 14:548 Schaums and Axiom agree +ff:=sinhcoshrule ee +--R +--R (9) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.549~~~~~$\displaystyle +\int{\frac{dx}{\sinh^2{ax}}}~dx$} +$$\int{\frac{1}{\sinh^2{ax}}}= +-\frac{\coth{ax}}{a} +$$ +<<*>>= +)clear all + +--S 28 +aa:=integrate(1/sinh(a*x)^2,x) +--R +--R +--R 2 +--R (1) - ------------------------------------------------------- +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Union(Expression Integer,...) +--E + +--S 29 +bb:=-coth(a*x)/a +--R +--R coth(a x) +--R (2) - --------- +--R a +--R Type: Expression Integer +--E + +--S 30 +cc:=aa-bb +--R +--R (3) +--R 2 +--R coth(a x)sinh(a x) + 2cosh(a x)coth(a x)sinh(a x) +--R + +--R 2 +--R (cosh(a x) - 1)coth(a x) - 2 +--R / +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E + +--S 31 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 32 +dd:=sinhsqrrule cc +--R +--R (5) +--R 2 +--R 4cosh(a x)coth(a x)sinh(a x) + (cosh(2a x) + 2cosh(a x) - 3)coth(a x) - 4 +--R -------------------------------------------------------------------------- +--R 2 +--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) - 3a +--R Type: Expression Integer +--E + +--S 33 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 34 +ee:=coshsqrrule dd +--R +--R 2cosh(a x)coth(a x)sinh(a x) + (cosh(2a x) - 1)coth(a x) - 2 +--R (7) ------------------------------------------------------------ +--R 2a cosh(a x)sinh(a x) + a cosh(2a x) - a +--R Type: Expression Integer +--E + +--S 35 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--I %B sinh(y + x) - %B sinh(y - x) +--I (8) %B cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 36 +ff:=sinhcoshrule ee +--R +--R coth(a x)sinh(2a x) + (cosh(2a x) - 1)coth(a x) - 2 +--R (9) --------------------------------------------------- +--R a sinh(2a x) + a cosh(2a x) - a +--R Type: Expression Integer +--E + +--S 37 +cothrule:=rule(coth(x) == cosh(x)/sinh(x)) +--R +--R cosh(x) +--R (10) coth(x) == ------- +--R sinh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 38 +gg:=cothrule ff +--R +--R cosh(a x)sinh(2a x) - 2sinh(a x) + cosh(a x)cosh(2a x) - cosh(a x) +--R (11) ------------------------------------------------------------------ +--R a sinh(a x)sinh(2a x) + (a cosh(2a x) - a)sinh(a x) +--R Type: Expression Integer +--E + +--S 39 +hh:=sinhcoshrule gg +--R +--R sinh(3a x) - 3sinh(a x) + 2cosh(a x)cosh(2a x) - 2cosh(a x) +--R (12) ----------------------------------------------------------- +--R a sinh(3a x) + 2a sinh(a x)sinh(2a x) - 3a sinh(a x) +--R Type: Expression Integer +--E + +--S 40 +sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) +--R +--I %M cosh(y + x) - %M cosh(y - x) +--I (13) %M sinh(x)sinh(y) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 41 +ii:=sinhsinhrule gg +--R +--R 2cosh(a x)sinh(2a x) - 4sinh(a x) + 2cosh(a x)cosh(2a x) - 2cosh(a x) +--R (14) --------------------------------------------------------------------- +--R (2a cosh(2a x) - 2a)sinh(a x) + a cosh(3a x) - a cosh(a x) +--R Type: Expression Integer +--E + +--S 42 +coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) +--R +--I %N cosh(y + x) + %N cosh(y - x) +--I (15) %N cosh(x)cosh(y) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 43 +jj:=coshcoshrule ii +--R +--R 2cosh(a x)sinh(2a x) - 4sinh(a x) + cosh(3a x) - cosh(a x) +--R (16) ---------------------------------------------------------- +--R (2a cosh(2a x) - 2a)sinh(a x) + a cosh(3a x) - a cosh(a x) +--R Type: Expression Integer +--E + +--S 44 14:549 Schaums and Axiom differ by a constant +kk:=sinhcoshrule jj +--R +--R 1 +--R (17) - +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.550~~~~~$\displaystyle +\int{\sinh{ax}\sinh{px}}~dx$} +$$\int{\sinh{ax}\sinh{px}}= +\frac{\sinh(a+p)x}{2(a+p)}-\frac{\sinh(a-p)x}{2(a-p)} +$$ +<<*>>= +)clear all + +--S 45 +aa:=integrate(sinh(a*x)*sinh(p*x),x) +--R +--R +--R a cosh(a x)sinh(p x) - p cosh(p x)sinh(a x) +--R (1) ------------------------------------------- +--R 2 2 2 2 2 2 +--R (p - a )sinh(a x) + (- p + a )cosh(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 46 +bb:=(sinh(a+p)*x)/(2*(a+p))-(sinh(a-p)*x)/(2*(a-p)) +--R +--R (p - a)x sinh(p + a) + (- p - a)x sinh(p - a) +--R (2) --------------------------------------------- +--R 2 2 +--R 2p - 2a +--R Type: Expression Integer +--E + +--S 47 14:550 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 2a cosh(a x)sinh(p x) +--R + +--R 2 +--R ((- p + a)x sinh(p + a) + (p + a)x sinh(p - a))sinh(a x) +--R + +--R 2 +--R - 2p cosh(p x)sinh(a x) + (p - a)x cosh(a x) sinh(p + a) +--R + +--R 2 +--R (- p - a)x cosh(a x) sinh(p - a) +--R / +--R 2 2 2 2 2 2 +--R (2p - 2a )sinh(a x) + (- 2p + 2a )cosh(a x) +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.551~~~~~$\displaystyle +\int{\sinh{ax}\sin{px}}~dx$} +$$\int{\sinh{ax}\sin{px}}= +\frac{a\cosh{ax}\sin{px}-p\sinh{ax}\cos{px}}{a^2+p^2} +$$ +<<*>>= +)clear all + +--S 48 +aa:=integrate(sinh(a*x)*sin(p*x),x) +--R +--R +--R (1) +--R 2 +--R (a sin(p x) - p cos(p x))sinh(a x) +--R + +--R (2a cosh(a x)sin(p x) - 2p cos(p x)cosh(a x))sinh(a x) +--R + +--R 2 2 +--R (a cosh(a x) + a)sin(p x) - p cos(p x)cosh(a x) + p cos(p x) +--R / +--R 2 2 2 2 +--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 49 +bb:=(a*cosh(a*x)*sin(p*x)-p*sinh(a*x)*cos(p*x))/(a^2+p^2) +--R +--R - p cos(p x)sinh(a x) + a cosh(a x)sin(p x) +--R (2) ------------------------------------------- +--R 2 2 +--R p + a +--R Type: Expression Integer +--E + +--S 50 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (a sin(p x) + p cos(p x))sinh(a x) + (- a cosh(a x) + a)sin(p x) +--R + +--R 2 +--R - p cos(p x)cosh(a x) + p cos(p x) +--R / +--R 2 2 2 2 +--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x) +--R Type: Expression Integer +--E + +--S 51 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 52 +dd:=sinhsqrrule cc +--R +--R (5) +--R 2 +--R (a cosh(2a x) - 2a cosh(a x) + a)sin(p x) + p cos(p x)cosh(2a x) +--R + +--R 2 +--R - 2p cos(p x)cosh(a x) + p cos(p x) +--R / +--R 2 2 2 2 +--R (4p + 4a )sinh(a x) + (4p + 4a )cosh(a x) +--R Type: Expression Integer +--E + +--S 53 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 54 14:551 Schaums and Axiom agree +ee:=coshsqrrule dd +--R +--R (7) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.552~~~~~$\displaystyle +\int{\sinh{ax}\cos{px}}~dx$} +$$\int{\sinh{ax}\cos{px}}= +\frac{a\cosh{ax}\cos{px}+p\sinh{ax}\sin{px}}{a^2+p^2} +$$ +<<*>>= +)clear all + +--S 55 +aa:=integrate(sinh(a*x)*cos(p*x),x) +--R +--R +--R (1) +--R 2 +--R (p sin(p x) + a cos(p x))sinh(a x) +--R + +--R (2p cosh(a x)sin(p x) + 2a cos(p x)cosh(a x))sinh(a x) +--R + +--R 2 2 +--R (p cosh(a x) - p)sin(p x) + a cos(p x)cosh(a x) + a cos(p x) +--R / +--R 2 2 2 2 +--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 56 +bb:=(a*cosh(a*x)*cos(p*x)+p*sinh(a*x)*sin(p*x))/(a^2+p^2) +--R +--R p sin(p x)sinh(a x) + a cos(p x)cosh(a x) +--R (2) ----------------------------------------- +--R 2 2 +--R p + a +--R Type: Expression Integer +--E + +--S 57 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (- p sin(p x) + a cos(p x))sinh(a x) + (p cosh(a x) - p)sin(p x) +--R + +--R 2 +--R - a cos(p x)cosh(a x) + a cos(p x) +--R / +--R 2 2 2 2 +--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x) +--R Type: Expression Integer +--E + +--S 58 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 59 +dd:=sinhsqrrule cc +--R +--R (5) +--R 2 +--R (- p cosh(2a x) + 2p cosh(a x) - p)sin(p x) + a cos(p x)cosh(2a x) +--R + +--R 2 +--R - 2a cos(p x)cosh(a x) + a cos(p x) +--R / +--R 2 2 2 2 +--R (4p + 4a )sinh(a x) + (4p + 4a )cosh(a x) +--R Type: Expression Integer +--E + +--S 60 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 61 14:552 Schaums and Axiom agree +ee:=coshsqrrule dd +--R +--R (7) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.553~~~~~$\displaystyle +\int{\frac{dx}{p+q\sinh{ax}}}~dx$} +$$\int{\frac{1}{p+q\sinh{ax}}}= +\frac{1}{a\sqrt{p^2+q^2}} +\ln\left(\frac{qe^{ax}+p-\sqrt{p^2+q^2}}{qe^{ax}+p+\sqrt{p^2+q^2}}\right) +$$ +<<*>>= +)clear all + +--S 62 +aa:=integrate(1/(p+q*sinh(a*x)),x) +--R +--R +--R (1) +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R / +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Union(Expression Integer,...) +--E + +--S 63 +bb:=1/(a*sqrt(p^2+q^2))*log((q*%e^(a*x)+p-sqrt(p^2+q^2))/(q*%e^(a*x)+p+sqrt(p^2+q^2))) +--R +--R +-------+ +--R | 2 2 a x +--R - \|q + p + q %e + p +--R log(--------------------------) +--R +-------+ +--R | 2 2 a x +--R \|q + p + q %e + p +--R (2) ------------------------------- +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Expression Integer +--E + +--S 64 14:553 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R + +--R +-------+ +--R | 2 2 a x +--R - \|q + p + q %e + p +--R - log(--------------------------) +--R +-------+ +--R | 2 2 a x +--R \|q + p + q %e + p +--R / +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.554~~~~~$\displaystyle +\int{\frac{dx}{(p+q\sinh{ax})^2}}~dx$} +$$\int{\frac{1}{(p+q\sinh{ax})^2}}= +\frac{-q\cosh{ax}}{a(p^2+q^2)(p+q\sinh{ax})} ++\frac{p}{p^2+q^2}\int{\frac{1}{p+q\sinh{ax}}} +$$ +<<*>>= +)clear all + +--S 65 +aa:=integrate(1/(p*q*sinh(a*x))^2,x) +--R +--R +--R (1) +--R 2 +--R - ------------------------------------------------------------------------ +--R 2 2 2 2 2 2 2 2 2 2 +--R a p q sinh(a x) + 2a p q cosh(a x)sinh(a x) + a p q cosh(a x) - a p q +--R Type: Union(Expression Integer,...) +--E + +--S 66 +t1:=integrate(1/(p+q*sinh(a*x)),x) +--R +--R (2) +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R / +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Union(Expression Integer,...) +--E + +--S 67 +bb:=(-q*cosh(a*x))/(a*(p^2+q^2)*(p+q*sinh(a*x)))+p/(p^2+q^2)*t1 +--R +--R (3) +--R 2 +--R (p q sinh(a x) + p ) +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R + +--R +-------+ +--R | 2 2 +--R - q cosh(a x)\|q + p +--R / +--R +-------+ +--R 3 2 2 3 | 2 2 +--R ((a q + a p q)sinh(a x) + a p q + a p )\|q + p +--R Type: Expression Integer +--E + +--S 68 14:554 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (4) +--R 3 3 3 3 3 4 2 2 +--R - p q sinh(a x) + (- 2p q cosh(a x) - p q )sinh(a x) +--R + +--R 3 3 2 4 2 3 3 4 2 2 +--R (- p q cosh(a x) - 2p q cosh(a x) + p q )sinh(a x) - p q cosh(a x) +--R + +--R 4 2 +--R p q +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R + +--R 2 3 2 2 3 2 3 2 +--R p q cosh(a x)sinh(a x) + (2p q cosh(a x) - 2q - 2p q)sinh(a x) +--R + +--R 2 3 3 2 3 2 3 +--R p q cosh(a x) - p q cosh(a x) - 2p q - 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R / +--R 2 5 4 3 3 +--R (a p q + a p q )sinh(a x) +--R + +--R 2 5 4 3 3 4 5 2 2 +--R ((2a p q + 2a p q )cosh(a x) + a p q + a p q )sinh(a x) +--R + +--R 2 5 4 3 2 3 4 5 2 2 5 +--R (a p q + a p q )cosh(a x) + (2a p q + 2a p q )cosh(a x) - a p q +--R + +--R 4 3 +--R - a p q +--R * +--R sinh(a x) +--R + +--R 3 4 5 2 2 3 4 5 2 +--R (a p q + a p q )cosh(a x) - a p q - a p q +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.555~~~~~$\displaystyle +\int{\frac{dx}{p^2+q^2\sinh^2{ax}}}$} +$$\int{\frac{1}{p^2+q^2\sinh^2{ax}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{ap\sqrt{q^2-p^2}}\tan^{-1}\frac{\sqrt{q^2-p^2}\tanh{ax}}{p}\\ +\\ +\displaystyle +\frac{1}{2ap\sqrt{p^2-q^2}}\ln\left(\frac{p+\sqrt{p^2-q^2}\tanh{ax}} +{p-\sqrt{p^2-q^2}\tanh{ax}}\right) +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 69 +aa:=integrate(1/(p^2+q^2*sinh(a*x)^2),x) +--R +--R +--R (1) +--R [ +--R log +--R 4 4 4 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 4 2 4 2 2 2 +--R (6q cosh(a x) - 2q + 4p q )sinh(a x) +--R + +--R 4 3 4 2 2 +--R (4q cosh(a x) + (- 4q + 8p q )cosh(a x))sinh(a x) +--R + +--R 4 4 4 2 2 2 4 2 2 4 +--R q cosh(a x) + (- 2q + 4p q )cosh(a x) + q - 8p q + 8p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 4 3 2 2 4 3 2 +--R (4p q - 4p q )sinh(a x) + (8p q - 8p q )cosh(a x)sinh(a x) +--R + +--R 4 3 2 2 4 3 2 5 +--R (4p q - 4p q )cosh(a x) - 4p q + 12p q - 8p +--R / +--R 2 4 2 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (6q cosh(a x) - 2q + 4p )sinh(a x) +--R + +--R 2 3 2 2 2 4 +--R (4q cosh(a x) + (- 4q + 8p )cosh(a x))sinh(a x) + q cosh(a x) +--R + +--R 2 2 2 2 +--R (- 2q + 4p )cosh(a x) + q +--R / +--R +---------+ +--R | 2 2 +--R 2a p\|- q + p +--R , +--R +--R atan +--R 2 2 2 2 2 2 2 +--R (q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) - q + 2p ) +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R 2 3 +--R 2p q - 2p +--R / +--R +-------+ +--R | 2 2 +--R a p\|q - p +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 70 +bb1:=1/(a*p*sqrt(q^2-p^2))*atan((sqrt(q^2-p^2)*tanh(a*x))/p) +--R +--R +-------+ +--R | 2 2 +--R tanh(a x)\|q - p +--R atan(-------------------) +--R p +--R (2) ------------------------- +--R +-------+ +--R | 2 2 +--R a p\|q - p +--R Type: Expression Integer +--E + +--S 71 +bb2:=1/(2*a*p*sqrt(p^2-q^2))*log((p+sqrt(p^2-q^2)*tanh(a*x))/(p-sqrt(p^2-q^2)*tanh(a*x))) +--R +--R +---------+ +--R | 2 2 +--R - tanh(a x)\|- q + p - p +--R log(---------------------------) +--R +---------+ +--R | 2 2 +--R tanh(a x)\|- q + p - p +--R (3) -------------------------------- +--R +---------+ +--R | 2 2 +--R 2a p\|- q + p +--R Type: Expression Integer +--E + +--S 72 +cc1:=aa.1-bb1 +--R +--R (4) +--R +-------+ +--R | 2 2 +--R \|q - p +--R * +--R log +--R 4 4 4 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 4 2 4 2 2 2 +--R (6q cosh(a x) - 2q + 4p q )sinh(a x) +--R + +--R 4 3 4 2 2 +--R (4q cosh(a x) + (- 4q + 8p q )cosh(a x))sinh(a x) +--R + +--R 4 4 4 2 2 2 4 2 2 4 +--R q cosh(a x) + (- 2q + 4p q )cosh(a x) + q - 8p q + 8p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 4 3 2 2 4 3 2 +--R (4p q - 4p q )sinh(a x) + (8p q - 8p q )cosh(a x)sinh(a x) +--R + +--R 4 3 2 2 4 3 2 5 +--R (4p q - 4p q )cosh(a x) - 4p q + 12p q - 8p +--R / +--R 2 4 2 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (6q cosh(a x) - 2q + 4p )sinh(a x) +--R + +--R 2 3 2 2 2 4 +--R (4q cosh(a x) + (- 4q + 8p )cosh(a x))sinh(a x) + q cosh(a x) +--R + +--R 2 2 2 2 +--R (- 2q + 4p )cosh(a x) + q +--R + +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 tanh(a x)\|q - p +--R - 2\|- q + p atan(-------------------) +--R p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 73 +cc2:=aa.2-bb1 +--R +--R (5) +--R +-------+ +--R | 2 2 +--R tanh(a x)\|q - p +--R - atan(-------------------) +--R p +--R + +--R atan +--R 2 2 2 2 2 2 2 +--R (q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) - q + 2p ) +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R 2 3 +--R 2p q - 2p +--R / +--R +-------+ +--R | 2 2 +--R a p\|q - p +--R Type: Expression Integer +--E + +--S 74 +cc3:=aa.2-bb1 +--R +--R (6) +--R +-------+ +--R | 2 2 +--R tanh(a x)\|q - p +--R - atan(-------------------) +--R p +--R + +--R atan +--R 2 2 2 2 2 2 2 +--R (q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) - q + 2p ) +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R 2 3 +--R 2p q - 2p +--R / +--R +-------+ +--R | 2 2 +--R a p\|q - p +--R Type: Expression Integer +--E + +--S 75 14:555 Axiom cannot simplify this expression +cc4:=aa.2-bb2 +--R +--R (7) +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 - tanh(a x)\|- q + p - p +--R - \|q - p log(---------------------------) +--R +---------+ +--R | 2 2 +--R tanh(a x)\|- q + p - p +--R + +--R +---------+ +--R | 2 2 +--R 2\|- q + p +--R * +--R atan +--R 2 2 2 2 2 2 2 +--R (q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) - q + 2p ) +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R 2 3 +--R 2p q - 2p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|- q + p \|q - p +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.556~~~~~$\displaystyle +\int{\frac{dx}{p^2-q^2\sinh^2{ax}}}~dx$} +$$\int{\frac{1}{p^2-q^2\sinh^2{ax}}}= +\frac{1}{2ap\sqrt{p^2+q^2}}\ln\left(\frac{p+\sqrt{p^2+q^2}\tanh{ax}} +{p-\sqrt{p^2+q^2}\tanh{ax}}\right) +$$ +<<*>>= +)clear all + +--S 76 +aa:=integrate(1/(p^2+q^2*sinh(a*x)^2),x) +--R +--R +--R (1) +--R [ +--R log +--R 4 4 4 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 4 2 4 2 2 2 +--R (6q cosh(a x) - 2q + 4p q )sinh(a x) +--R + +--R 4 3 4 2 2 +--R (4q cosh(a x) + (- 4q + 8p q )cosh(a x))sinh(a x) +--R + +--R 4 4 4 2 2 2 4 2 2 4 +--R q cosh(a x) + (- 2q + 4p q )cosh(a x) + q - 8p q + 8p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 4 3 2 2 4 3 2 +--R (4p q - 4p q )sinh(a x) + (8p q - 8p q )cosh(a x)sinh(a x) +--R + +--R 4 3 2 2 4 3 2 5 +--R (4p q - 4p q )cosh(a x) - 4p q + 12p q - 8p +--R / +--R 2 4 2 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (6q cosh(a x) - 2q + 4p )sinh(a x) +--R + +--R 2 3 2 2 2 4 +--R (4q cosh(a x) + (- 4q + 8p )cosh(a x))sinh(a x) + q cosh(a x) +--R + +--R 2 2 2 2 +--R (- 2q + 4p )cosh(a x) + q +--R / +--R +---------+ +--R | 2 2 +--R 2a p\|- q + p +--R , +--R +--R atan +--R 2 2 2 2 2 2 2 +--R (q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) - q + 2p ) +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R 2 3 +--R 2p q - 2p +--R / +--R +-------+ +--R | 2 2 +--R a p\|q - p +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 77 +bb:=1/(2*a*p*sqrt(p^2+q^2))*log((p+sqrt(p^2+q^2)*tanh(a*x))/(p-sqrt(p^2+q^2)*tanh(a*x))) +--R +--R +-------+ +--R | 2 2 +--R - tanh(a x)\|q + p - p +--R log(-------------------------) +--R +-------+ +--R | 2 2 +--R tanh(a x)\|q + p - p +--R (2) ------------------------------ +--R +-------+ +--R | 2 2 +--R 2a p\|q + p +--R Type: Expression Integer +--E + +--S 78 +cc1:=aa.1-bb +--R +--R (3) +--R +-------+ +--R | 2 2 +--R \|q + p +--R * +--R log +--R 4 4 4 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 4 2 4 2 2 2 +--R (6q cosh(a x) - 2q + 4p q )sinh(a x) +--R + +--R 4 3 4 2 2 +--R (4q cosh(a x) + (- 4q + 8p q )cosh(a x))sinh(a x) +--R + +--R 4 4 4 2 2 2 4 2 2 4 +--R q cosh(a x) + (- 2q + 4p q )cosh(a x) + q - 8p q + 8p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 4 3 2 2 4 3 2 +--R (4p q - 4p q )sinh(a x) + (8p q - 8p q )cosh(a x)sinh(a x) +--R + +--R 4 3 2 2 4 3 2 5 +--R (4p q - 4p q )cosh(a x) - 4p q + 12p q - 8p +--R / +--R 2 4 2 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (6q cosh(a x) - 2q + 4p )sinh(a x) +--R + +--R 2 3 2 2 2 4 +--R (4q cosh(a x) + (- 4q + 8p )cosh(a x))sinh(a x) + q cosh(a x) +--R + +--R 2 2 2 2 +--R (- 2q + 4p )cosh(a x) + q +--R + +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 - tanh(a x)\|q + p - p +--R - \|- q + p log(-------------------------) +--R +-------+ +--R | 2 2 +--R tanh(a x)\|q + p - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|- q + p \|q + p +--R Type: Expression Integer +--E + +--S 79 14:556 Axiom cannot simplify this expression +cc2:=aa.2-bb +--R +--R (4) +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 - tanh(a x)\|q + p - p +--R - \|q - p log(-------------------------) +--R +-------+ +--R | 2 2 +--R tanh(a x)\|q + p - p +--R + +--R +-------+ +--R | 2 2 +--R 2\|q + p +--R * +--R atan +--R 2 2 2 2 2 2 2 +--R (q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) - q + 2p ) +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R 2 3 +--R 2p q - 2p +--R / +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|q - p \|q + p +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.557~~~~~$\displaystyle +\int{x^m\sinh{ax}}~dx$} +$$\int{x^m\sinh{ax}}= +\frac{x^m\cosh{ax}}{a}-\frac{m}{a}\int{x^{m-1}\cosh{ax}} +$$ +<<*>>= +)clear all + +--S 80 14:557 Axiom cannot compute this integral +aa:=integrate(x^m*sinh(a*x),x) +--R +--R +--R x +--R ++ m +--I (1) | sinh(%N a)%N d%N +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.558~~~~~$\displaystyle +\int{\sinh^n}~dx$} +$$\int{\sinh^n}= +\frac{\sinh^{n-1}{ax}\cosh{ax}}{an}-\frac{n-1}{n}\int{\sinh^{n-2}{ax}} +$$ +<<*>>= +)clear all + +--S 81 14:558 Axiom cannot compute this integral +aa:=integrate(sinh(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | sinh(%N a) d%N +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.559~~~~~$\displaystyle +\int{\frac{\sinh{ax}}{x^n}}~dx$} +$$\int{\frac{\sinh{ax}}{x^n}}= +\frac{-\sinh{ax}}{(n-1)x^{n-1}}+\frac{a}{n-1}\int{\frac{\cosh{ax}}{n^{n-1}}} +$$ +<<*>>= +)clear all + +--S 82 14:559 Axiom cannot compute this integral +aa:=integrate(sinh(a*x)/x^n,x) +--R +--R x +--I ++ sinh(%T a) +--I (3) | ---------- d%T +--R ++ n +--I %T +--R Type: Union(Expression Integer,...) +--E + +@ + +\section{\cite{1}:14.560~~~~~$\displaystyle +\int{\frac{dx}{\sinh^n{ax}}}~dx$} +$$\int{\frac{1}{\sinh^n{ax}}}= +\frac{-\cosh{ax}}{a(n-1)\sinh^{n-1}{ax}} +-\frac{n-2}{n-1}\int{\frac{1}{\sinh^{n-2}{ax}}} +$$ +<<*>>= +)clear all + +--S 83 14:560 Axiom cannot compute this integral +aa:=integrate(1/sinh(a*x)^n,x) +--R +--R +--R x +--R ++ 1 +--I (1) | ----------- d%N +--R ++ n +--I sinh(%N a) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.561~~~~~$\displaystyle +\int{\frac{x~dx}{\sinh^n{ax}}}~dx$} +$$\int{\frac{x}{\sinh^n{ax}}}= +\frac{-x\cosh{ax}}{a(n-1)\sinh^{n-1}{ax}} +-\frac{1}{a^2(n-1)(n-2)\sinh^{n-2}{ax}} +-\frac{n-2}{n-1}\int{\frac{x}{\sinh^{n-2}{ax}}} +$$ +<<*>>= +)clear all + +--S 84 14:561 Axiom cannot compute this integral +aa:=integrate(x/sinh(a*x)^n,x) +--R +--R +--R x +--I ++ %N +--I (1) | ----------- d%N +--R ++ n +--I sinh(%N a) +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p86 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum27.input.pdf b/src/axiom-website/CATS/schaum27.input.pdf new file mode 100644 index 0000000..7b98372 --- /dev/null +++ b/src/axiom-website/CATS/schaum27.input.pdf @@ -0,0 +1,3038 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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+/Info 1 0 R +>> +startxref +137051 +%%EOF diff --git a/src/axiom-website/CATS/schaum28.input.pamphlet b/src/axiom-website/CATS/schaum28.input.pamphlet new file mode 100644 index 0000000..49efb0e --- /dev/null +++ b/src/axiom-website/CATS/schaum28.input.pamphlet @@ -0,0 +1,2854 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum28.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.562~~~~~$\displaystyle +\int{\cosh{ax}}~dx$} +$$\int{\cosh{ax}}= +\frac{\sinh{ax}}{a} +$$ +<<*>>= +)spool schaum28.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(cosh(a*x),x) +--R +--R +--R sinh(a x) +--R (1) --------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=sinh(a*x)/a +--R +--R sinh(a x) +--R (2) --------- +--R a +--R Type: Expression Integer +--E + +--S 3 14:562 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.563~~~~~$\displaystyle +\int{x\cosh{ax}}~dx$} +$$\int{x\cosh{ax}}= +\frac{x\sinh{ax}}{a}-\frac{\cosh{ax}}{a^2} +$$ +<<*>>= +)clear all + +--S 4 +aa:=integrate(x*cosh(a*x),x) +--R +--R +--R a x sinh(a x) - cosh(a x) +--R (1) ------------------------- +--R 2 +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 5 +bb:=(x*sinh(a*x))/a-cosh(a*x)/a^2 +--R +--R a x sinh(a x) - cosh(a x) +--R (2) ------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 6 14:563 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.564~~~~~$\displaystyle +\int{x^2\cosh{ax}}~dx$} +$$\int{x^2\cosh{ax}}= +-\frac{2x\cosh{ax}}{a^2}+\left(\frac{x^2}{a}+\frac{2}{a^3}\right)\sinh{ax} +$$ +<<*>>= +)clear all + +--S 7 +aa:=integrate(x^2*cosh(a*x),x) +--R +--R +--R 2 2 +--R (a x + 2)sinh(a x) - 2a x cosh(a x) +--R (1) ------------------------------------ +--R 3 +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 8 +bb:=-(2*x*cosh(a*x))/a^2+(x^2/a+2/a^3)*sinh(a*x) +--R +--R 2 2 +--R (a x + 2)sinh(a x) - 2a x cosh(a x) +--R (2) ------------------------------------ +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 9 14:564 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.565~~~~~$\displaystyle +\int{\frac{\cosh{ax}}{x}}~dx$} +$$\int{\frac{\cosh{ax}}{x}}= +\ln{x}+\frac{(ax)^2}{2\cdot 2!} ++\frac{(ax)^4}{4\cdot 4!} ++\frac{(ax)^6}{6\cdot 6!}+\cdots +$$ +<<*>>= +)clear all + +--S 10 14:565 Axiom cannot compute this integral +aa:=integrate(cosh(a*x)/x,x) +--R +--R +--R x +--I ++ cosh(%N a) +--I (1) | ---------- d%N +--I ++ %N +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.566~~~~~$\displaystyle +\int{\frac{\cosh{ax}}{x^2}}~dx$} +$$\int{\frac{\cosh{ax}}{x^2}}= +-\frac{\cosh{ax}}{x}+a\int{\frac{\sinh{ax}}{a}} +$$ +<<*>>= +)clear all + +--S 11 14:566 Axiom cannot compute this integral +aa:=integrate(cosh(a*x)/x^2,x) +--R +--R +--R x +--I ++ cosh(%N a) +--I (1) | ---------- d%N +--R ++ 2 +--I %N +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.567~~~~~$\displaystyle +\int{\frac{dx}{\cosh{ax}}}~dx$} +$$\int{\frac{1}{\cosh{ax}}}= +\frac{2}{a}\tan^{-1}e^{ax} +$$ +<<*>>= +)clear all + +--S 12 +aa:=integrate(1/cosh(a*x),x) +--R +--R +--R 2atan(sinh(a x) + cosh(a x)) +--R (1) ---------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 13 +bb:=2/a*atan(%e^(a*x)) +--R +--R a x +--R 2atan(%e ) +--R (2) ------------ +--R a +--R Type: Expression Integer +--E + +--S 14 +cc:=aa-bb +--R +--R a x +--R 2atan(sinh(a x) + cosh(a x)) - 2atan(%e ) +--R (3) ------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 15 14:567 Schaums and Axiom agree +dd:=complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.568~~~~~$\displaystyle +\int{\frac{x~dx}{\cosh{ax}}}~dx$} +$$\int{\frac{x}{\cosh{ax}}}= +\frac{1}{a^2}\left\{\frac{(ax)^2}{2}-\frac{(ax)^4}{8}+\frac{5(ax)^6}{144} ++\cdots+\frac{(-1)^nE_n(ax)^{2n+2}}{(2n+2)(2n)!}+\cdots\right\} +$$ +<<*>>= +)clear all + +--S 16 14:568 Axiom cannot compute this integral +aa:=integrate(x/cosh(a*x),x) +--R +--R +--R x +--I ++ %N +--I (1) | ---------- d%N +--I ++ cosh(%N a) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.569~~~~~$\displaystyle +\int{\cosh^2{ax}}~dx$} +$$\int{\cosh^2{ax}}= +\frac{x}{2}+\frac{\sinh{ax}\cosh{ax}}{2a} +$$ +Note that the Schaums print edition (1968 printing 3) has a typo: +$$\int{\cosh^2{ax}}= +\frac{x}{2}+\frac{\sinh{ax}\cosh{ax}}{2} +$$ +<<*>>= +)clear all + +--S 17 +aa:=integrate(cosh(a*x)^2,x) +--R +--R +--R cosh(a x)sinh(a x) + a x +--R (1) ------------------------ +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 18 +bb:=x/2+(sinh(a*x)*cosh(a*x))/(2*a) +--R +--R cosh(a x)sinh(a x) + a x +--R (2) ------------------------ +--R 2a +--R Type: Expression Integer +--E + +--S 19 14:569 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.570~~~~~$\displaystyle +\int{x\cosh^2{ax}}~dx$} +$$\int{x\cosh^2{ax}}= +\frac{x^2}{4}+\frac{x\sinh{2ax}}{4a}-\frac{\cosh{2ax}}{8a^2} +$$ +<<*>>= +)clear all + +--S 20 +aa:=integrate(x*cosh(a*x)^2,x) +--R +--R +--R 2 2 2 2 +--R - sinh(a x) + 4a x cosh(a x)sinh(a x) - cosh(a x) + 2a x +--R (1) ----------------------------------------------------------- +--R 2 +--R 8a +--R Type: Union(Expression Integer,...) +--E + +--S 21 +bb:=x^2/4+(x*sinh(2*a*x))/(4*a)-cosh(2*a*x)/(8*a^2) +--R +--R 2 2 +--R 2a x sinh(2a x) - cosh(2a x) + 2a x +--R (2) ------------------------------------ +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 22 +cc:=aa-bb +--R +--R (3) +--R 2 +--R - 2a x sinh(2a x) - sinh(a x) + 4a x cosh(a x)sinh(a x) + cosh(2a x) +--R + +--R 2 +--R - cosh(a x) +--R / +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 23 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 24 +dd:=sinhsqrrule cc +--R +--R (5) +--R 2 +--R - 4a x sinh(2a x) + 8a x cosh(a x)sinh(a x) + cosh(2a x) - 2cosh(a x) + 1 +--R -------------------------------------------------------------------------- +--R 2 +--R 16a +--R Type: Expression Integer +--E + +--S 25 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 26 +ee:=coshsqrrule dd +--R +--R - x sinh(2a x) + 2x cosh(a x)sinh(a x) +--R (7) -------------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 27 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %S sinh(y + x) - %S sinh(y - x) +--I (8) %S cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 28 14:570 Schaums and Axiom agree +ff:=sinhcoshrule ee +--R +--R (9) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.571~~~~~$\displaystyle +\int{\frac{dx}{\cosh^2{ax}}}~dx$} +$$\int{\frac{1}{\cosh^2{ax}}}= +\frac{\tanh{ax}}{a} +$$ +<<*>>= +)clear all + +--S 29 +aa:=integrate(1/cosh(a*x)^2,x) +--R +--R +--R 2 +--R (1) - ------------------------------------------------------- +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Union(Expression Integer,...) +--E + +--S 30 +bb:=tanh(a*x)/a +--R +--R tanh(a x) +--R (2) --------- +--R a +--R Type: Expression Integer +--E + +--S 31 +cc:=aa-bb +--R +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1)tanh(a x) - 2 +--R (3) ------------------------------------------------------------------ +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 32 14:571 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 1 +--R (4) - - +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.572~~~~~$\displaystyle +\int{\cosh{ax}\cosh{px}}~dx$} +$$\int{\cosh{ax}\cosh{px}}= +\frac{\sinh(a-p)x}{2(a-p)}+\frac{\sinh(a+p)x}{2(a+p)} +$$ +<<*>>= +)clear all + +--S 33 +aa:=integrate(cosh(a*x)*cosh(p*x),x) +--R +--R +--R - p cosh(a x)sinh(p x) + a cosh(p x)sinh(a x) +--R (1) --------------------------------------------- +--R 2 2 2 2 2 2 +--R (p - a )sinh(a x) + (- p + a )cosh(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 34 +bb:=(sinh(a-p)*x)/(2*(a-p))+(sinh(a+p)*x)/(2*(a+p)) +--R +--R (p - a)x sinh(p + a) + (p + a)x sinh(p - a) +--R (2) ------------------------------------------- +--R 2 2 +--R 2p - 2a +--R Type: Expression Integer +--E + +--S 35 +cc:=aa-bb +--R +--R (3) +--R - 2p cosh(a x)sinh(p x) +--R + +--R 2 +--R ((- p + a)x sinh(p + a) + (- p - a)x sinh(p - a))sinh(a x) +--R + +--R 2 +--R 2a cosh(p x)sinh(a x) + (p - a)x cosh(a x) sinh(p + a) +--R + +--R 2 +--R (p + a)x cosh(a x) sinh(p - a) +--R / +--R 2 2 2 2 2 2 +--R (2p - 2a )sinh(a x) + (- 2p + 2a )cosh(a x) +--R Type: Expression Integer +--E + +--S 36 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 37 +dd:=sinhsqrrule cc +--R +--R (5) +--R - 4p cosh(a x)sinh(p x) + 4a cosh(p x)sinh(a x) +--R + +--R 2 +--R ((- p + a)x cosh(2a x) + (2p - 2a)x cosh(a x) + (p - a)x)sinh(p + a) +--R + +--R 2 +--R ((- p - a)x cosh(2a x) + (2p + 2a)x cosh(a x) + (p + a)x)sinh(p - a) +--R / +--R 2 2 2 2 2 2 2 +--R (2p - 2a )cosh(2a x) + (- 4p + 4a )cosh(a x) - 2p + 2a +--R Type: Expression Integer +--E + +--S 38 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 39 +ee:=coshsqrrule dd +--R +--R (7) +--R 2p cosh(a x)sinh(p x) - 2a cosh(p x)sinh(a x) + (- p + a)x sinh(p + a) +--R + +--R (- p - a)x sinh(p - a) +--R / +--R 2 2 +--R 2p - 2a +--R Type: Expression Integer +--E + +--S 40 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %V sinh(y + x) - %V sinh(y - x) +--I (8) %V cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 41 14:572 Axiom cannot simplify this expression +ff:=sinhcoshrule ee +--R +--R (9) +--R (p - a)sinh((p + a)x) + (p + a)sinh((p - a)x) + (- p + a)x sinh(p + a) +--R + +--R (- p - a)x sinh(p - a) +--R / +--R 2 2 +--R 2p - 2a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.573~~~~~$\displaystyle +\int{\cosh{ax}\sin{px}}~dx$} +$$\int{\cosh{ax}\sin{px}}= +\frac{a\sinh{ax}\sin{px}-p\cosh{ax}\cos{px}}{a^2+p^2} +$$ +<<*>>= +)clear all + +--S 42 +aa:=integrate(cosh(a*x)*sin(p*x),x) +--R +--R +--R (1) +--R 2 +--R (a sin(p x) - p cos(p x))sinh(a x) +--R + +--R (2a cosh(a x)sin(p x) - 2p cos(p x)cosh(a x))sinh(a x) +--R + +--R 2 2 +--R (a cosh(a x) - a)sin(p x) - p cos(p x)cosh(a x) - p cos(p x) +--R / +--R 2 2 2 2 +--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 43 +bb:=(a*sinh(a*x)*sin(p*x)-p*cosh(a*x)*cos(p*x))/(a^2+p^2) +--R +--R a sin(p x)sinh(a x) - p cos(p x)cosh(a x) +--R (2) ----------------------------------------- +--R 2 2 +--R p + a +--R Type: Expression Integer +--E + +--S 44 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (- a sin(p x) - p cos(p x))sinh(a x) + (a cosh(a x) - a)sin(p x) +--R + +--R 2 +--R p cos(p x)cosh(a x) - p cos(p x) +--R / +--R 2 2 2 2 +--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x) +--R Type: Expression Integer +--E + +--S 45 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (4) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 46 +dd:=coshsqrrule cc +--R +--R (5) +--R 2 +--R (- 2a sin(p x) - 2p cos(p x))sinh(a x) + (a cosh(2a x) - a)sin(p x) +--R + +--R p cos(p x)cosh(2a x) - p cos(p x) +--R / +--R 2 2 2 2 +--R (4p + 4a )sinh(a x) + (4p + 4a )cosh(a x) +--R Type: Expression Integer +--E + +--S 47 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (6) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 48 14:573 Schaums and Axiom agree +ee:=sinhsqrrule dd +--R +--R (7) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.574~~~~~$\displaystyle +\int{\cosh{ax}\cos{px}}~dx$} +$$\int{\cosh{ax}\cos{px}}= +\frac{a\sinh{ax}\cos{px}+p\cosh{ax}\sin{px}}{a^2+p^2} +$$ +<<*>>= +)clear all + +--S 49 +aa:=integrate(cosh(a*x)*cos(p*x),x) +--R +--R +--R (1) +--R 2 +--R (p sin(p x) + a cos(p x))sinh(a x) +--R + +--R (2p cosh(a x)sin(p x) + 2a cos(p x)cosh(a x))sinh(a x) +--R + +--R 2 2 +--R (p cosh(a x) + p)sin(p x) + a cos(p x)cosh(a x) - a cos(p x) +--R / +--R 2 2 2 2 +--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 50 +bb:=(a*sinh(a*x)*cos(p*x)+p*cosh(a*x)*sin(p*x))/(a^2+p^2) +--R +--R a cos(p x)sinh(a x) + p cosh(a x)sin(p x) +--R (2) ----------------------------------------- +--R 2 2 +--R p + a +--R Type: Expression Integer +--E + +--S 51 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (p sin(p x) - a cos(p x))sinh(a x) + (- p cosh(a x) + p)sin(p x) +--R + +--R 2 +--R a cos(p x)cosh(a x) - a cos(p x) +--R / +--R 2 2 2 2 +--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x) +--R Type: Expression Integer +--E + +--S 52 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (4) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 53 +dd:=coshsqrrule cc +--R +--R (5) +--R 2 +--R (2p sin(p x) - 2a cos(p x))sinh(a x) + (- p cosh(2a x) + p)sin(p x) +--R + +--R a cos(p x)cosh(2a x) - a cos(p x) +--R / +--R 2 2 2 2 +--R (4p + 4a )sinh(a x) + (4p + 4a )cosh(a x) +--R Type: Expression Integer +--E + +--S 54 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (6) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 55 14:574 Schaums and Axiom agree +ee:=sinhsqrrule dd +--R +--R (7) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.575~~~~~$\displaystyle +\int{\frac{dx}{\cosh{ax}+1}}$} +$$\int{\frac{1}{\cosh{ax}+1}}= +\frac{1}{a}\tanh{\frac{ax}{2}} +$$ +<<*>>= +)clear all + +--S 56 +aa:=integrate(1/(cosh(a*x)+1),x) +--R +--R +--R 2 +--R (1) - ----------------------------- +--R a sinh(a x) + a cosh(a x) + a +--R Type: Union(Expression Integer,...) +--E + +--S 57 +bb:=1/a*tanh((a*x)/2) +--R +--R a x +--R tanh(---) +--R 2 +--R (2) --------- +--R a +--R Type: Expression Integer +--E + +--S 58 +cc:=aa-bb +--R +--R a x +--R (- sinh(a x) - cosh(a x) - 1)tanh(---) - 2 +--R 2 +--R (3) ------------------------------------------ +--R a sinh(a x) + a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 59 +tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) +--R +--R sinh(x) +--R (4) tanh(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 60 +dd:=tanhrule cc +--R +--R a x a x a x +--R - sinh(---)sinh(a x) + (- cosh(a x) - 1)sinh(---) - 2cosh(---) +--R 2 2 2 +--R (5) -------------------------------------------------------------- +--R a x a x a x +--R a cosh(---)sinh(a x) + a cosh(---)cosh(a x) + a cosh(---) +--R 2 2 2 +--R Type: Expression Integer +--E + +--S 61 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %BC sinh(y + x) - %BC sinh(y - x) +--I (6) %BC cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 62 +ee:=sinhcoshrule dd +--R +--R 3a x a x a x a x +--R - sinh(----) - 2sinh(---)sinh(a x) - sinh(---) - 4cosh(---) +--R 2 2 2 2 +--R (7) ----------------------------------------------------------------- +--R 3a x a x a x a x +--R a sinh(----) + a sinh(---) + 2a cosh(---)cosh(a x) + 2a cosh(---) +--R 2 2 2 2 +--R Type: Expression Integer +--E + +--S 63 +sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) +--R +--I %BD sinh(y + x) - %BD sinh(y - x) +--I (8) %BD cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 64 +ff:=sinhsinhrule ee +--R +--R 3a x a x 3a x a x +--R - sinh(----) - sinh(---) - cosh(----) - 3cosh(---) +--R 2 2 2 2 +--R (9) ----------------------------------------------------------------- +--R 3a x a x a x a x +--R a sinh(----) + a sinh(---) + 2a cosh(---)cosh(a x) + 2a cosh(---) +--R 2 2 2 2 +--R Type: Expression Integer +--E + +--S 65 +coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) +--R +--I %BC cosh(y + x) + %BC cosh(y - x) +--I (10) %BC cosh(x)cosh(y) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 66 14:575 Schaums and Axiom differ by a constant +gg:=coshcoshrule ff +--R +--R 1 +--R (11) - - +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.576~~~~~$\displaystyle +\int{\frac{dx}{\cosh{ax}-1}}$} +$$\int{\frac{1}{\cosh{ax}-1}}= +-\frac{1}{a}\coth{\frac{ax}{2}} +$$ +<<*>>= +)clear all + +--S 67 +aa:=integrate(1/(cosh(a*x)-1),x) +--R +--R +--R 2 +--R (1) - ----------------------------- +--R a sinh(a x) + a cosh(a x) - a +--R Type: Union(Expression Integer,...) +--E + +--S 68 +bb:=-1/a*coth((a*x)/2) +--R +--R a x +--R coth(---) +--R 2 +--R (2) - --------- +--R a +--R Type: Expression Integer +--E + +--S 69 +cc:=aa-bb +--R +--R a x a x +--R coth(---)sinh(a x) + (cosh(a x) - 1)coth(---) - 2 +--R 2 2 +--R (3) ------------------------------------------------- +--R a sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E + +--S 70 +cothrule:=rule(coth(x) == cosh(x)/sinh(x)) +--R +--R cosh(x) +--R (4) coth(x) == ------- +--R sinh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 71 +dd:=cothrule cc +--R +--R a x a x a x a x +--R cosh(---)sinh(a x) - 2sinh(---) + cosh(---)cosh(a x) - cosh(---) +--R 2 2 2 2 +--R (5) ---------------------------------------------------------------- +--R a x a x +--R a sinh(---)sinh(a x) + (a cosh(a x) - a)sinh(---) +--R 2 2 +--R Type: Expression Integer +--E + +--S 72 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %BD sinh(y + x) - %BD sinh(y - x) +--I (6) %BD cosh(y)sinh(x) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 73 +ee:=sinhcoshrule dd +--R +--R 3a x a x a x a x +--R sinh(----) - 3sinh(---) + 2cosh(---)cosh(a x) - 2cosh(---) +--R 2 2 2 2 +--R (7) ---------------------------------------------------------- +--R 3a x a x a x +--R a sinh(----) + 2a sinh(---)sinh(a x) - 3a sinh(---) +--R 2 2 2 +--R Type: Expression Integer +--E + +--S 74 +sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) +--R +--I %BE cosh(y + x) - %BE cosh(y - x) +--I (8) %BE sinh(x)sinh(y) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 75 +ff:=sinhsinhrule ee +--R +--R 3a x a x a x a x +--R sinh(----) - 3sinh(---) + 2cosh(---)cosh(a x) - 2cosh(---) +--R 2 2 2 2 +--R (9) ---------------------------------------------------------- +--R 3a x a x 3a x a x +--R a sinh(----) - 3a sinh(---) + a cosh(----) - a cosh(---) +--R 2 2 2 2 +--R Type: Expression Integer +--E + +--S 76 +coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) +--R +--I %BF cosh(y + x) + %BF cosh(y - x) +--I (10) %BF cosh(x)cosh(y) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 77 14:576 Schaums and Axiom differ by a constant +gg:=coshcoshrule ff +--R +--R 1 +--R (11) - +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.577~~~~~$\displaystyle +\int{\frac{x~dx}{\cosh{ax}+1}}~dx$} +$$\int{\frac{x}{\cosh{ax}+1}}= +\frac{x}{a}\tanh\frac{ax}{2}-\frac{2}{a^2}\ln\cosh\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 78 +aa:=integrate(x/(cosh(a*x)+1),x) +--R +--R +--R (1) +--R (- 2sinh(a x) - 2cosh(a x) - 2)log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2a x sinh(a x) + 2a x cosh(a x) +--R / +--R 2 2 2 +--R a sinh(a x) + a cosh(a x) + a +--R Type: Union(Expression Integer,...) +--E + +--S 79 +bb:=x/a*tanh((a*x)/2)-2/a^2*log(cosh((a*x)/2)) +--R +--R a x a x +--R - 2log(cosh(---)) + a x tanh(---) +--R 2 2 +--R (2) --------------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 80 +cc:=aa-bb +--R +--R (3) +--R (- 2sinh(a x) - 2cosh(a x) - 2)log(sinh(a x) + cosh(a x) + 1) +--R + +--R a x +--R (2sinh(a x) + 2cosh(a x) + 2)log(cosh(---)) +--R 2 +--R + +--R a x +--R (- a x sinh(a x) - a x cosh(a x) - a x)tanh(---) + 2a x sinh(a x) +--R 2 +--R + +--R 2a x cosh(a x) +--R / +--R 2 2 2 +--R a sinh(a x) + a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 81 +tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) +--R +--R sinh(x) +--R (4) tanh(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 82 +dd:=tanhrule cc +--R +--R (5) +--R a x a x a x +--R (- 2cosh(---)sinh(a x) - 2cosh(---)cosh(a x) - 2cosh(---)) +--R 2 2 2 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R a x a x a x a x +--R (2cosh(---)sinh(a x) + 2cosh(---)cosh(a x) + 2cosh(---))log(cosh(---)) +--R 2 2 2 2 +--R + +--R a x a x +--R (- a x sinh(---) + 2a x cosh(---))sinh(a x) +--R 2 2 +--R + +--R a x a x +--R (- a x cosh(a x) - a x)sinh(---) + 2a x cosh(---)cosh(a x) +--R 2 2 +--R / +--R 2 a x 2 a x 2 a x +--R a cosh(---)sinh(a x) + a cosh(---)cosh(a x) + a cosh(---) +--R 2 2 2 +--R Type: Expression Integer +--E + +--S 83 +coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) +--R +--I %BG cosh(y + x) + %BG cosh(y - x) +--I (6) %BG cosh(x)cosh(y) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 84 +ee:=coshcoshrule dd +--R +--R (7) +--R a x 3a x a x +--R (- 4cosh(---)sinh(a x) - 2cosh(----) - 6cosh(---)) +--R 2 2 2 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R a x 3a x a x a x +--R (4cosh(---)sinh(a x) + 2cosh(----) + 6cosh(---))log(cosh(---)) +--R 2 2 2 2 +--R + +--R a x a x +--R (- 2a x sinh(---) + 4a x cosh(---))sinh(a x) +--R 2 2 +--R + +--R a x 3a x a x +--R (- 2a x cosh(a x) - 2a x)sinh(---) + 2a x cosh(----) + 2a x cosh(---) +--R 2 2 2 +--R / +--R 2 a x 2 3a x 2 a x +--R 2a cosh(---)sinh(a x) + a cosh(----) + 3a cosh(---) +--R 2 2 2 +--R Type: Expression Integer +--E + +--S 85 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %BH sinh(y + x) - %BH sinh(y - x) +--I (8) %BH cosh(y)sinh(x) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 86 +ff:=sinhcoshrule ee +--R +--R (9) +--R 3a x a x 3a x a x +--R (- 2sinh(----) - 2sinh(---) - 2cosh(----) - 6cosh(---)) +--R 2 2 2 2 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 3a x a x 3a x a x a x +--R (2sinh(----) + 2sinh(---) + 2cosh(----) + 6cosh(---))log(cosh(---)) +--R 2 2 2 2 2 +--R + +--R 3a x a x a x +--R a x sinh(----) - 2a x sinh(---)sinh(a x) + a x sinh(---) +--R 2 2 2 +--R + +--R 3a x a x +--R 2a x cosh(----) + 2a x cosh(---) +--R 2 2 +--R / +--R 2 3a x 2 a x 2 3a x 2 a x +--R a sinh(----) + a sinh(---) + a cosh(----) + 3a cosh(---) +--R 2 2 2 2 +--R Type: Expression Integer +--E + +--S 87 +sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) +--R +--I %BI cosh(y + x) - %BI cosh(y - x) +--I (10) %BI sinh(x)sinh(y) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 88 +gg:=sinhsinhrule ff +--R +--R a x +--R - 2log(sinh(a x) + cosh(a x) + 1) + 2log(cosh(---)) + a x +--R 2 +--R (11) --------------------------------------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 89 14:577 Schaums and Axiom differ by a constant +complexNormalize gg +--R +--R 2log(2) +--R (12) - ------- +--R 2 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.578~~~~~$\displaystyle +\int{\frac{x~dx}{\cosh{ax}-1}}$} +$$\int{\frac{x}{\cosh{ax}-1}} +-\frac{x}{a}\coth\frac{ax}{2}+\frac{2}{a^2}\ln\sinh\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 90 +aa:=integrate(x/(cosh(a*x)-1),x) +--R +--R +--R (1) +--R (2sinh(a x) + 2cosh(a x) - 2)log(sinh(a x) + cosh(a x) - 1) +--R + +--R - 2a x sinh(a x) - 2a x cosh(a x) +--R / +--R 2 2 2 +--R a sinh(a x) + a cosh(a x) - a +--R Type: Union(Expression Integer,...) +--E + +--S 91 +bb:=-x/a*coth((a*x)/2)+2/a^2*log(sinh((a*x)/2)) +--R +--R a x a x +--R 2log(sinh(---)) - a x coth(---) +--R 2 2 +--R (2) ------------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 92 +cc:=aa-bb +--R +--R (3) +--R (2sinh(a x) + 2cosh(a x) - 2)log(sinh(a x) + cosh(a x) - 1) +--R + +--R a x +--R (- 2sinh(a x) - 2cosh(a x) + 2)log(sinh(---)) +--R 2 +--R + +--R a x a x +--R (a x coth(---) - 2a x)sinh(a x) + (a x cosh(a x) - a x)coth(---) +--R 2 2 +--R + +--R - 2a x cosh(a x) +--R / +--R 2 2 2 +--R a sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E + +--S 93 +cothrule:=rule(coth(x) == cosh(x)/sinh(x)) +--R +--R cosh(x) +--R (4) coth(x) == ------- +--R sinh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 94 +dd:=cothrule cc +--R +--R (5) +--R a x a x +--R (2sinh(---)sinh(a x) + (2cosh(a x) - 2)sinh(---)) +--R 2 2 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R a x a x a x +--R (- 2sinh(---)sinh(a x) + (- 2cosh(a x) + 2)sinh(---))log(sinh(---)) +--R 2 2 2 +--R + +--R a x a x a x +--R (- 2a x sinh(---) + a x cosh(---))sinh(a x) - 2a x cosh(a x)sinh(---) +--R 2 2 2 +--R + +--R a x a x +--R a x cosh(---)cosh(a x) - a x cosh(---) +--R 2 2 +--R / +--R 2 a x 2 2 a x +--R a sinh(---)sinh(a x) + (a cosh(a x) - a )sinh(---) +--R 2 2 +--R Type: Expression Integer +--E + +--S 95 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %BJ sinh(y + x) - %BJ sinh(y - x) +--I (6) %BJ cosh(y)sinh(x) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 96 +ee:=sinhcoshrule dd +--R +--R (7) +--R 3a x a x a x +--R (2sinh(----) + 4sinh(---)sinh(a x) - 6sinh(---)) +--R 2 2 2 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 3a x a x a x a x +--R (- 2sinh(----) - 4sinh(---)sinh(a x) + 6sinh(---))log(sinh(---)) +--R 2 2 2 2 +--R + +--R 3a x a x a x +--R - a x sinh(----) - 4a x sinh(---)sinh(a x) + 3a x sinh(---) +--R 2 2 2 +--R + +--R a x a x +--R 2a x cosh(---)cosh(a x) - 2a x cosh(---) +--R 2 2 +--R / +--R 2 3a x 2 a x 2 a x +--R a sinh(----) + 2a sinh(---)sinh(a x) - 3a sinh(---) +--R 2 2 2 +--R Type: Expression Integer +--E + +--S 97 +sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) +--R +--I %BK cosh(y + x) - %BK cosh(y - x) +--I (8) %BK sinh(x)sinh(y) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 98 +ff:=sinhsinhrule ee +--R +--R (9) +--R 3a x a x 3a x a x +--R (2sinh(----) - 6sinh(---) + 2cosh(----) - 2cosh(---)) +--R 2 2 2 2 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 3a x a x 3a x a x a x +--R (- 2sinh(----) + 6sinh(---) - 2cosh(----) + 2cosh(---))log(sinh(---)) +--R 2 2 2 2 2 +--R + +--R 3a x a x 3a x +--R - a x sinh(----) + 3a x sinh(---) - 2a x cosh(----) +--R 2 2 2 +--R + +--R a x +--R 2a x cosh(---)cosh(a x) +--R 2 +--R / +--R 2 3a x 2 a x 2 3a x 2 a x +--R a sinh(----) - 3a sinh(---) + a cosh(----) - a cosh(---) +--R 2 2 2 2 +--R Type: Expression Integer +--E + +--S 99 +coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) +--R +--I %BL cosh(y + x) + %BL cosh(y - x) +--I (10) %BL cosh(x)cosh(y) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 100 +gg:=coshcoshrule ff +--R +--R a x +--R 2log(sinh(a x) + cosh(a x) - 1) - 2log(sinh(---)) - a x +--R 2 +--R (11) ------------------------------------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 101 14:578 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R 2log(2) +--R (12) ------- +--R 2 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.579~~~~~$\displaystyle +\int{\frac{dx}{(\cosh{ax}+1)^2}}$} +$$\int{\frac{1}{(\cosh{ax}+1)^2}}= +\frac{1}{2a}\tanh{\frac{ax}{2}}-\frac{1}{6a}\tanh^3{\frac{ax}{2}} +$$ +<<*>>= +)clear all + +--S 102 +aa:=integrate(1/(cosh(a*x)+1)^2,x) +--R +--R +--R (1) +--R - 6sinh(a x) - 6cosh(a x) - 2 +--R / +--R 3 2 +--R 3a sinh(a x) + (9a cosh(a x) + 9a)sinh(a x) +--R + +--R 2 3 +--R (9a cosh(a x) + 18a cosh(a x) + 9a)sinh(a x) + 3a cosh(a x) +--R + +--R 2 +--R 9a cosh(a x) + 9a cosh(a x) + 3a +--R Type: Union(Expression Integer,...) +--E + +--S 103 +bb:=1/(2*a)*tanh((a*x)/2)-1/(6*a)*tanh((a*x)/2)^3 +--R +--R a x 3 a x +--R - tanh(---) + 3tanh(---) +--R 2 2 +--R (2) ------------------------- +--R 6a +--R Type: Expression Integer +--E + +--S 104 14:579 Axiom cannot compute this integral +cc:=aa-bb +--R +--R (3) +--R 3 2 +--R sinh(a x) + (3cosh(a x) + 3)sinh(a x) +--R + +--R 2 3 2 +--R (3cosh(a x) + 6cosh(a x) + 3)sinh(a x) + cosh(a x) + 3cosh(a x) +--R + +--R 3cosh(a x) + 1 +--R * +--R a x 3 +--R tanh(---) +--R 2 +--R + +--R 3 2 +--R - 3sinh(a x) + (- 9cosh(a x) - 9)sinh(a x) +--R + +--R 2 3 +--R (- 9cosh(a x) - 18cosh(a x) - 9)sinh(a x) - 3cosh(a x) +--R + +--R 2 +--R - 9cosh(a x) - 9cosh(a x) - 3 +--R * +--R a x +--R tanh(---) +--R 2 +--R + +--R - 12sinh(a x) - 12cosh(a x) - 4 +--R / +--R 3 2 +--R 6a sinh(a x) + (18a cosh(a x) + 18a)sinh(a x) +--R + +--R 2 3 +--R (18a cosh(a x) + 36a cosh(a x) + 18a)sinh(a x) + 6a cosh(a x) +--R + +--R 2 +--R 18a cosh(a x) + 18a cosh(a x) + 6a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.580~~~~~$\displaystyle +\int{\frac{dx}{(\cosh{ax}-1)^2}}$} +$$\int{\frac{1}{(\cosh{ax}-1)^2}}= +\frac{1}{2a}\coth{\frac{ax}{2}}-\frac{1}{6a}\coth^3{\frac{ax}{2}} +$$ +<<*>>= +)clear all + +--S 105 +aa:=integrate(1/(cosh(a*x)-1)^2,x) +--R +--R +--R (1) +--R - 6sinh(a x) - 6cosh(a x) + 2 +--R / +--R 3 2 +--R 3a sinh(a x) + (9a cosh(a x) - 9a)sinh(a x) +--R + +--R 2 3 +--R (9a cosh(a x) - 18a cosh(a x) + 9a)sinh(a x) + 3a cosh(a x) +--R + +--R 2 +--R - 9a cosh(a x) + 9a cosh(a x) - 3a +--R Type: Union(Expression Integer,...) +--E + +--S 106 +bb:=1/(2*a)*coth((a*x)/2)-1/(6*a)*coth((a*x)/2)^3 +--R +--R a x 3 a x +--R - coth(---) + 3coth(---) +--R 2 2 +--R (2) ------------------------- +--R 6a +--R Type: Expression Integer +--E + +--S 107 14:580 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R a x 3 a x 3 +--R (coth(---) - 3coth(---))sinh(a x) +--R 2 2 +--R + +--R a x 3 a x 2 +--R ((3cosh(a x) - 3)coth(---) + (- 9cosh(a x) + 9)coth(---))sinh(a x) +--R 2 2 +--R + +--R 2 a x 3 +--R (3cosh(a x) - 6cosh(a x) + 3)coth(---) +--R 2 +--R + +--R 2 a x +--R (- 9cosh(a x) + 18cosh(a x) - 9)coth(---) - 12 +--R 2 +--R * +--R sinh(a x) +--R + +--R 3 2 a x 3 +--R (cosh(a x) - 3cosh(a x) + 3cosh(a x) - 1)coth(---) +--R 2 +--R + +--R 3 2 a x +--R (- 3cosh(a x) + 9cosh(a x) - 9cosh(a x) + 3)coth(---) - 12cosh(a x) + 4 +--R 2 +--R / +--R 3 2 +--R 6a sinh(a x) + (18a cosh(a x) - 18a)sinh(a x) +--R + +--R 2 3 +--R (18a cosh(a x) - 36a cosh(a x) + 18a)sinh(a x) + 6a cosh(a x) +--R + +--R 2 +--R - 18a cosh(a x) + 18a cosh(a x) - 6a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.581~~~~~$\displaystyle +\int{\frac{dx}{p+q\cosh{ax}}}$} +$$\int{\frac{1}{p+q\cosh{ax}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{2}{a\sqrt{q^2-p^2}}\tan^{-1}\frac{qe^{ax}+p}{\sqrt{q^2-p^2}}\\ +\\ +\displaystyle +\frac{1}{a\sqrt{p^2-a^2}}\ln\left(\frac{qe^{ax}+p-\sqrt{p^2-q^2}} +{qe^{ax}+p+\sqrt{p^2-q^2}}\right) +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 108 +aa:=integrate(1/(p+q*cosh(a*x)),x) +--R +--R +--R (1) +--R [ +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R / +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R , +--R +-------+ +--R | 2 2 +--R (q sinh(a x) + q cosh(a x) + p)\|q - p +--R 2atan(-----------------------------------------) +--R 2 2 +--R q - p +--R ------------------------------------------------] +--R +-------+ +--R | 2 2 +--R a\|q - p +--R Type: Union(List Expression Integer,...) +--E + +--S 109 +bb1:=2/(a*sqrt(q^2-p^2))*atan((q*%e^(a*x)+p)/sqrt(q^2-p^2)) +--R +--R a x +--R q %e + p +--R 2atan(-----------) +--R +-------+ +--R | 2 2 +--R \|q - p +--R (2) ------------------ +--R +-------+ +--R | 2 2 +--R a\|q - p +--R Type: Expression Integer +--E + +--S 110 +bb2:=1/(a*sqrt(p^2-q^2))*log((q*%e^(a*x)+p-sqrt(p^2-q^2))/(q*%e^(a*x)+p+sqrt(p^2-q^2))) +--R +--R +---------+ +--R | 2 2 a x +--R - \|- q + p + q %e + p +--R log(----------------------------) +--R +---------+ +--R | 2 2 a x +--R \|- q + p + q %e + p +--R (3) --------------------------------- +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Expression Integer +--E + +--S 111 +cc1:=aa.1-bb1 +--R +--R (4) +--R +-------+ +--R | 2 2 +--R \|q - p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R +---------+ a x +--R | 2 2 q %e + p +--R - 2\|- q + p atan(-----------) +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 112 +cc2:=aa.2-bb1 +--R +--R +-------+ +--R | 2 2 a x +--R (q sinh(a x) + q cosh(a x) + p)\|q - p q %e + p +--R 2atan(-----------------------------------------) - 2atan(-----------) +--R 2 2 +-------+ +--R q - p | 2 2 +--R \|q - p +--R (5) --------------------------------------------------------------------- +--R +-------+ +--R | 2 2 +--R a\|q - p +--R Type: Expression Integer +--E + +--S 113 +cc3:=aa.1-bb2 +--R +--R (6) +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R +---------+ +--R | 2 2 a x +--R - \|- q + p + q %e + p +--R - log(----------------------------) +--R +---------+ +--R | 2 2 a x +--R \|- q + p + q %e + p +--R / +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Expression Integer +--E + +--S 114 14:581 Axiom cannot simplify this expression +cc4:=aa.2-bb2 +--R +--R (7) +--R +---------+ +--R +-------+ | 2 2 a x +--R | 2 2 - \|- q + p + q %e + p +--R - \|q - p log(----------------------------) +--R +---------+ +--R | 2 2 a x +--R \|- q + p + q %e + p +--R + +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 (q sinh(a x) + q cosh(a x) + p)\|q - p +--R 2\|- q + p atan(-----------------------------------------) +--R 2 2 +--R q - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a\|- q + p \|q - p +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.582~~~~~$\displaystyle +\int{\frac{dx}{(p+q\cosh{ax})^2}}~dx$} +$$\int{\frac{1}{(p+q\cosh{ax})^2}}= +\frac{q\sinh{ax}}{a(q^2-p^2)(p+q\cosh{ax})} +-\frac{p}{q^2-p^2}\int{\frac{1}{p+q\cosh{ax}}} +$$ +<<*>>= +)clear all + +--S 115 +aa:=integrate(1/(p+q*cosh(a*x))^2,x) +--R +--R +--R (1) +--R [ +--R 2 2 2 +--R p q sinh(a x) + (2p q cosh(a x) + 2p )sinh(a x) + p q cosh(a x) +--R + +--R 2 +--R 2p cosh(a x) + p q +--R * +--R log +--R 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) +--R + +--R 2 2 2 2 +--R q cosh(a x) + 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q + 2p q)sinh(a x) + (- 2q + 2p q)cosh(a x) - 2p q + 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R +---------+ +--R | 2 2 +--R (- 2p sinh(a x) - 2p cosh(a x) - 2q)\|- q + p +--R / +--R 3 2 2 +--R (a q - a p q)sinh(a x) +--R + +--R 3 2 2 3 +--R ((2a q - 2a p q)cosh(a x) + 2a p q - 2a p )sinh(a x) +--R + +--R 3 2 2 2 3 3 2 +--R (a q - a p q)cosh(a x) + (2a p q - 2a p )cosh(a x) + a q - a p q +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R , +--R +--R 2 2 +--R - 2p q sinh(a x) + (- 4p q cosh(a x) - 4p )sinh(a x) +--R + +--R 2 2 +--R - 2p q cosh(a x) - 4p cosh(a x) - 2p q +--R * +--R +-------+ +--R | 2 2 +--R (q sinh(a x) + q cosh(a x) + p)\|q - p +--R atan(-----------------------------------------) +--R 2 2 +--R q - p +--R + +--R +-------+ +--R | 2 2 +--R (- 2p sinh(a x) - 2p cosh(a x) - 2q)\|q - p +--R / +--R 3 2 2 +--R (a q - a p q)sinh(a x) +--R + +--R 3 2 2 3 +--R ((2a q - 2a p q)cosh(a x) + 2a p q - 2a p )sinh(a x) +--R + +--R 3 2 2 2 3 3 2 +--R (a q - a p q)cosh(a x) + (2a p q - 2a p )cosh(a x) + a q - a p q +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 116 +t1:=integrate(1/(p+q*cosh(a*x)),x) +--R +--R (2) +--R [ +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R / +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R , +--R +-------+ +--R | 2 2 +--R (q sinh(a x) + q cosh(a x) + p)\|q - p +--R 2atan(-----------------------------------------) +--R 2 2 +--R q - p +--R ------------------------------------------------] +--R +-------+ +--R | 2 2 +--R a\|q - p +--R Type: Union(List Expression Integer,...) +--E + +--S 117 +bb1:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.1 +--R +--R (3) +--R 2 +--R (- p q cosh(a x) - p ) +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R +---------+ +--R | 2 2 +--R q sinh(a x)\|- q + p +--R / +--R +---------+ +--R 3 2 2 3 | 2 2 +--R ((a q - a p q)cosh(a x) + a p q - a p )\|- q + p +--R Type: Expression Integer +--E + +--S 118 +bb2:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.2 +--R +--R (4) +--R +-------+ +--R | 2 2 +--R 2 (q sinh(a x) + q cosh(a x) + p)\|q - p +--R (- 2p q cosh(a x) - 2p )atan(-----------------------------------------) +--R 2 2 +--R q - p +--R + +--R +-------+ +--R | 2 2 +--R q sinh(a x)\|q - p +--R / +--R +-------+ +--R 3 2 2 3 | 2 2 +--R ((a q - a p q)cosh(a x) + a p q - a p )\|q - p +--R Type: Expression Integer +--E + +--S 119 +cc1:=aa.1-bb1 +--R +--R (5) +--R 2 2 2 +--R (p q cosh(a x) + p q)sinh(a x) +--R + +--R 2 2 2 3 2 3 +--R (2p q cosh(a x) + 4p q cosh(a x) + 2p )sinh(a x) + p q cosh(a x) +--R + +--R 2 2 2 3 2 +--R 3p q cosh(a x) + (p q + 2p )cosh(a x) + p q +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R 2 2 2 +--R (p q cosh(a x) + p q)sinh(a x) +--R + +--R 2 2 2 3 2 3 +--R (2p q cosh(a x) + 4p q cosh(a x) + 2p )sinh(a x) + p q cosh(a x) +--R + +--R 2 2 2 3 2 +--R 3p q cosh(a x) + (p q + 2p )cosh(a x) + p q +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q + 2p q)sinh(a x) + (- 2q + 2p q)cosh(a x) - 2p q + 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R 2 3 2 2 +--R - q sinh(a x) + (- 2q cosh(a x) - 2p q)sinh(a x) +--R + +--R 2 2 2 2 +--R (- q cosh(a x) - 4p q cosh(a x) - q - 2p )sinh(a x) +--R + +--R 2 2 2 +--R - 2p q cosh(a x) + (- 2q - 2p )cosh(a x) - 2p q +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R / +--R 4 2 2 3 3 2 +--R ((a q - a p q )cosh(a x) + a p q - a p q)sinh(a x) +--R + +--R 4 2 2 2 3 3 2 2 +--R (2a q - 2a p q )cosh(a x) + (4a p q - 4a p q)cosh(a x) + 2a p q +--R + +--R 4 +--R - 2a p +--R * +--R sinh(a x) +--R + +--R 4 2 2 3 3 3 2 +--R (a q - a p q )cosh(a x) + (3a p q - 3a p q)cosh(a x) +--R + +--R 4 2 2 4 3 3 +--R (a q + a p q - 2a p )cosh(a x) + a p q - a p q +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R Type: Expression Integer +--E + +--S 120 +cc2:=aa.2-bb1 +--R +--R (6) +--R 2 2 2 +--R (p q cosh(a x) + p q)sinh(a x) +--R + +--R 2 2 2 3 2 3 +--R (2p q cosh(a x) + 4p q cosh(a x) + 2p )sinh(a x) + p q cosh(a x) +--R + +--R 2 2 2 3 2 +--R 3p q cosh(a x) + (p q + 2p )cosh(a x) + p q +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R 2 2 2 +--R (- 2p q cosh(a x) - 2p q)sinh(a x) +--R + +--R 2 2 2 3 2 3 +--R (- 4p q cosh(a x) - 8p q cosh(a x) - 4p )sinh(a x) - 2p q cosh(a x) +--R + +--R 2 2 2 3 2 +--R - 6p q cosh(a x) + (- 2p q - 4p )cosh(a x) - 2p q +--R * +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 (q sinh(a x) + q cosh(a x) + p)\|q - p +--R \|- q + p atan(-----------------------------------------) +--R 2 2 +--R q - p +--R + +--R 2 3 2 2 +--R - q sinh(a x) + (- 2q cosh(a x) - 2p q)sinh(a x) +--R + +--R 2 2 2 2 +--R (- q cosh(a x) - 4p q cosh(a x) - q - 2p )sinh(a x) +--R + +--R 2 2 2 +--R - 2p q cosh(a x) + (- 2q - 2p )cosh(a x) - 2p q +--R * +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R \|- q + p \|q - p +--R / +--R 4 2 2 3 3 2 +--R ((a q - a p q )cosh(a x) + a p q - a p q)sinh(a x) +--R + +--R 4 2 2 2 3 3 2 2 +--R (2a q - 2a p q )cosh(a x) + (4a p q - 4a p q)cosh(a x) + 2a p q +--R + +--R 4 +--R - 2a p +--R * +--R sinh(a x) +--R + +--R 4 2 2 3 3 3 2 +--R (a q - a p q )cosh(a x) + (3a p q - 3a p q)cosh(a x) +--R + +--R 4 2 2 4 3 3 +--R (a q + a p q - 2a p )cosh(a x) + a p q - a p q +--R * +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R \|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 121 +cc3:=aa.1-bb2 +--R +--R (7) +--R 2 2 2 +--R (p q cosh(a x) + p q)sinh(a x) +--R + +--R 2 2 2 3 2 3 +--R (2p q cosh(a x) + 4p q cosh(a x) + 2p )sinh(a x) + p q cosh(a x) +--R + +--R 2 2 2 3 2 +--R 3p q cosh(a x) + (p q + 2p )cosh(a x) + p q +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q + 2p q)sinh(a x) + (- 2q + 2p q)cosh(a x) - 2p q + 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R 2 2 2 +--R (2p q cosh(a x) + 2p q)sinh(a x) +--R + +--R 2 2 2 3 2 3 +--R (4p q cosh(a x) + 8p q cosh(a x) + 4p )sinh(a x) + 2p q cosh(a x) +--R + +--R 2 2 2 3 2 +--R 6p q cosh(a x) + (2p q + 4p )cosh(a x) + 2p q +--R * +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 (q sinh(a x) + q cosh(a x) + p)\|q - p +--R \|- q + p atan(-----------------------------------------) +--R 2 2 +--R q - p +--R + +--R 2 3 2 2 +--R - q sinh(a x) + (- 2q cosh(a x) - 2p q)sinh(a x) +--R + +--R 2 2 2 2 +--R (- q cosh(a x) - 4p q cosh(a x) - q - 2p )sinh(a x) +--R + +--R 2 2 2 +--R - 2p q cosh(a x) + (- 2q - 2p )cosh(a x) - 2p q +--R * +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R \|- q + p \|q - p +--R / +--R 4 2 2 3 3 2 +--R ((a q - a p q )cosh(a x) + a p q - a p q)sinh(a x) +--R + +--R 4 2 2 2 3 3 2 2 +--R (2a q - 2a p q )cosh(a x) + (4a p q - 4a p q)cosh(a x) + 2a p q +--R + +--R 4 +--R - 2a p +--R * +--R sinh(a x) +--R + +--R 4 2 2 3 3 3 2 +--R (a q - a p q )cosh(a x) + (3a p q - 3a p q)cosh(a x) +--R + +--R 4 2 2 4 3 3 +--R (a q + a p q - 2a p )cosh(a x) + a p q - a p q +--R * +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R \|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 122 14:582 Axiom cannot simplify this expression +cc4:=aa.2-bb2 +--R +--R (8) +--R 2 3 2 2 +--R - q sinh(a x) + (- 2q cosh(a x) - 2p q)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (- q cosh(a x) - 4p q cosh(a x) - q - 2p )sinh(a x) - 2p q cosh(a x) +--R + +--R 2 2 +--R (- 2q - 2p )cosh(a x) - 2p q +--R / +--R 4 2 2 3 3 2 +--R ((a q - a p q )cosh(a x) + a p q - a p q)sinh(a x) +--R + +--R 4 2 2 2 3 3 2 2 +--R (2a q - 2a p q )cosh(a x) + (4a p q - 4a p q)cosh(a x) + 2a p q +--R + +--R 4 +--R - 2a p +--R * +--R sinh(a x) +--R + +--R 4 2 2 3 3 3 2 +--R (a q - a p q )cosh(a x) + (3a p q - 3a p q)cosh(a x) +--R + +--R 4 2 2 4 3 3 +--R (a q + a p q - 2a p )cosh(a x) + a p q - a p q +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.583~~~~~$\displaystyle +\int{\frac{dx}{p^2-q^2\cosh^2{ax}}}$} +$$\int{\frac{1}{p^2-q^2\cosh^2{ax}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{2ap\sqrt{p^2-q^2}}\ln\left(\frac{p\tanh{ax}+\sqrt{p^2-q^2}} +{p\tanh{ax}-\sqrt{p^2-q^2}}\right)\\ +\\ +\displaystyle +\frac{1}{ap\sqrt{q^2-p^2}}\tan^{-1}\frac{p\tanh{ax}}{\sqrt{q^2-p^2}}\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 123 +aa:=integrate(1/(p^2-q^2*cosh(a*x)^2),x) +--R +--R +--R (1) +--R [ +--R log +--R 4 4 4 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 4 2 4 2 2 2 +--R (6q cosh(a x) + 2q - 4p q )sinh(a x) +--R + +--R 4 3 4 2 2 +--R (4q cosh(a x) + (4q - 8p q )cosh(a x))sinh(a x) +--R + +--R 4 4 4 2 2 2 4 2 2 4 +--R q cosh(a x) + (2q - 4p q )cosh(a x) + q - 8p q + 8p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 4 3 2 2 4 3 2 +--R (- 4p q + 4p q )sinh(a x) + (- 8p q + 8p q )cosh(a x)sinh(a x) +--R + +--R 4 3 2 2 4 3 2 5 +--R (- 4p q + 4p q )cosh(a x) - 4p q + 12p q - 8p +--R / +--R 2 4 2 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (6q cosh(a x) + 2q - 4p )sinh(a x) +--R + +--R 2 3 2 2 2 4 +--R (4q cosh(a x) + (4q - 8p )cosh(a x))sinh(a x) + q cosh(a x) +--R + +--R 2 2 2 2 +--R (2q - 4p )cosh(a x) + q +--R / +--R +---------+ +--R | 2 2 +--R 2a p\|- q + p +--R , +--R +--R - +--R atan +--R 2 2 2 2 2 2 +--R q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) + q +--R + +--R 2 +--R - 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R 2 3 +--R 2p q - 2p +--R / +--R +-------+ +--R | 2 2 +--R a p\|q - p +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 124 +bb1:=1/(2*a*p*sqrt(p^2-q^2))*log((p*tanh(a*x)+sqrt(p^2-q^2))/(p*tanh(a*x)-sqrt(p^2-q^2))) +--R +--R +---------+ +--R | 2 2 +--R - \|- q + p - p tanh(a x) +--R log(----------------------------) +--R +---------+ +--R | 2 2 +--R \|- q + p - p tanh(a x) +--R (2) --------------------------------- +--R +---------+ +--R | 2 2 +--R 2a p\|- q + p +--R Type: Expression Integer +--E + +--S 125 +bb2:=-1/(a*p*sqrt(q^2-p^2))*atan((p*tanh(a*x))/sqrt(q^2-p^2)) +--R +--R p tanh(a x) +--R atan(-----------) +--R +-------+ +--R | 2 2 +--R \|q - p +--R (3) - ----------------- +--R +-------+ +--R | 2 2 +--R a p\|q - p +--R Type: Expression Integer +--E + +--S 126 +cc1:=aa.1-bb1 +--R +--R (4) +--R log +--R 4 4 4 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 4 2 4 2 2 2 +--R (6q cosh(a x) + 2q - 4p q )sinh(a x) +--R + +--R 4 3 4 2 2 +--R (4q cosh(a x) + (4q - 8p q )cosh(a x))sinh(a x) +--R + +--R 4 4 4 2 2 2 4 2 2 4 +--R q cosh(a x) + (2q - 4p q )cosh(a x) + q - 8p q + 8p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 4 3 2 2 4 3 2 +--R (- 4p q + 4p q )sinh(a x) + (- 8p q + 8p q )cosh(a x)sinh(a x) +--R + +--R 4 3 2 2 4 3 2 5 +--R (- 4p q + 4p q )cosh(a x) - 4p q + 12p q - 8p +--R / +--R 2 4 2 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (6q cosh(a x) + 2q - 4p )sinh(a x) +--R + +--R 2 3 2 2 2 4 +--R (4q cosh(a x) + (4q - 8p )cosh(a x))sinh(a x) + q cosh(a x) +--R + +--R 2 2 2 2 +--R (2q - 4p )cosh(a x) + q +--R + +--R +---------+ +--R | 2 2 +--R - \|- q + p - p tanh(a x) +--R - log(----------------------------) +--R +---------+ +--R | 2 2 +--R \|- q + p - p tanh(a x) +--R / +--R +---------+ +--R | 2 2 +--R 2a p\|- q + p +--R Type: Expression Integer +--E + +--S 127 +cc2:=aa.2-bb1 +--R +--R (5) +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 - \|- q + p - p tanh(a x) +--R - \|q - p log(----------------------------) +--R +---------+ +--R | 2 2 +--R \|- q + p - p tanh(a x) +--R + +--R - +--R +---------+ +--R | 2 2 +--R 2\|- q + p +--R * +--R atan +--R 2 2 2 2 2 2 +--R q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) + q +--R + +--R 2 +--R - 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R 2 3 +--R 2p q - 2p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 128 +cc3:=aa.1-bb2 +--R +--R (6) +--R +-------+ +--R | 2 2 +--R \|q - p +--R * +--R log +--R 4 4 4 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 4 2 4 2 2 2 +--R (6q cosh(a x) + 2q - 4p q )sinh(a x) +--R + +--R 4 3 4 2 2 +--R (4q cosh(a x) + (4q - 8p q )cosh(a x))sinh(a x) +--R + +--R 4 4 4 2 2 2 4 2 2 4 +--R q cosh(a x) + (2q - 4p q )cosh(a x) + q - 8p q + 8p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 4 3 2 2 +--R (- 4p q + 4p q )sinh(a x) +--R + +--R 4 3 2 +--R (- 8p q + 8p q )cosh(a x)sinh(a x) +--R + +--R 4 3 2 2 4 3 2 5 +--R (- 4p q + 4p q )cosh(a x) - 4p q + 12p q - 8p +--R / +--R 2 4 2 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (6q cosh(a x) + 2q - 4p )sinh(a x) +--R + +--R 2 3 2 2 2 4 +--R (4q cosh(a x) + (4q - 8p )cosh(a x))sinh(a x) + q cosh(a x) +--R + +--R 2 2 2 2 +--R (2q - 4p )cosh(a x) + q +--R + +--R +---------+ +--R | 2 2 p tanh(a x) +--R 2\|- q + p atan(-----------) +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 129 14:583 Axiom cannot simplify this expression +cc4:=aa.2-bb2 +--R +--R (7) +--R - +--R atan +--R 2 2 2 2 2 2 +--R q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) + q +--R + +--R 2 +--R - 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R 2 3 +--R 2p q - 2p +--R + +--R p tanh(a x) +--R atan(-----------) +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R +-------+ +--R | 2 2 +--R a p\|q - p +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.584~~~~~$\displaystyle +\int{\frac{dx}{p^2+q^2\cosh^2{ax}}}$} +$$\int{\frac{1}{p^2+q^2\cosh^2{ax}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{2ap\sqrt{p^2+q^2}}\ln\left(\frac{p\tanh{ax}+\sqrt{p^2+q^2}} +{p\tanh{ax}-\sqrt{p^2+q^2}}\right)\\ +\\ +\displaystyle +\frac{1}{ap\sqrt{p^2+q^2}}\tan^{-1}\frac{p\tanh{ax}}{\sqrt{p^2+q^2}}\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 130 +aa:=integrate(1/(p^2+q^2*cosh(a*x)^2),x) +--R +--R +--R (1) +--R log +--R 4 4 4 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 4 2 4 2 2 2 +--R (6q cosh(a x) + 2q + 4p q )sinh(a x) +--R + +--R 4 3 4 2 2 4 4 +--R (4q cosh(a x) + (4q + 8p q )cosh(a x))sinh(a x) + q cosh(a x) +--R + +--R 4 2 2 2 4 2 2 4 +--R (2q + 4p q )cosh(a x) + q + 8p q + 8p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 4 3 2 2 4 3 2 +--R (- 4p q - 4p q )sinh(a x) + (- 8p q - 8p q )cosh(a x)sinh(a x) +--R + +--R 4 3 2 2 4 3 2 5 +--R (- 4p q - 4p q )cosh(a x) - 4p q - 12p q - 8p +--R / +--R 2 4 2 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (6q cosh(a x) + 2q + 4p )sinh(a x) +--R + +--R 2 3 2 2 2 4 +--R (4q cosh(a x) + (4q + 8p )cosh(a x))sinh(a x) + q cosh(a x) +--R + +--R 2 2 2 2 +--R (2q + 4p )cosh(a x) + q +--R / +--R +-------+ +--R | 2 2 +--R 2a p\|q + p +--R Type: Union(Expression Integer,...) +--E + +--S 131 +bb1:=1/(2*a*p*sqrt(p^2+q^2))*log((p*tanh(a*x)+sqrt(p^2+q^2))/(p*tanh(a*x)-sqrt(p^2+q^2))) +--R +--R +-------+ +--R | 2 2 +--R - \|q + p - p tanh(a x) +--R log(--------------------------) +--R +-------+ +--R | 2 2 +--R \|q + p - p tanh(a x) +--R (2) ------------------------------- +--R +-------+ +--R | 2 2 +--R 2a p\|q + p +--R Type: Expression Integer +--E + +--S 132 +bb2:=1/(a*p*sqrt(p^2+q^2))*atan((p*tanh(a*x))/sqrt(p^2+q^2)) +--R +--R p tanh(a x) +--R atan(-----------) +--R +-------+ +--R | 2 2 +--R \|q + p +--R (3) ----------------- +--R +-------+ +--R | 2 2 +--R a p\|q + p +--R Type: Expression Integer +--E + +--S 133 +cc1:=aa-bb1 +--R +--R (4) +--R log +--R 4 4 4 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 4 2 4 2 2 2 +--R (6q cosh(a x) + 2q + 4p q )sinh(a x) +--R + +--R 4 3 4 2 2 +--R (4q cosh(a x) + (4q + 8p q )cosh(a x))sinh(a x) +--R + +--R 4 4 4 2 2 2 4 2 2 4 +--R q cosh(a x) + (2q + 4p q )cosh(a x) + q + 8p q + 8p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 4 3 2 2 4 3 2 +--R (- 4p q - 4p q )sinh(a x) + (- 8p q - 8p q )cosh(a x)sinh(a x) +--R + +--R 4 3 2 2 4 3 2 5 +--R (- 4p q - 4p q )cosh(a x) - 4p q - 12p q - 8p +--R / +--R 2 4 2 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (6q cosh(a x) + 2q + 4p )sinh(a x) +--R + +--R 2 3 2 2 2 4 +--R (4q cosh(a x) + (4q + 8p )cosh(a x))sinh(a x) + q cosh(a x) +--R + +--R 2 2 2 2 +--R (2q + 4p )cosh(a x) + q +--R + +--R +-------+ +--R | 2 2 +--R - \|q + p - p tanh(a x) +--R - log(--------------------------) +--R +-------+ +--R | 2 2 +--R \|q + p - p tanh(a x) +--R / +--R +-------+ +--R | 2 2 +--R 2a p\|q + p +--R Type: Expression Integer +--E + +--S 134 14:584 Axiom cannot simplify this expression +cc2:=aa-bb2 +--R +--R (5) +--R log +--R 4 4 4 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 4 2 4 2 2 2 +--R (6q cosh(a x) + 2q + 4p q )sinh(a x) +--R + +--R 4 3 4 2 2 +--R (4q cosh(a x) + (4q + 8p q )cosh(a x))sinh(a x) +--R + +--R 4 4 4 2 2 2 4 2 2 4 +--R q cosh(a x) + (2q + 4p q )cosh(a x) + q + 8p q + 8p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 4 3 2 2 4 3 2 +--R (- 4p q - 4p q )sinh(a x) + (- 8p q - 8p q )cosh(a x)sinh(a x) +--R + +--R 4 3 2 2 4 3 2 5 +--R (- 4p q - 4p q )cosh(a x) - 4p q - 12p q - 8p +--R / +--R 2 4 2 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (6q cosh(a x) + 2q + 4p )sinh(a x) +--R + +--R 2 3 2 2 2 4 +--R (4q cosh(a x) + (4q + 8p )cosh(a x))sinh(a x) + q cosh(a x) +--R + +--R 2 2 2 2 +--R (2q + 4p )cosh(a x) + q +--R + +--R p tanh(a x) +--R - 2atan(-----------) +--R +-------+ +--R | 2 2 +--R \|q + p +--R / +--R +-------+ +--R | 2 2 +--R 2a p\|q + p +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.585~~~~~$\displaystyle +\int{x^m\cosh{ax}}~dx$} +$$\int{x^m\cosh{ax}}= +\frac{x^m\sinh{ax}}{a}-\frac{m}{a}\int{x^{m-1}\sinh{ax}} +$$ +<<*>>= +)clear all + +--S 135 14:585 Axiom cannot compute this integral +aa:=integrate(x^m*cosh(a*x),x) +--R +--R +--R x +--R ++ m +--I (1) | cosh(%N a)%N d%N +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.586~~~~~$\displaystyle +\int{\cosh^n{ax}}~dx$} +$$\int{\cosh^n{ax}}= +\frac{\cosh^{n-1}{ax}\sinh{ax}}{an}+\frac{n-1}{n}\int{\cosh^{n-2}{ax}} +$$ +<<*>>= +)clear all + +--S 136 14:586 Axiom cannot compute this integral +aa:=integrate(cosh(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | cosh(%N a) d%N +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.587~~~~~$\displaystyle +\int{\frac{\cosh{ax}}{x^n}}~dx$} +$$\int{\frac{\cosh{ax}}{x^n}}= +\frac{-\cosh{ax}}{(n-1)x^{n-1}} ++\frac{a}{n-1}\int{\frac{\sinh{ax}}{x^{n-1}}} +$$ +<<*>>= +)clear all + +--S 137 14:587 Axiom cannot compute this integral +aa:=integrate(cosh(a*x)/x^n,x) +--R +--R +--R x +--I ++ cosh(%N a) +--I (1) | ---------- d%N +--R ++ n +--I %N +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.588~~~~~$\displaystyle +\int{\frac{dx}{\cosh^n{ax}}}~dx$} +$$\int{\frac{1}{\cosh^n{ax}}}= +\frac{\sinh{ax}}{a(n-1)\cosh^{n-1}{ax}} ++\frac{n-2}{n-1}\int{\frac{1}{\cosh^{n-2}{ax}}} +$$ +<<*>>= +)clear all + +--S 138 14:588 Axiom cannot compute this integral +aa:=integrate(1/cosh(a*x)^n,x) +--R +--R +--R x +--R ++ 1 +--I (1) | ----------- d%N +--R ++ n +--I cosh(%N a) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.589~~~~~$\displaystyle +\int{\frac{x}{\cosh^n{ax}}}~dx$} +$$\int{\frac{x}{\cosh^n{ax}}}= +\frac{x\sinh{ax}}{a(n-1)\cosh^{n-1}{ax}} ++\frac{1}{(n-1)(n-2)a^2\cosh^{n-2}{ax}} ++\frac{n-2}{n-1}\int{\frac{x}{\cosh^{n-2}}} +$$ +<<*>>= +)clear all + +--S 139 14:589 Axiom cannot compute this integral +aa:=integrate(1/cosh(a*x)^n,x) +--R +--R +--R x +--R ++ 1 +--I (1) | ----------- d%N +--R ++ n +--I cosh(%N a) +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp88-89 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum28.input.pdf b/src/axiom-website/CATS/schaum28.input.pdf new file mode 100644 index 0000000..8280248 --- /dev/null +++ b/src/axiom-website/CATS/schaum28.input.pdf @@ -0,0 +1,4223 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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+0000158881 00000 n +0000159603 00000 n +0000159520 00000 n +0000160134 00000 n +0000160856 00000 n +0000160773 00000 n +0000163559 00000 n +0000161485 00000 n +0000162207 00000 n +0000162124 00000 n +0000162738 00000 n +0000163460 00000 n +0000163377 00000 n +0000166354 00000 n +0000164089 00000 n +0000164811 00000 n +0000164728 00000 n +0000165533 00000 n +0000165450 00000 n +0000166255 00000 n +0000166172 00000 n +0000166739 00000 n +0000166762 00000 n +0000166784 00000 n +trailer +<< +/Size 381 +/Root 2 0 R +/Info 1 0 R +>> +startxref +166905 +%%EOF diff --git a/src/axiom-website/CATS/schaum29.input.pamphlet b/src/axiom-website/CATS/schaum29.input.pamphlet new file mode 100644 index 0000000..9c79630 --- /dev/null +++ b/src/axiom-website/CATS/schaum29.input.pamphlet @@ -0,0 +1,1431 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum29.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.590~~~~~$\displaystyle +\int{\sinh{ax}\cosh{ax}}~dx$} +$$\int{\sinh{ax}\cosh{ax}}= +\frac{\sinh^2{ax}}{2a} +$$ +<<*>>= +)spool schaum29.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(sinh(a*x)*cosh(a*x),x) +--R +--R +--R 2 2 +--R sinh(a x) + cosh(a x) +--R (1) ----------------------- +--R 4a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=sinh(a*x)^2/(2*a) +--R +--R 2 +--R sinh(a x) +--R (2) ---------- +--R 2a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R 2 2 +--R - sinh(a x) + cosh(a x) +--R (3) ------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 4 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 5 +dd:=sinhsqrrule cc +--R +--R 2 +--R - cosh(2a x) + 2cosh(a x) + 1 +--R (5) ------------------------------ +--R 8a +--R Type: Expression Integer +--E + +--S 6 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 7 14:590 Schaums and Axiom differ by a constant +ee:=coshsqrrule dd +--R +--R 1 +--R (7) -- +--R 4a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.591~~~~~$\displaystyle +\int{\sinh{px}\cosh{qx}}~dx$} +$$\int{\sinh{px}\cosh{qx}}= +\frac{\cosh(p+q)x}{2(p+q)}+\frac{\cosh(p-q)x}{2(p-q)} +$$ +<<*>>= +)clear all + +--S 8 +aa:=integrate(sinh(p*x)*cosh(q*x),x) +--R +--R +--R - q sinh(p x)sinh(q x) + p cosh(p x)cosh(q x) +--R (1) --------------------------------------------- +--R 2 2 2 2 2 2 +--R (q - p )sinh(p x) + (- q + p )cosh(p x) +--R Type: Union(Expression Integer,...) +--E + +--S 9 +bb:=(cosh(p+q)*x)/(2*(p+q))+(cosh(p-q)*x)/(2*(p-q)) +--R +--R (q - p)x cosh(q + p) + (- q - p)x cosh(q - p) +--R (2) --------------------------------------------- +--R 2 2 +--R 2q - 2p +--R Type: Expression Integer +--E + +--S 10 14:591 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R - 2q sinh(p x)sinh(q x) +--R + +--R 2 +--R ((- q + p)x cosh(q + p) + (q + p)x cosh(q - p))sinh(p x) +--R + +--R 2p cosh(p x)cosh(q x) +--R + +--R 2 +--R ((q - p)x cosh(q + p) + (- q - p)x cosh(q - p))cosh(p x) +--R / +--R 2 2 2 2 2 2 +--R (2q - 2p )sinh(p x) + (- 2q + 2p )cosh(p x) +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.592~~~~~$\displaystyle +\int{\sinh^n{ax}\cosh{ax}}~dx$} +$$\int{\sinh^n{ax}\cosh{ax}}= +\frac{\sinh^{n+1}{ax}}{(n+1)a} +$$ +<<*>>= +)clear all + +--S 11 +aa:=integrate(sinh(a*x)^n*cosh(a*x),x) +--R +--R +--R - sinh(a x)sinh(n log(sinh(a x))) - sinh(a x)cosh(n log(sinh(a x))) +--R (1) ------------------------------------------------------------------- +--R 2 2 +--R (a n + a)sinh(a x) + (- a n - a)cosh(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 12 +bb:=sinh(a*x)/((n+1)*a) +--R +--R sinh(a x) +--R (2) --------- +--R a n + a +--R Type: Expression Integer +--E + +--S 13 14:592 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R - sinh(a x)sinh(n log(sinh(a x))) - sinh(a x)cosh(n log(sinh(a x))) +--R + +--R 3 2 +--R - sinh(a x) + cosh(a x) sinh(a x) +--R / +--R 2 2 +--R (a n + a)sinh(a x) + (- a n - a)cosh(a x) +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.593~~~~~$\displaystyle +\int{\cosh^n{ax}\sinh{ax}}~dx$} +$$\int{\cosh^n{ax}\sinh{ax}}= +\frac{\cosh^{n+1}{ax}}{(n+1)a} +$$ +<<*>>= +)clear all + +--S 14 +aa:=integrate(cosh(a*x)^n*sinh(a*x),x) +--R +--R +--R - cosh(a x)sinh(n log(cosh(a x))) - cosh(a x)cosh(n log(cosh(a x))) +--R (1) ------------------------------------------------------------------- +--R 2 2 +--R (a n + a)sinh(a x) + (- a n - a)cosh(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 15 +bb:=cosh(a*x)^(n+1)/((n+1)*a) +--R +--R n + 1 +--R cosh(a x) +--R (2) -------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 16 14:593 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R - cosh(a x)sinh(n log(cosh(a x))) - cosh(a x)cosh(n log(cosh(a x))) +--R + +--R 2 2 n + 1 +--R (- sinh(a x) + cosh(a x) )cosh(a x) +--R / +--R 2 2 +--R (a n + a)sinh(a x) + (- a n - a)cosh(a x) +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.594~~~~~$\displaystyle +\int{\sinh^2{ax}\cosh^2{ax}}~dx$} +$$\int{\sinh^2{ax}\cosh^2{ax}}= +\frac{\sinh{4ax}}{32a}-\frac{x}{8} +$$ +<<*>>= +)clear all + +--S 17 +aa:=integrate(sinh(a*x)^2*cosh(a*x)^2,x) +--R +--R +--R 3 3 +--R cosh(a x)sinh(a x) + cosh(a x) sinh(a x) - a x +--R (1) ----------------------------------------------- +--R 8a +--R Type: Union(Expression Integer,...) +--E + +--S 18 +bb:=sinh(4*a*x)/(32*a)-x/8 +--R +--R sinh(4a x) - 4a x +--R (2) ----------------- +--R 32a +--R Type: Expression Integer +--E + +--S 19 14:594 Schaums and Axiom agree +cc:=complexNormalize(aa-bb) +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.595~~~~~$\displaystyle +\int{\frac{dx}{\sinh{ax}\cosh{ax}}}$} +$$\int{\frac{1}{\sinh{ax}\cosh{ax}}}= +\frac{1}{a}\ln\tanh{ax} +$$ +<<*>>= +)clear all + +--S 20 +aa:=integrate(1/(sinh(a*x)*cosh(a*x)),x) +--R +--R +--R 2cosh(a x) 2sinh(a x) +--R - log(- ---------------------) + log(- ---------------------) +--R sinh(a x) - cosh(a x) sinh(a x) - cosh(a x) +--R (1) ------------------------------------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 21 +bb:=1/a*log(tanh(a*x)) +--R +--R log(tanh(a x)) +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 22 +cc:=aa-bb +--R +--R (3) +--R 2cosh(a x) +--R - log(tanh(a x)) - log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R / +--R a +--R Type: Expression Integer +--E + +--S 23 +dd:=expandLog cc +--R +--R - log(tanh(a x)) + log(sinh(a x)) - log(cosh(a x)) +--R (4) -------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 24 +tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) +--R +--R sinh(x) +--R (5) tanh(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 25 +ee:=tanhrule dd +--R +--R sinh(a x) +--R log(sinh(a x)) - log(---------) - log(cosh(a x)) +--R cosh(a x) +--R (6) ------------------------------------------------ +--R a +--R Type: Expression Integer +--E + +--S 26 14:595 Schaums and Axiom agree +ff:=expandLog ee +--R +--R (7) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.596~~~~~$\displaystyle +\int{\frac{dx}{\sinh^2{ax}\cosh{ax}}}$} +$$\int{\frac{1}{\sinh^2{ax}\cosh{ax}}}= +-\frac{1}{a}\tan^{-1}\sinh{ax}-\frac{{\rm csch~}{ax}}{a} +$$ +<<*>>= +)clear all + +--S 27 +aa:=integrate(1/(sinh(a*x)^2*cosh(a*x)),x) +--R +--R (1) +--R 2 2 +--R (- 2sinh(a x) - 4cosh(a x)sinh(a x) - 2cosh(a x) + 2) +--R * +--R atan(sinh(a x) + cosh(a x)) +--R + +--R - 2sinh(a x) - 2cosh(a x) +--R / +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Union(Expression Integer,...) +--E + +--S 28 +bb:=-1/a*atan(sinh(a*x)-csch(a*x))/a +--R +--R atan(sinh(a x) - csch(a x)) +--R (2) - --------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 29 14:596 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (- 2a sinh(a x) - 4a cosh(a x)sinh(a x) - 2a cosh(a x) + 2a) +--R * +--R atan(sinh(a x) + cosh(a x)) +--R + +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1) +--R * +--R atan(sinh(a x) - csch(a x)) +--R + +--R - 2a sinh(a x) - 2a cosh(a x) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.597~~~~~$\displaystyle +\int{\frac{dx}{\sinh{ax}\cosh^2{ax}}}$} +$$\int{\frac{1}{\sinh{ax}\cosh^2{ax}}}= +\frac{{\rm sech~}{ax}}{a}+\frac{1}{a}\ln\tanh{\frac{ax}{2}} +$$ +<<*>>= +)clear all + +--S 30 +aa:=integrate(1/(sinh(a*x)*cosh(a*x)^2),x) +--R +--R +--R (1) +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1) +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1) +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2sinh(a x) + 2cosh(a x) +--R / +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Union(Expression Integer,...) +--E + +--S 31 +bb:=sech(a*x)/a+1/a*log(tanh((a*x)/2)) +--R +--R a x +--R log(tanh(---)) + sech(a x) +--R 2 +--R (2) -------------------------- +--R a +--R Type: Expression Integer +--E + +--S 32 +cc:=aa-bb +--R +--R (3) +--R 2 2 a x +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1)log(tanh(---)) +--R 2 +--R + +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1) +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1) +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 +--R - sech(a x)sinh(a x) + (- 2cosh(a x)sech(a x) + 2)sinh(a x) +--R + +--R 2 +--R (- cosh(a x) - 1)sech(a x) + 2cosh(a x) +--R / +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 33 +sechrule:=rule(sech(x) == 1/cosh(x)) +--R +--R 1 +--R (4) sech(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 34 +dd:=sechrule cc +--R +--R (5) +--R 2 2 3 +--R (- cosh(a x)sinh(a x) - 2cosh(a x) sinh(a x) - cosh(a x) - cosh(a x)) +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 2 3 +--R (- cosh(a x)sinh(a x) - 2cosh(a x) sinh(a x) - cosh(a x) - cosh(a x)) +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 2 3 +--R (cosh(a x)sinh(a x) + 2cosh(a x) sinh(a x) + cosh(a x) + cosh(a x)) +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 2 +--R - sinh(a x) + cosh(a x) - 1 +--R / +--R 2 2 3 +--R a cosh(a x)sinh(a x) + 2a cosh(a x) sinh(a x) + a cosh(a x) + a cosh(a x) +--R Type: Expression Integer +--E + +--S 35 +tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) +--R +--R sinh(x) +--R (6) tanh(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 36 +ee:=tanhrule dd +--R +--R (7) +--R 2 2 3 +--R (- cosh(a x)sinh(a x) - 2cosh(a x) sinh(a x) - cosh(a x) - cosh(a x)) +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 2 3 +--R (cosh(a x)sinh(a x) + 2cosh(a x) sinh(a x) + cosh(a x) + cosh(a x)) +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 2 3 +--R (- cosh(a x)sinh(a x) - 2cosh(a x) sinh(a x) - cosh(a x) - cosh(a x)) +--R * +--R a x +--R sinh(---) +--R 2 +--R log(---------) +--R a x +--R cosh(---) +--R 2 +--R + +--R 2 2 +--R - sinh(a x) + cosh(a x) - 1 +--R / +--R 2 2 3 +--R a cosh(a x)sinh(a x) + 2a cosh(a x) sinh(a x) + a cosh(a x) + a cosh(a x) +--R Type: Expression Integer +--E + +--S 37 +coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x)) +--R +--R 3 cosh(3x) - 3cosh(x) +--R (8) cosh(x) == ------------------- +--R 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 38 +ff:=coshcuberule ee +--R +--R (9) +--R 2 2 +--R - 4cosh(a x)sinh(a x) - 8cosh(a x) sinh(a x) - cosh(3a x) +--R + +--R - cosh(a x) +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 2 +--R (4cosh(a x)sinh(a x) + 8cosh(a x) sinh(a x) + cosh(3a x) + cosh(a x)) +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 2 +--R - 4cosh(a x)sinh(a x) - 8cosh(a x) sinh(a x) - cosh(3a x) +--R + +--R - cosh(a x) +--R * +--R a x +--R sinh(---) +--R 2 +--R log(---------) +--R a x +--R cosh(---) +--R 2 +--R + +--R 2 2 +--R - 4sinh(a x) + 4cosh(a x) - 4 +--R / +--R 2 2 +--R 4a cosh(a x)sinh(a x) + 8a cosh(a x) sinh(a x) + a cosh(3a x) +--R + +--R a cosh(a x) +--R Type: Expression Integer +--E + +--S 39 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (10) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 40 +gg:=coshsqrrule ff +--R +--R (11) +--R 2 +--R - 4cosh(a x)sinh(a x) + (- 4cosh(2a x) - 4)sinh(a x) - cosh(3a x) +--R + +--R - cosh(a x) +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 +--R 4cosh(a x)sinh(a x) + (4cosh(2a x) + 4)sinh(a x) + cosh(3a x) +--R + +--R cosh(a x) +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 +--R - 4cosh(a x)sinh(a x) + (- 4cosh(2a x) - 4)sinh(a x) - cosh(3a x) +--R + +--R - cosh(a x) +--R * +--R a x +--R sinh(---) +--R 2 +--R log(---------) +--R a x +--R cosh(---) +--R 2 +--R + +--R 2 +--R - 4sinh(a x) + 2cosh(2a x) - 2 +--R / +--R 2 +--R 4a cosh(a x)sinh(a x) + (4a cosh(2a x) + 4a)sinh(a x) + a cosh(3a x) +--R + +--R a cosh(a x) +--R Type: Expression Integer +--E + +--S 41 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (12) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 42 +hh:=sinhsqrrule gg +--R +--R (13) +--R - log(sinh(a x) + cosh(a x) + 1) + log(sinh(a x) + cosh(a x) - 1) +--R + +--R a x +--R sinh(---) +--R 2 +--R - log(---------) +--R a x +--R cosh(---) +--R 2 +--R / +--R a +--R Type: Expression Integer +--E + +--S 43 +ii:=expandLog hh +--R +--R (14) +--R - log(sinh(a x) + cosh(a x) + 1) + log(sinh(a x) + cosh(a x) - 1) +--R + +--R a x a x +--R - log(sinh(---)) + log(cosh(---)) +--R 2 2 +--R / +--R a +--R Type: Expression Integer +--E + +--S 44 14:597 Schaums and Axiom agree +jj:=complexNormalize ii +--R +--R (15) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.598~~~~~$\displaystyle +\int{\frac{dx}{\sinh^2{ax}\cosh^2{ax}}}$} +$$\int{\frac{1}{\sinh^2{ax}\cosh^2{ax}}}= +-\frac{2\coth{2ax}}{a} +$$ +<<*>>= +)clear all + +--S 45 +aa:=integrate(1/(sinh(a*x)^2*cosh(a*x)^2),x) +--R +--R +--R (1) +--R - +--R 4 +--R / +--R 4 3 2 2 +--R a sinh(a x) + 4a cosh(a x)sinh(a x) + 6a cosh(a x) sinh(a x) +--R + +--R 3 4 +--R 4a cosh(a x) sinh(a x) + a cosh(a x) - a +--R Type: Union(Expression Integer,...) +--E + +--S 46 +bb:=-(2*coth(2*a*x))/a +--R +--R 2coth(2a x) +--R (2) - ----------- +--R a +--R Type: Expression Integer +--E + +--S 47 14:598 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 4 3 +--R 2coth(2a x)sinh(a x) + 8cosh(a x)coth(2a x)sinh(a x) +--R + +--R 2 2 3 +--R 12cosh(a x) coth(2a x)sinh(a x) + 8cosh(a x) coth(2a x)sinh(a x) +--R + +--R 4 +--R (2cosh(a x) - 2)coth(2a x) - 4 +--R / +--R 4 3 2 2 +--R a sinh(a x) + 4a cosh(a x)sinh(a x) + 6a cosh(a x) sinh(a x) +--R + +--R 3 4 +--R 4a cosh(a x) sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.599~~~~~$\displaystyle +\int{\frac{\sinh^2{ax}}{\cosh{ax}}}~dx$} +$$\int{\frac{\sinh^2{ax}}{\cosh{ax}}}~dx= +\frac{\sinh{ax}}{a}-\frac{1}{a}\tan^{-1}\sinh{ax} +$$ +<<*>>= +)clear all + +--S 48 +aa:=integrate(sinh(a*x)^2/cosh(a*x),x) +--R +--R +--R (1) +--R 2 +--R (- 4sinh(a x) - 4cosh(a x))atan(sinh(a x) + cosh(a x)) + sinh(a x) +--R + +--R 2 +--R 2cosh(a x)sinh(a x) + cosh(a x) - 1 +--R / +--R 2a sinh(a x) + 2a cosh(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 49 +bb:=sinh(a*x)/a-1/a*atan(sinh(a*x)) +--R +--R - atan(sinh(a x)) + sinh(a x) +--R (2) ----------------------------- +--R a +--R Type: Expression Integer +--E + +--S 50 14:599 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R (- 4sinh(a x) - 4cosh(a x))atan(sinh(a x) + cosh(a x)) +--R + +--R 2 2 +--R (2sinh(a x) + 2cosh(a x))atan(sinh(a x)) - sinh(a x) + cosh(a x) - 1 +--R / +--R 2a sinh(a x) + 2a cosh(a x) +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.600~~~~~$\displaystyle +\int{\frac{\cosh^2{ax}}{\sinh{ax}}}~dx$} +$$\int{\frac{\cosh^2{ax}}{\sinh{ax}}}= +\frac{\cosh{ax}}{a}+\frac{1}{a}\ln\tanh{\frac{ax}{2}} +$$ +<<*>>= +)clear all + +--S 51 +aa:=integrate(cosh(a*x)^2/sinh(a*x),x) +--R +--R +--R (1) +--R (- 2sinh(a x) - 2cosh(a x))log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 +--R (2sinh(a x) + 2cosh(a x))log(sinh(a x) + cosh(a x) - 1) + sinh(a x) +--R + +--R 2 +--R 2cosh(a x)sinh(a x) + cosh(a x) + 1 +--R / +--R 2a sinh(a x) + 2a cosh(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 52 +bb:=cosh(a*x)/a+1/a*log(tanh((a*x)/2)) +--R +--R a x +--R log(tanh(---)) + cosh(a x) +--R 2 +--R (2) -------------------------- +--R a +--R Type: Expression Integer +--E + +--S 53 14:600 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R a x +--R (- 2sinh(a x) - 2cosh(a x))log(tanh(---)) +--R 2 +--R + +--R (- 2sinh(a x) - 2cosh(a x))log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 +--R (2sinh(a x) + 2cosh(a x))log(sinh(a x) + cosh(a x) - 1) + sinh(a x) +--R + +--R 2 +--R - cosh(a x) + 1 +--R / +--R 2a sinh(a x) + 2a cosh(a x) +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.601~~~~~$\displaystyle +\int{\frac{dx}{\cosh{ax}(1+\sinh{ax})}}$} +$$\int{\frac{1}{\cosh{ax}(1+\sinh{ax})}}= +\frac{1}{2a}\ln\left(\frac{1+\sinh{ax}}{\cosh{ax}}\right) ++\frac{1}{a}\tan^{-1}{e^{ax}} +$$ +<<*>>= +)clear all + +--S 54 +aa:=integrate(1/(cosh(a*x)*(1+sinh(a*x))),x) +--R +--R +--R (1) +--R 2cosh(a x) - 2sinh(a x) - 2 +--R - log(- ---------------------) + log(---------------------) +--R sinh(a x) - cosh(a x) sinh(a x) - cosh(a x) +--R + +--R 2atan(sinh(a x) + cosh(a x)) +--R / +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 55 +bb:=1/(2*a)*log((1+sinh(a*x))/cosh(a*x))+1/a*atan(%e^(a*x)) +--R +--R sinh(a x) + 1 a x +--R log(-------------) + 2atan(%e ) +--R cosh(a x) +--R (2) --------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 56 +cc:=aa-bb +--R +--R (3) +--R sinh(a x) + 1 2cosh(a x) +--R - log(-------------) - log(- ---------------------) +--R cosh(a x) sinh(a x) - cosh(a x) +--R + +--R - 2sinh(a x) - 2 a x +--R log(---------------------) + 2atan(sinh(a x) + cosh(a x)) - 2atan(%e ) +--R sinh(a x) - cosh(a x) +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 57 +dd:=expandLog cc +--R +--R a x +--R atan(sinh(a x) + cosh(a x)) - atan(%e ) +--R (4) ----------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 58 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (5) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 59 +ee:=atanrule dd +--R +--R a x +--R - %e + %i - sinh(a x) - cosh(a x) + %i +--R %i log(------------) - %i log(----------------------------) +--R a x sinh(a x) + cosh(a x) + %i +--R %e + %i +--R (6) ----------------------------------------------------------- +--R 2a +--R Type: Expression Complex Integer +--E + +--S 60 +ff:=expandLog ee +--R +--R (7) +--R %i log(sinh(a x) + cosh(a x) + %i) - %i log(sinh(a x) + cosh(a x) - %i) +--R + +--R a x a x +--R - %i log(%e + %i) + %i log(%e - %i) +--R / +--R 2a +--R Type: Expression Complex Integer +--E + +--S 61 14:601 Schaums and Axiom agree +gg:=complexNormalize ff +--R +--R (8) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.602~~~~~$\displaystyle +\int{\frac{dx}{\sinh{ax}(\cosh{ax}+1)}}$} +$$\int{\frac{1}{\sinh{ax}(\cosh{ax}+1)}}= +\frac{1}{2a}\ln\tanh\frac{ax}{2}+\frac{1}{2a(\cosh{ax}+1)} +$$ +<<*>>= +)clear all + +--S 62 +aa:=integrate(1/(sinh(a*x)*(cosh(a*x)+1)),x) +--R +--R +--R (1) +--R 2 2 +--R - sinh(a x) + (- 2cosh(a x) - 2)sinh(a x) - cosh(a x) - 2cosh(a x) +--R + +--R - 1 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 2 +--R (sinh(a x) + (2cosh(a x) + 2)sinh(a x) + cosh(a x) + 2cosh(a x) + 1) +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2sinh(a x) + 2cosh(a x) +--R / +--R 2 2 +--R 2a sinh(a x) + (4a cosh(a x) + 4a)sinh(a x) + 2a cosh(a x) +--R + +--R 4a cosh(a x) + 2a +--R Type: Union(Expression Integer,...) +--E + +--S 63 +bb:=1/(2*a)*log(tanh((a*x)/2))+1/(2*a*(cosh(a*x)+1)) +--R +--R a x +--R (cosh(a x) + 1)log(tanh(---)) + 1 +--R 2 +--R (2) --------------------------------- +--R 2a cosh(a x) + 2a +--R Type: Expression Integer +--E + +--S 64 +cc:=aa-bb +--R +--R (3) +--R 2 +--R (- cosh(a x) - 1)sinh(a x) +--R + +--R 2 3 2 +--R (- 2cosh(a x) - 4cosh(a x) - 2)sinh(a x) - cosh(a x) - 3cosh(a x) +--R + +--R - 3cosh(a x) - 1 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 +--R (- cosh(a x) - 1)sinh(a x) +--R + +--R 2 3 2 +--R (- 2cosh(a x) - 4cosh(a x) - 2)sinh(a x) - cosh(a x) - 3cosh(a x) +--R + +--R - 3cosh(a x) - 1 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 2 +--R (cosh(a x) + 1)sinh(a x) + (2cosh(a x) + 4cosh(a x) + 2)sinh(a x) +--R + +--R 3 2 +--R cosh(a x) + 3cosh(a x) + 3cosh(a x) + 1 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 2 +--R - sinh(a x) + cosh(a x) - 1 +--R / +--R 2 +--R (2a cosh(a x) + 2a)sinh(a x) +--R + +--R 2 3 +--R (4a cosh(a x) + 8a cosh(a x) + 4a)sinh(a x) + 2a cosh(a x) +--R + +--R 2 +--R 6a cosh(a x) + 6a cosh(a x) + 2a +--R Type: Expression Integer +--E + +--S 65 +coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x)) +--R +--R 3 cosh(3x) - 3cosh(x) +--R (4) cosh(x) == ------------------- +--R 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 66 +dd:=coshcuberule cc +--R +--R (5) +--R 2 +--R (- 4cosh(a x) - 4)sinh(a x) +--R + +--R 2 +--R (- 8cosh(a x) - 16cosh(a x) - 8)sinh(a x) - cosh(3a x) +--R + +--R 2 +--R - 12cosh(a x) - 9cosh(a x) - 4 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 +--R (- 4cosh(a x) - 4)sinh(a x) +--R + +--R 2 +--R (- 8cosh(a x) - 16cosh(a x) - 8)sinh(a x) - cosh(3a x) +--R + +--R 2 +--R - 12cosh(a x) - 9cosh(a x) - 4 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 2 +--R (4cosh(a x) + 4)sinh(a x) + (8cosh(a x) + 16cosh(a x) + 8)sinh(a x) +--R + +--R 2 +--R cosh(3a x) + 12cosh(a x) + 9cosh(a x) + 4 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 2 +--R - 4sinh(a x) + 4cosh(a x) - 4 +--R / +--R 2 +--R (8a cosh(a x) + 8a)sinh(a x) +--R + +--R 2 +--R (16a cosh(a x) + 32a cosh(a x) + 16a)sinh(a x) + 2a cosh(3a x) +--R + +--R 2 +--R 24a cosh(a x) + 18a cosh(a x) + 8a +--R Type: Expression Integer +--E + +--S 67 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (6) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 68 +ee:=sinhsqrrule dd +--R +--R (7) +--R 2 +--R (- 8cosh(a x) - 16cosh(a x) - 8)sinh(a x) - cosh(3a x) +--R + +--R 2 +--R (- 2cosh(a x) - 2)cosh(2a x) - 12cosh(a x) - 7cosh(a x) - 2 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 +--R (- 8cosh(a x) - 16cosh(a x) - 8)sinh(a x) - cosh(3a x) +--R + +--R 2 +--R (- 2cosh(a x) - 2)cosh(2a x) - 12cosh(a x) - 7cosh(a x) - 2 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 +--R (8cosh(a x) + 16cosh(a x) + 8)sinh(a x) + cosh(3a x) +--R + +--R 2 +--R (2cosh(a x) + 2)cosh(2a x) + 12cosh(a x) + 7cosh(a x) + 2 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 +--R - 2cosh(2a x) + 4cosh(a x) - 2 +--R / +--R 2 +--R (16a cosh(a x) + 32a cosh(a x) + 16a)sinh(a x) + 2a cosh(3a x) +--R + +--R 2 +--R (4a cosh(a x) + 4a)cosh(2a x) + 24a cosh(a x) + 14a cosh(a x) + 4a +--R Type: Expression Integer +--E + +--S 69 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (8) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 70 +ff:=coshsqrrule ee +--R +--R (9) +--R a x +--R - log(tanh(---)) - log(sinh(a x) + cosh(a x) + 1) +--R 2 +--R + +--R log(sinh(a x) + cosh(a x) - 1) +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 71 14:602 Schaums and Axiom agree +gg:=complexNormalize ff +--R +--R (10) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.603~~~~~$\displaystyle +\int{\frac{dx}{\sinh{ax}(\cosh{ax}-1)}}$} +$$\int{\frac{1}{\sinh{ax}(\cosh{ax}-1)}}= +-\frac{1}{2a}\ln\tanh\frac{ax}{2}-\frac{1}{2a(cosh{ax}-1)} +$$ +<<*>>= +)clear all + +--S 72 +aa:=integrate(1/(sinh(a*x)*(cosh(a*x)-1)),x) +--R +--R +--R (1) +--R 2 2 +--R (sinh(a x) + (2cosh(a x) - 2)sinh(a x) + cosh(a x) - 2cosh(a x) + 1) +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 2 +--R - sinh(a x) + (- 2cosh(a x) + 2)sinh(a x) - cosh(a x) + 2cosh(a x) +--R + +--R - 1 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R - 2sinh(a x) - 2cosh(a x) +--R / +--R 2 2 +--R 2a sinh(a x) + (4a cosh(a x) - 4a)sinh(a x) + 2a cosh(a x) +--R + +--R - 4a cosh(a x) + 2a +--R Type: Union(Expression Integer,...) +--E + +--S 73 +bb:=-1/(2*a)*log(tanh((a*x)/2))-1/(2*a*(cosh(a*x)-1)) +--R +--R a x +--R (- cosh(a x) + 1)log(tanh(---)) - 1 +--R 2 +--R (2) ----------------------------------- +--R 2a cosh(a x) - 2a +--R Type: Expression Integer +--E + +--S 74 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (cosh(a x) - 1)sinh(a x) + (2cosh(a x) - 4cosh(a x) + 2)sinh(a x) +--R + +--R 3 2 +--R cosh(a x) - 3cosh(a x) + 3cosh(a x) - 1 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 2 +--R (cosh(a x) - 1)sinh(a x) + (2cosh(a x) - 4cosh(a x) + 2)sinh(a x) +--R + +--R 3 2 +--R cosh(a x) - 3cosh(a x) + 3cosh(a x) - 1 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 +--R (- cosh(a x) + 1)sinh(a x) +--R + +--R 2 3 2 +--R (- 2cosh(a x) + 4cosh(a x) - 2)sinh(a x) - cosh(a x) + 3cosh(a x) +--R + +--R - 3cosh(a x) + 1 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 2 +--R sinh(a x) - cosh(a x) + 1 +--R / +--R 2 +--R (2a cosh(a x) - 2a)sinh(a x) +--R + +--R 2 3 +--R (4a cosh(a x) - 8a cosh(a x) + 4a)sinh(a x) + 2a cosh(a x) +--R + +--R 2 +--R - 6a cosh(a x) + 6a cosh(a x) - 2a +--R Type: Expression Integer +--E + +--S 75 +coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x)) +--R +--R 3 cosh(3x) - 3cosh(x) +--R (4) cosh(x) == ------------------- +--R 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 76 +dd:=coshcuberule cc +--R +--R (5) +--R 2 2 +--R (4cosh(a x) - 4)sinh(a x) + (8cosh(a x) - 16cosh(a x) + 8)sinh(a x) +--R + +--R 2 +--R cosh(3a x) - 12cosh(a x) + 9cosh(a x) - 4 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 2 +--R (4cosh(a x) - 4)sinh(a x) + (8cosh(a x) - 16cosh(a x) + 8)sinh(a x) +--R + +--R 2 +--R cosh(3a x) - 12cosh(a x) + 9cosh(a x) - 4 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 +--R (- 4cosh(a x) + 4)sinh(a x) +--R + +--R 2 +--R (- 8cosh(a x) + 16cosh(a x) - 8)sinh(a x) - cosh(3a x) +--R + +--R 2 +--R 12cosh(a x) - 9cosh(a x) + 4 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 2 +--R 4sinh(a x) - 4cosh(a x) + 4 +--R / +--R 2 +--R (8a cosh(a x) - 8a)sinh(a x) +--R + +--R 2 +--R (16a cosh(a x) - 32a cosh(a x) + 16a)sinh(a x) + 2a cosh(3a x) +--R + +--R 2 +--R - 24a cosh(a x) + 18a cosh(a x) - 8a +--R Type: Expression Integer +--E + +--S 77 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (6) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 78 +ee:=sinhsqrrule dd +--R +--R (7) +--R 2 +--R (8cosh(a x) - 16cosh(a x) + 8)sinh(a x) + cosh(3a x) +--R + +--R 2 +--R (2cosh(a x) - 2)cosh(2a x) - 12cosh(a x) + 7cosh(a x) - 2 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 +--R (8cosh(a x) - 16cosh(a x) + 8)sinh(a x) + cosh(3a x) +--R + +--R 2 +--R (2cosh(a x) - 2)cosh(2a x) - 12cosh(a x) + 7cosh(a x) - 2 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 +--R (- 8cosh(a x) + 16cosh(a x) - 8)sinh(a x) - cosh(3a x) +--R + +--R 2 +--R (- 2cosh(a x) + 2)cosh(2a x) + 12cosh(a x) - 7cosh(a x) + 2 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 +--R 2cosh(2a x) - 4cosh(a x) + 2 +--R / +--R 2 +--R (16a cosh(a x) - 32a cosh(a x) + 16a)sinh(a x) + 2a cosh(3a x) +--R + +--R 2 +--R (4a cosh(a x) - 4a)cosh(2a x) - 24a cosh(a x) + 14a cosh(a x) - 4a +--R Type: Expression Integer +--E + +--S 79 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (8) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 80 +ff:=coshsqrrule ee +--R +--R (9) +--R a x +--R log(tanh(---)) + log(sinh(a x) + cosh(a x) + 1) +--R 2 +--R + +--R - log(sinh(a x) + cosh(a x) - 1) +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 81 14:603 Schaums and Axiom agree +gg:=complexNormalize ff +--R +--R (10) 0 +--R Type: Expression Integer +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp89-90 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum29.input.pdf b/src/axiom-website/CATS/schaum29.input.pdf new file mode 100644 index 0000000..6bff5fd --- /dev/null +++ b/src/axiom-website/CATS/schaum29.input.pdf @@ -0,0 +1,2801 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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0000000..bed98c8 --- /dev/null +++ b/src/axiom-website/CATS/schaum3.input.pamphlet @@ -0,0 +1,398 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum3.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.105~~~~~$\displaystyle\int{\frac{dx}{(ax+b)(px+q)}}$} +$$\int{\frac{1}{(ax+b)(px+q)}}= +\frac{1}{bp-aq}~\ln\left(\frac{px+q}{ax+b}\right)$$ +<<*>>= +)spool schaum3.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(1/((a*x+b)*(p*x+q)),x) +--R +--R +--R - log(p x + q) + log(a x + b) +--R (1) ----------------------------- +--R a q - b p +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=1/(b*p-a*q)*log((p*x+q)/(a*x+b)) +--R +--R +--R p x + q +--R log(-------) +--R a x + b +--R (2) - ------------ +--R a q - b p +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R +--R p x + q +--R - log(p x + q) + log(a x + b) + log(-------) +--R a x + b +--R (3) -------------------------------------------- +--R a q - b p +--R Type: Expression Integer +--E + +--S 4 +logdiv:=rule(log(a)-log(b) == log(a/b)) +--R +--R a +--I (4) - log(b) + log(a) + %I == log(-) + %I +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 5 +dd:=logdiv cc +--R +--R 1 +--R log(a x + b) + log(-------) +--R a x + b +--R (5) --------------------------- +--R a q - b p +--R Type: Expression Integer +--E + +--S 6 +logmul:=rule(log(a)+log(b) == log(a*b)) +--R +--I (6) log(b) + log(a) + %J == log(a b) + %J +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 7 14:105 Schaums and Axiom agree +ee:=logmul dd +--R +--R (7) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.106~~~~~$\displaystyle\int{\frac{x~dx}{(ax+b)(px+q)}}$} +$$\int{\frac{x}{(ax+b)(px+q)}}= +\frac{1}{bp-aq}\left\{\frac{b}{a}~\ln(ax+b)-\frac{q}{p}~\ln(px+q)\right\}$$ +<<*>>= +)clear all + +--S 8 +aa:=integrate(x/((a*x+b)*(p*x+q)),x) +--R +--R +--R a q log(p x + q) - b p log(a x + b) +--R (1) ----------------------------------- +--R 2 2 +--R a p q - a b p +--R Type: Union(Expression Integer,...) +--E + +--S 9 +bb:=1/(b*p-a*q)*(b/a*log(a*x+b)-q/p*log(p*x+q)) +--R +--R +--R a q log(p x + q) - b p log(a x + b) +--R (2) ----------------------------------- +--R 2 2 +--R a p q - a b p +--R Type: Expression Integer +--E + +--S 10 14:106 Schaums and Axiom agree +cc:=aa-bb +--R +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.107~~~~~$\displaystyle\int{\frac{dx}{(ax+b)^2(px+q)}}$} +$$\int{\frac{1}{(ax+b)^2(px+q)}}= +\frac{1}{bp-aq} +\left\{\frac{1}{ax+b}+ +\frac{p}{bp-aq}~\ln\left(\frac{px+q}{ax+b}\right)\right\}$$ +<<*>>= +)clear all + +--S 11 +aa:=integrate(1/((a*x+b)^2*(p*x+q)),x) +--R +--R +--R (a p x + b p)log(p x + q) + (- a p x - b p)log(a x + b) - a q + b p +--R (1) ------------------------------------------------------------------- +--R 3 2 2 2 2 2 2 2 3 2 +--R (a q - 2a b p q + a b p )x + a b q - 2a b p q + b p +--R Type: Union(Expression Integer,...) +--E + +--S 12 +bb:=1/(b*p-a*q)*(1/(a*x+b)+p/(b*p-a*q)*log((p*x+q)/(a*x+b))) +--R +--R +--R p x + q +--R (a p x + b p)log(-------) - a q + b p +--R a x + b +--R (2) ------------------------------------------------------ +--R 3 2 2 2 2 2 2 2 3 2 +--R (a q - 2a b p q + a b p )x + a b q - 2a b p q + b p +--R Type: Expression Integer +--E + +--S 13 +cc:=aa-bb +--R +--R +--R p x + q +--R p log(p x + q) - p log(a x + b) - p log(-------) +--R a x + b +--R (3) ------------------------------------------------ +--R 2 2 2 2 +--R a q - 2a b p q + b p +--R Type: Expression Integer +--E + +--S 14 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 15 14:107 Schaums and Axiom agree +dd:=divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.108~~~~~$\displaystyle\int{\frac{x~dx}{(ax+b)^2(px+q)}}$} +$$\int{\frac{x}{(ax+b)^2(px+q)}}= +\frac{1}{bp-aq} +\left\{\frac{q}{bp-aq} +~\ln\left(\frac{ax+b}{px+q}\right)-\frac{b}{a(ax+b)}\right\}$$ + +<<*>>= +)clear all + +--S 16 +aa:=integrate(x/((a*x+b)^2*(p*x+q)),x) +--R +--R +--R (1) +--R 2 2 2 +--R (- a q x - a b q)log(p x + q) + (a q x + a b q)log(a x + b) + a b q - b p +--R ------------------------------------------------------------------------- +--R 4 2 3 2 2 2 3 2 2 2 3 2 +--R (a q - 2a b p q + a b p )x + a b q - 2a b p q + a b p +--R Type: Union(Expression Integer,...) +--E + +--S 17 +bb:=1/(b*p-a*q)*(q/(b*p-a*q)*log((a*x+b)/(p*x+q))-b/(a*(a*x+b))) +--R +--R +--R 2 a x + b 2 +--R (a q x + a b q)log(-------) + a b q - b p +--R p x + q +--R (2) -------------------------------------------------------- +--R 4 2 3 2 2 2 3 2 2 2 3 2 +--R (a q - 2a b p q + a b p )x + a b q - 2a b p q + a b p +--R Type: Expression Integer +--E + +--S 18 +cc:=aa-bb +--R +--R +--R a x + b +--R - q log(p x + q) + q log(a x + b) - q log(-------) +--R p x + q +--R (3) -------------------------------------------------- +--R 2 2 2 2 +--R a q - 2a b p q + b p +--R Type: Expression Integer +--E + +--S 19 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 20 14:108 Schaums and Axiom agree +dd:=divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.109~~~~~$\displaystyle +\int{\frac{x^2~dx}{(ax+b)^2(px+q)}}$} +$$\int{\frac{x^2}{(ax+b)^2(px+q)}}=$$ +$$\frac{b^2}{(bp-aq)a^2(ax+b)}+\frac{1}{(bp-aq)^2} +\left\{\frac{q^2}{p}~\ln(px+q)+\frac{b(bp-2aq)}{a^2}~\ln(ax+b)\right\}$$ +<<*>>= +)clear all + +--S 21 +aa:=integrate(x^2/((a*x+b)^2*(p*x+q)),x) +--R +--R +--R (1) +--R 3 2 2 2 +--R (a q x + a b q )log(p x + q) +--R + +--R 2 2 2 2 3 2 2 3 2 +--R ((- 2a b p q + a b p )x - 2a b p q + b p )log(a x + b) - a b p q + b p +--R / +--R 5 2 4 2 3 2 3 4 2 3 2 2 2 3 3 +--R (a p q - 2a b p q + a b p )x + a b p q - 2a b p q + a b p +--R Type: Union(Expression Integer,...) +--E + +--S 22 +bb:=b^2/((b*p-a*q)*a^2*(a*x+b))+_ + 1/(b*p-a*q)^2*(q^2/p*log(p*x+q)+((b*(b*p-2*a*q))/a^2)*log(a*x+b)) +--R +--R +--R (2) +--R 3 2 2 2 +--R (a q x + a b q )log(p x + q) +--R + +--R 2 2 2 2 3 2 2 3 2 +--R ((- 2a b p q + a b p )x - 2a b p q + b p )log(a x + b) - a b p q + b p +--R / +--R 5 2 4 2 3 2 3 4 2 3 2 2 2 3 3 +--R (a p q - 2a b p q + a b p )x + a b p q - 2a b p q + a b p +--R Type: Expression Integer +--E + +--S 23 14:109 Schaums and Axiom agree +cc:=aa-bb +--R +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.110~~~~~$\displaystyle\int{\frac{dx}{(ax+b)^m(px+q)^n}}$} +$$\int{\frac{1}{(ax+b)^m(px+q)^n}}=$$ +$$\frac{-1}{(n-1)(bp-aq)} +\left\{\frac{1}{(ax+b)^{m-1}(px+q)^{n-1}}+ +a(m+n-2)~\int{\frac{1}{(ax+b)^m(px+q)^{n-1}}}\right\}$$ +<<*>>= +)clear all + +--S 24 14:110 Axiom cannot do this integral +aa:=integrate(1/((a*x+b)^m*(p*x+q)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | ---------------------- d%L +--R ++ m n +--I (b + %L a) (q + %L p) +--R Type: Union(Expression Integer,...) +--E + +@ +\section{\cite{1}:14.111~~~~~$\displaystyle\int{\frac{ax+b}{px+q}~dx}$} +$$\int{\frac{ax+b}{px+q}}=\frac{ax}{p}+\frac{bp-aq}{p^2}~\ln(px+q)$$ +<<*>>= +)clear all + +--S 25 +aa:=integrate((a*x+b)/(p*x+q),x) +--R +--R +--R (- a q + b p)log(p x + q) + a p x +--R (1) --------------------------------- +--R 2 +--R p +--R Type: Union(Expression Integer,...) +--E + +--S 26 +bb:=(a*x)/p+(b*p-a*q)/p^2*log(p*x+q) +--R +--R +--R (- a q + b p)log(p x + q) + a p x +--R (2) --------------------------------- +--R 2 +--R p +--R Type: Expression Integer +--E + +--S 27 14:111 Schaums and Axiom agree +cc:=aa-bb +--R +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.112~~~~~$\displaystyle\int{\frac{(ax+b)^m}{(px+q)^n}~dx}$} +$$\int{\frac{(ax+b)^m}{(px+q)^n}}=\left\{ +\begin{array}{c} +\frac{-1}{(n-1)(bp-aq)} +\left\{\frac{(ax+b)^{m+1}}{(px+q)^{n-1}}+(n-m-2)a +\int{\frac{(ax+b)^m}{(px+q)^{n-1}}}\right\}\\ +\frac{-1}{(n-m-1)p}+\left\{\frac{(ax+b)^m}{(px+q)^{n-1}}+m(bp-aq) +\int{\frac{(ax+b)^{m-1}}{(px+q)^n}}\right\}\\ +\frac{-1}{(n-1)p}\left\{\frac{(ax+b)^m}{(px+q)^{n-1}}-ma +\int{\frac{(ax+b)^{m-1}}{(px+q)^{n-1}}}\right\} +\end{array} +\right.$$ +<<*>>= +)clear all + +--S 28 14:112 Axiom cannot do this integral +aa:=integrate((a*x+b)^m/(p*x+q)^n,x) +--R +--R +--R x m +--I ++ (b + %L a) +--I (1) | ----------- d%L +--R ++ n +--I (q + %L p) +--R Type: Union(Expression Integer,...) +--E +<<*>>= +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp62-63 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum3.input.pdf b/src/axiom-website/CATS/schaum3.input.pdf new file mode 100644 index 0000000..e958a81 --- /dev/null +++ b/src/axiom-website/CATS/schaum3.input.pdf @@ -0,0 +1,1944 @@ 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n +0000034102 00000 n +0000035600 00000 n +0000115887 00000 n +0000037183 00000 n +0000108386 00000 n +0000108191 00000 n +0000035863 00000 n +0000036786 00000 n +0000037125 00000 n +0000114569 00000 n +0000115074 00000 n +0000115482 00000 n +0000115995 00000 n +0000116377 00000 n +0000116400 00000 n +0000116422 00000 n +trailer +<< +/Size 130 +/Root 2 0 R +/Info 1 0 R +>> +startxref +116543 +%%EOF diff --git a/src/axiom-website/CATS/schaum30.input.pamphlet b/src/axiom-website/CATS/schaum30.input.pamphlet new file mode 100644 index 0000000..5df48bc --- /dev/null +++ b/src/axiom-website/CATS/schaum30.input.pamphlet @@ -0,0 +1,744 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum30.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.604~~~~~$\displaystyle +\int{\tanh{ax}}~dx$} +$$\int{\tanh{ax}}= +\frac{1}{a}\ln\cosh{ax} +$$ +<<*>>= +)spool schaum30.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(tanh(a*x),x) +--R +--R +--R 2cosh(a x) +--R log(- ---------------------) - a x +--R sinh(a x) - cosh(a x) +--R (1) ---------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=1/a*log(cosh(a*x)) +--R +--R log(cosh(a x)) +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R 2cosh(a x) +--R - log(cosh(a x)) + log(- ---------------------) - a x +--R sinh(a x) - cosh(a x) +--R (3) ----------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 4 +dd:=expandLog cc +--R +--R - log(sinh(a x) - cosh(a x)) + log(- 2) - a x +--R (4) --------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 5 14:604 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R - log(- 1) + log(- 2) +--R (5) --------------------- +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.605~~~~~$\displaystyle +\int{\tanh^2{ax}}~dx$} +$$\int{\tanh^2{ax}}= +x-\frac{\tanh{ax}}{a} +$$ +<<*>>= +)clear all + +--S 6 +aa:=integrate(tanh(a*x)^2,x) +--R +--R +--R - sinh(a x) + (a x + 1)cosh(a x) +--R (1) -------------------------------- +--R a cosh(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 7 +bb:=x-tanh(a*x)/a +--R +--R - tanh(a x) + a x +--R (2) ----------------- +--R a +--R Type: Expression Integer +--E + +--S 8 +cc:=aa-bb +--R +--R cosh(a x)tanh(a x) - sinh(a x) + cosh(a x) +--R (3) ------------------------------------------ +--R a cosh(a x) +--R Type: Expression Integer +--E + +--S 9 +tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) +--R +--R sinh(x) +--R (4) tanh(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 10 14:605 Schaums and Axiom differ by a constant +dd:=tanhrule cc +--R +--R 1 +--R (5) - +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.606~~~~~$\displaystyle +\int{\tanh^3{ax}}~dx$} +$$\int{\tanh^3{ax}}= +\frac{1}{a}\ln\cosh{ax}-\frac{\tanh^2{ax}}{2a} +$$ +<<*>>= +)clear all + +--S 11 +aa:=integrate(tanh(a*x)^3,x) +--R +--R +--R (1) +--R 4 3 2 2 +--R sinh(a x) + 4cosh(a x)sinh(a x) + (6cosh(a x) + 2)sinh(a x) +--R + +--R 3 4 2 +--R (4cosh(a x) + 4cosh(a x))sinh(a x) + cosh(a x) + 2cosh(a x) + 1 +--R * +--R 2cosh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 4 3 +--R - a x sinh(a x) - 4a x cosh(a x)sinh(a x) +--R + +--R 2 2 +--R (- 6a x cosh(a x) - 2a x + 2)sinh(a x) +--R + +--R 3 4 +--R (- 4a x cosh(a x) + (- 4a x + 4)cosh(a x))sinh(a x) - a x cosh(a x) +--R + +--R 2 +--R (- 2a x + 2)cosh(a x) - a x +--R / +--R 4 3 2 2 +--R a sinh(a x) + 4a cosh(a x)sinh(a x) + (6a cosh(a x) + 2a)sinh(a x) +--R + +--R 3 4 2 +--R (4a cosh(a x) + 4a cosh(a x))sinh(a x) + a cosh(a x) + 2a cosh(a x) + a +--R Type: Union(Expression Integer,...) +--E + +--S 12 +bb:=1/a*log(cosh(a*x))-tanh(a*x)^2/(2*a) +--R +--R 2 +--R 2log(cosh(a x)) - tanh(a x) +--R (2) ---------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 13 14:606 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 4 3 2 2 +--R - 2sinh(a x) - 8cosh(a x)sinh(a x) + (- 12cosh(a x) - 4)sinh(a x) +--R + +--R 3 4 2 +--R (- 8cosh(a x) - 8cosh(a x))sinh(a x) - 2cosh(a x) - 4cosh(a x) - 2 +--R * +--R log(cosh(a x)) +--R + +--R 4 3 2 2 +--R 2sinh(a x) + 8cosh(a x)sinh(a x) + (12cosh(a x) + 4)sinh(a x) +--R + +--R 3 4 2 +--R (8cosh(a x) + 8cosh(a x))sinh(a x) + 2cosh(a x) + 4cosh(a x) + 2 +--R * +--R 2cosh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 4 3 2 2 +--R sinh(a x) + 4cosh(a x)sinh(a x) + (6cosh(a x) + 2)sinh(a x) +--R + +--R 3 4 2 +--R (4cosh(a x) + 4cosh(a x))sinh(a x) + cosh(a x) + 2cosh(a x) + 1 +--R * +--R 2 +--R tanh(a x) +--R + +--R 4 3 +--R - 2a x sinh(a x) - 8a x cosh(a x)sinh(a x) +--R + +--R 2 2 +--R (- 12a x cosh(a x) - 4a x + 4)sinh(a x) +--R + +--R 3 4 +--R (- 8a x cosh(a x) + (- 8a x + 8)cosh(a x))sinh(a x) - 2a x cosh(a x) +--R + +--R 2 +--R (- 4a x + 4)cosh(a x) - 2a x +--R / +--R 4 3 2 2 +--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) + (12a cosh(a x) + 4a)sinh(a x) +--R + +--R 3 4 2 +--R (8a cosh(a x) + 8a cosh(a x))sinh(a x) + 2a cosh(a x) + 4a cosh(a x) +--R + +--R 2a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.607~~~~~$\displaystyle +\int{\tanh^n{ax}{{\rm ~sech}^2{ax}}}~dx$} +$$\int{\tanh^n{ax}{{\rm ~sech}^2{ax}}}= +\frac{\tanh^{n+1}{ax}}{(n+1)a} +$$ +<<*>>= +)clear all + +--S 14 +aa:=integrate(tanh(a*x)^n*sech(a*x)^2,x) +--R +--R +--R sinh(a x) sinh(a x) +--R sinh(a x)sinh(n log(---------)) + sinh(a x)cosh(n log(---------)) +--R cosh(a x) cosh(a x) +--R (1) ----------------------------------------------------------------- +--R (a n + a)cosh(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 15 +bb:=tanh(a*x)^(n+1)/((n+1)*a) +--R +--R n + 1 +--R tanh(a x) +--R (2) -------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 16 14:607 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R sinh(a x) sinh(a x) +--R sinh(a x)sinh(n log(---------)) + sinh(a x)cosh(n log(---------)) +--R cosh(a x) cosh(a x) +--R + +--R n + 1 +--R - cosh(a x)tanh(a x) +--R / +--R (a n + a)cosh(a x) +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.608~~~~~$\displaystyle +\int{\frac{{\rm sech}^2{ax}}{\tanh{ax}}}~dx$} +$$\int{\frac{{\rm sech}^2{ax}}{\tanh{ax}}}= +\frac{1}{a}\ln\tanh{ax} +$$ +<<*>>= +)clear all + +--S 17 +aa:=integrate(sech(a*x)^2/tanh(a*x),x) +--R +--R +--R 2cosh(a x) 2sinh(a x) +--R - log(- ---------------------) + log(- ---------------------) +--R sinh(a x) - cosh(a x) sinh(a x) - cosh(a x) +--R (1) ------------------------------------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 18 +bb:=1/a*log(tanh(a*x)) +--R +--R log(tanh(a x)) +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 19 +cc:=aa-bb +--R +--R (3) +--R 2cosh(a x) +--R - log(tanh(a x)) - log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R / +--R a +--R Type: Expression Integer +--E + +--S 20 +tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) +--R +--R sinh(x) +--R (4) tanh(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 21 +dd:=tanhrule cc +--R +--R (5) +--R sinh(a x) 2cosh(a x) +--R - log(---------) - log(- ---------------------) +--R cosh(a x) sinh(a x) - cosh(a x) +--R + +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R / +--R a +--R Type: Expression Integer +--E + +--S 22 14:608 Schaums and Axiom agree +ee:=expandLog dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.609~~~~~$\displaystyle +\int{\frac{dx}{\tanh{ax}}}~dx$} +$$\int{\frac{1}{\tanh{ax}}}= +\frac{1}{a}\ln\sinh{ax} +$$ +<<*>>= +)clear all + +--S 23 +aa:=integrate(1/tanh(a*x),x) +--R +--R +--R 2sinh(a x) +--R log(- ---------------------) - a x +--R sinh(a x) - cosh(a x) +--R (1) ---------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 24 +bb:=1/a*log(sinh(a*x)) +--R +--R log(sinh(a x)) +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 25 +cc:=aa-bb +--R +--R 2sinh(a x) +--R - log(sinh(a x)) + log(- ---------------------) - a x +--R sinh(a x) - cosh(a x) +--R (3) ----------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 26 +dd:=expandLog cc +--R +--R - log(sinh(a x) - cosh(a x)) + log(- 2) - a x +--R (4) --------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 27 14:609 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R - log(- 1) + log(- 2) +--R (5) --------------------- +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.610~~~~~$\displaystyle +\int{x\tanh{ax}}~dx$} +$$\int{x\tanh{ax}}= +\frac{1}{a^2}\left\{ +\frac{(ax)^3}{3}-\frac{(ax)^5}{15}+\frac{2(ax)^7}{105}-\cdots +\frac{(-1)^{n-1}2^{2n}(2^{2n}-1)B_n(ax)^{2n+1}}{(2n+1)!}+\cdots\right\} +$$ +<<*>>= +)clear all + +--S 28 14:610 Axiom cannot compute this integral +aa:=integrate(x*tanh(a*x),x) +--R +--R +--R x +--R ++ +--I (1) | %O tanh(%O a)d%O +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.611~~~~~$\displaystyle +\int{x\tanh^2{ax}}~dx$} +$$\int{x\tanh^2{ax}}= +\frac{x^2}{2}-\frac{x\tanh{ax}}{a}+\frac{1}{a^2}\ln\cosh{ax} +$$ +<<*>>= +)clear all + +--S 29 +aa:=integrate(x*tanh(a*x)^2,x) +--R +--R +--R (1) +--R 2 2 +--R (2sinh(a x) + 4cosh(a x)sinh(a x) + 2cosh(a x) + 2) +--R * +--R 2cosh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 2 2 2 2 +--R (a x - 4a x)sinh(a x) + (2a x - 8a x)cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (a x - 4a x)cosh(a x) + a x +--R / +--R 2 2 2 2 2 2 +--R 2a sinh(a x) + 4a cosh(a x)sinh(a x) + 2a cosh(a x) + 2a +--R Type: Union(Expression Integer,...) +--E + +--S 30 +bb:=x^2/2-(x*tanh(a*x))/a+1/a^2*log(cosh(a*x)) +--R +--R 2 2 +--R 2log(cosh(a x)) - 2a x tanh(a x) + a x +--R (2) --------------------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 31 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1)log(cosh(a x)) +--R + +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1) +--R * +--R 2cosh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 2 +--R (a x sinh(a x) + 2a x cosh(a x)sinh(a x) + a x cosh(a x) + a x) +--R * +--R tanh(a x) +--R + +--R 2 2 +--R - 2a x sinh(a x) - 4a x cosh(a x)sinh(a x) - 2a x cosh(a x) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 32 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 33 +dd:=sinhsqrrule cc +--R +--R (5) +--R 2 +--R (- 4cosh(a x)sinh(a x) - cosh(2a x) - 2cosh(a x) - 1)log(cosh(a x)) +--R + +--R 2 +--R (4cosh(a x)sinh(a x) + cosh(2a x) + 2cosh(a x) + 1) +--R * +--R 2cosh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 +--R (4a x cosh(a x)sinh(a x) + a x cosh(2a x) + 2a x cosh(a x) + a x) +--R * +--R tanh(a x) +--R + +--R 2 +--R - 8a x cosh(a x)sinh(a x) - 2a x cosh(2a x) - 4a x cosh(a x) + 2a x +--R / +--R 2 2 2 2 2 +--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 34 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 35 +ee:=coshsqrrule dd +--R +--R (7) +--R (- 2cosh(a x)sinh(a x) - cosh(2a x) - 1)log(cosh(a x)) +--R + +--R 2cosh(a x) +--R (2cosh(a x)sinh(a x) + cosh(2a x) + 1)log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R (2a x cosh(a x)sinh(a x) + a x cosh(2a x) + a x)tanh(a x) +--R + +--R - 4a x cosh(a x)sinh(a x) - 2a x cosh(2a x) +--R / +--R 2 2 2 +--R 2a cosh(a x)sinh(a x) + a cosh(2a x) + a +--R Type: Expression Integer +--E + +--S 36 +ff:=expandLog ee +--R +--R (8) +--R (- 2cosh(a x)sinh(a x) - cosh(2a x) - 1)log(sinh(a x) - cosh(a x)) +--R + +--R (2a x cosh(a x)sinh(a x) + a x cosh(2a x) + a x)tanh(a x) +--R + +--R (2log(- 2) - 4a x)cosh(a x)sinh(a x) + (log(- 2) - 2a x)cosh(2a x) +--R + +--R log(- 2) +--R / +--R 2 2 2 +--R 2a cosh(a x)sinh(a x) + a cosh(2a x) + a +--R Type: Expression Integer +--E + +--S 37 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %N sinh(y + x) - %N sinh(y - x) +--I (9) %N cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 38 +gg:=sinhcoshrule ff +--R +--R (10) +--R (- sinh(2a x) - cosh(2a x) - 1)log(sinh(a x) - cosh(a x)) +--R + +--R (a x sinh(2a x) + a x cosh(2a x) + a x)tanh(a x) +--R + +--R (log(- 2) - 2a x)sinh(2a x) + (log(- 2) - 2a x)cosh(2a x) + log(- 2) +--R / +--R 2 2 2 +--R a sinh(2a x) + a cosh(2a x) + a +--R Type: Expression Integer +--E + +--S 39 14:611 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R - log(- 1) + log(- 2) +--R (11) --------------------- +--R 2 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.612~~~~~$\displaystyle +\int{\frac{\tanh{ax}}{x}}~dx$} +$$\int{\frac{\tanh{ax}}{x}}= +ax-\frac{(ax)^3}{9}+\frac{2(ax)^5}{75}-\cdots +\frac{(-1)^{n-1}2^{2n}(2^{2n}-1)B_n(ax)^{2n-1}}{(2n-1)(2n)!}+\cdots +$$ +<<*>>= +)clear all + +--S 40 14:612 Axiom cannot compute this integral +aa:=integrate(tanh(a*x)/x,x) +--R +--R +--R x +--I ++ tanh(%O a) +--I (1) | ---------- d%O +--I ++ %O +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.613~~~~~$\displaystyle +\int{\frac{dx}{p+q\tanh{ax}}}~dx$} +$$\int{\frac{1}{p+q\tanh{ax}}}= +\frac{px}{p^2-q^2}-\frac{q}{a(p^2-q^2)}\ln(q\sinh{ax}+p\cosh{ax}) +$$ +<<*>>= +)clear all + +--S 41 +aa:=integrate(1/(p+q*tanh(a*x)),x) +--R +--R +--R - 2q sinh(a x) - 2p cosh(a x) +--R q log(-----------------------------) + (- a q - a p)x +--R sinh(a x) - cosh(a x) +--R (1) ----------------------------------------------------- +--R 2 2 +--R a q - a p +--R Type: Union(Expression Integer,...) +--E + +--S 42 +bb:=(p*x)/(p^2-q^2)-q/(a*(p^2-q^2))*log(q*sinh(a*x)+p*cosh(a*x)) +--R +--R q log(q sinh(a x) + p cosh(a x)) - a p x +--R (2) ---------------------------------------- +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 43 +cc:=aa-bb +--R +--R (3) +--R - 2q sinh(a x) - 2p cosh(a x) +--R - q log(q sinh(a x) + p cosh(a x)) + q log(-----------------------------) +--R sinh(a x) - cosh(a x) +--R + +--R - a q x +--R / +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 44 +dd:=expandLog cc +--R +--R (4) +--R - q log(q sinh(a x) + p cosh(a x)) - q log(sinh(a x) - cosh(a x)) +--R + +--R q log(- q sinh(a x) - p cosh(a x)) + q log(2) - a q x +--R / +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 45 14:613 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R q log(2) - 2q log(- 1) +--R (5) ---------------------- +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.614~~~~~$\displaystyle +\int{\tanh^n{ax}}~dx$} +$$\int{\tanh^n{ax}}= +\frac{-\tanh^{n-1}{ax}}{a(n-1)}+\int{\tanh^{n-2}{ax}} +$$ +<<*>>= +)clear all + +--S 46 14:614 Axiom cannot compute this integral +aa:=integrate(tanh(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | tanh(%O a) d%O +--R ++ +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp89-90 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x +--R sinh(a x) - cosh(a x) +--R (1) ---------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=1/a*log(sinh(a*x)) +--R +--R log(sinh(a x)) +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R 2sinh(a x) +--R - log(sinh(a x)) + log(- ---------------------) - a x +--R sinh(a x) - cosh(a x) +--R (3) ----------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 4 +dd:=expandLog cc +--R +--R - log(sinh(a x) - cosh(a x)) + log(- 2) - a x +--R (4) --------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 5 14:615 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R - log(- 1) + log(- 2) +--R (5) --------------------- +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.616~~~~~$\displaystyle +\int{\coth^2{ax}}~dx$} +$$\int{\coth^2{ax}}= +x-\frac{\coth{ax}}{a} +$$ +<<*>>= +)clear all + +--S 6 +aa:=integrate(coth(a*x)^2,x) +--R +--R +--R (a x + 1)sinh(a x) - cosh(a x) +--R (1) ------------------------------ +--R a sinh(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 7 +bb:=x-coth(a*x)/a +--R +--R - coth(a x) + a x +--R (2) ----------------- +--R a +--R Type: Expression Integer +--E + +--S 8 +cc:=aa-bb +--R +--R (coth(a x) + 1)sinh(a x) - cosh(a x) +--R (3) ------------------------------------ +--R a sinh(a x) +--R Type: Expression Integer +--E + +--S 9 14:616 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 1 +--R (4) - +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.617~~~~~$\displaystyle +\int{\coth^3{ax}}~dx$} +$$\int{\coth^3{ax}}= +\frac{1}{a}\ln\sinh{ax}-\frac{\coth^2{ax}}{2a} +$$ +<<*>>= +)clear all + +--S 10 +aa:=integrate(coth(a*x)^3,x) +--R +--R +--R (1) +--R 4 3 2 2 +--R sinh(a x) + 4cosh(a x)sinh(a x) + (6cosh(a x) - 2)sinh(a x) +--R + +--R 3 4 2 +--R (4cosh(a x) - 4cosh(a x))sinh(a x) + cosh(a x) - 2cosh(a x) + 1 +--R * +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 4 3 +--R - a x sinh(a x) - 4a x cosh(a x)sinh(a x) +--R + +--R 2 2 +--R (- 6a x cosh(a x) + 2a x - 2)sinh(a x) +--R + +--R 3 4 +--R (- 4a x cosh(a x) + (4a x - 4)cosh(a x))sinh(a x) - a x cosh(a x) +--R + +--R 2 +--R (2a x - 2)cosh(a x) - a x +--R / +--R 4 3 2 2 +--R a sinh(a x) + 4a cosh(a x)sinh(a x) + (6a cosh(a x) - 2a)sinh(a x) +--R + +--R 3 4 2 +--R (4a cosh(a x) - 4a cosh(a x))sinh(a x) + a cosh(a x) - 2a cosh(a x) + a +--R Type: Union(Expression Integer,...) +--E + +--S 11 +bb:=1/a*log(sinh(a*x)-coth(a*x)^2)/(2*a) +--R +--R 2 +--R log(sinh(a x) - coth(a x) ) +--R (2) --------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 12 14:617 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 4 3 2 2 +--R - sinh(a x) - 4cosh(a x)sinh(a x) + (- 6cosh(a x) + 2)sinh(a x) +--R + +--R 3 4 2 +--R (- 4cosh(a x) + 4cosh(a x))sinh(a x) - cosh(a x) + 2cosh(a x) - 1 +--R * +--R 2 +--R log(sinh(a x) - coth(a x) ) +--R + +--R 4 3 +--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) +--R + +--R 2 2 +--R (12a cosh(a x) - 4a)sinh(a x) +--R + +--R 3 4 +--R (8a cosh(a x) - 8a cosh(a x))sinh(a x) + 2a cosh(a x) +--R + +--R 2 +--R - 4a cosh(a x) + 2a +--R * +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 4 2 3 +--R - 2a x sinh(a x) - 8a x cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 +--R (- 12a x cosh(a x) + 4a x - 4a)sinh(a x) +--R + +--R 2 3 2 2 4 +--R (- 8a x cosh(a x) + (8a x - 8a)cosh(a x))sinh(a x) - 2a x cosh(a x) +--R + +--R 2 2 2 +--R (4a x - 4a)cosh(a x) - 2a x +--R / +--R 2 4 2 3 2 2 2 2 +--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) + (12a cosh(a x) - 4a )sinh(a x) +--R + +--R 2 3 2 2 4 2 2 +--R (8a cosh(a x) - 8a cosh(a x))sinh(a x) + 2a cosh(a x) - 4a cosh(a x) +--R + +--R 2 +--R 2a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.618~~~~~$\displaystyle +\int{\coth^n{ax}{{\rm ~csch}^2{ax}}}~dx$} +$$\int{\coth^n{ax}{{\rm ~csch}^2{ax}}}= +-\frac{\coth^{n+1}{ax}}{(n+1)a} +$$ +<<*>>= +)clear all + +--S 13 +aa:=integrate(coth(a*x)^n*csch(a*x)^2,x) +--R +--R +--R cosh(a x) cosh(a x) +--R - cosh(a x)sinh(n log(---------)) - cosh(a x)cosh(n log(---------)) +--R sinh(a x) sinh(a x) +--R (1) ------------------------------------------------------------------- +--R (a n + a)sinh(a x) +--R Type: Union(Expression Integer,...) +--E + +--S 14 +bb:=-coth(a*x)^(n+1)/((n+1)*a) +--R +--R n + 1 +--R coth(a x) +--R (2) - -------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 15 +cc:=aa-bb +--R +--R (3) +--R cosh(a x) cosh(a x) +--R - cosh(a x)sinh(n log(---------)) - cosh(a x)cosh(n log(---------)) +--R sinh(a x) sinh(a x) +--R + +--R n + 1 +--R sinh(a x)coth(a x) +--R / +--R (a n + a)sinh(a x) +--R Type: Expression Integer +--E + +--S 16 +dd:=expandLog cc +--R +--R (4) +--R cosh(a x)sinh(n log(sinh(a x)) - n log(cosh(a x))) +--R + +--R - cosh(a x)cosh(n log(sinh(a x)) - n log(cosh(a x))) +--R + +--R n + 1 +--R sinh(a x)coth(a x) +--R / +--R (a n + a)sinh(a x) +--R Type: Expression Integer +--E + +--S 17 14:618 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.619~~~~~$\displaystyle +\int{\frac{{\rm csch}^2{ax}}{\coth{ax}}}~dx$} +$$\int{\frac{{\rm csch}^2{ax}}{\coth{ax}}}= +-\frac{1}{a}\ln\coth{ax} +$$ +<<*>>= +)clear all + +--S 18 +aa:=integrate(csch(a*x)^2/coth(a*x),x) +--R +--R +--R 2cosh(a x) 2sinh(a x) +--R - log(- ---------------------) + log(- ---------------------) +--R sinh(a x) - cosh(a x) sinh(a x) - cosh(a x) +--R (1) ------------------------------------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 19 +bb:=-1/a*log(coth(a*x)) +--R +--R log(coth(a x)) +--R (2) - -------------- +--R a +--R Type: Expression Integer +--E + +--S 20 +cc:=aa-bb +--R +--R (3) +--R 2cosh(a x) 2sinh(a x) +--R log(coth(a x)) - log(- ---------------------) + log(- ---------------------) +--R sinh(a x) - cosh(a x) sinh(a x) - cosh(a x) +--R ---------------------------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 21 +dd:=expandLog cc +--R +--R log(sinh(a x)) + log(coth(a x)) - log(cosh(a x)) +--R (4) ------------------------------------------------ +--R a +--R Type: Expression Integer +--E + +--S 22 14:619 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.620~~~~~$\displaystyle +\int{\frac{dx}{\coth{ax}}}~dx$} +$$\int{\frac{1}{\coth{ax}}}= +\frac{1}{a}\ln\cosh{ax} +$$ +<<*>>= +)clear all + +--S 23 +aa:=integrate(1/coth(a*x),x) +--R +--R +--R 2cosh(a x) +--R log(- ---------------------) - a x +--R sinh(a x) - cosh(a x) +--R (1) ---------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 24 +bb:=1/a*log(cosh(a*x)) +--R +--R log(cosh(a x)) +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 25 +cc:=aa-bb +--R +--R 2cosh(a x) +--R - log(cosh(a x)) + log(- ---------------------) - a x +--R sinh(a x) - cosh(a x) +--R (3) ----------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 26 +dd:=expandLog cc +--R +--R - log(sinh(a x) - cosh(a x)) + log(- 2) - a x +--R (4) --------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 27 14:620 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R - log(- 1) + log(- 2) +--R (5) --------------------- +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.621~~~~~$\displaystyle +\int{x\coth{ax}}~dx$} +$$\int{x\coth{ax}}= +\frac{1}{a^2}\left\{ +ax+\frac{(ax)^3}{9}-\frac{(ax)^5}{225}+\cdots +\frac{(-1)^{n-1}2^{2n}B_n(ax)^{2n+1}}{(2n+1)!}+\cdots\right\} +$$ +<<*>>= +)clear all + +--S 28 14:621 Axiom cannot compute this integral +aa:=integrate(x*coth(a*x),x) +--R +--R +--R x +--R ++ +--I (1) | %O coth(%O a)d%O +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.622~~~~~$\displaystyle +\int{x\coth^2{ax}}~dx$} +$$\int{x\coth^2{ax}}= +\frac{x^2}{2}-\frac{x\coth{ax}}{a}+\frac{1}{a^2}\ln\sinh{ax} +$$ +<<*>>= +)clear all + +--S 29 +aa:=integrate(x*coth(a*x)^2,x) +--R +--R +--R (1) +--R 2 2 +--R (2sinh(a x) + 4cosh(a x)sinh(a x) + 2cosh(a x) - 2) +--R * +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 2 2 2 2 +--R (a x - 4a x)sinh(a x) + (2a x - 8a x)cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (a x - 4a x)cosh(a x) - a x +--R / +--R 2 2 2 2 2 2 +--R 2a sinh(a x) + 4a cosh(a x)sinh(a x) + 2a cosh(a x) - 2a +--R Type: Union(Expression Integer,...) +--E + +--S 30 +bb:=x^2/2-(x*coth(a*x)/a)+1/a^2*log(sinh(a*x)) +--R +--R 2 2 +--R 2log(sinh(a x)) - 2a x coth(a x) + a x +--R (2) --------------------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 31 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) + 1)log(sinh(a x)) +--R + +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1) +--R * +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 +--R (a x coth(a x) - 2a x)sinh(a x) +--R + +--R (2a x cosh(a x)coth(a x) - 4a x cosh(a x))sinh(a x) +--R + +--R 2 2 +--R (a x cosh(a x) - a x)coth(a x) - 2a x cosh(a x) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E + +--S 32 +dd:=expandLog cc +--R +--R (4) +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) + 1) +--R * +--R log(sinh(a x) - cosh(a x)) +--R + +--R 2 +--R (a x coth(a x) + log(- 2) - 2a x)sinh(a x) +--R + +--R (2a x cosh(a x)coth(a x) + (2log(- 2) - 4a x)cosh(a x))sinh(a x) +--R + +--R 2 2 +--R (a x cosh(a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(a x) - log(- 2) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E + +--S 33 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (5) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 34 +ee:=sinhsqrrule dd +--R +--R (6) +--R 2 +--R (- 4cosh(a x)sinh(a x) - cosh(2a x) - 2cosh(a x) + 3) +--R * +--R log(sinh(a x) - cosh(a x)) +--R + +--R (4a x cosh(a x)coth(a x) + (4log(- 2) - 8a x)cosh(a x))sinh(a x) +--R + +--R 2 +--R (a x cosh(2a x) + 2a x cosh(a x) - 3a x)coth(a x) +--R + +--R 2 +--R (log(- 2) - 2a x)cosh(2a x) + (2log(- 2) - 4a x)cosh(a x) - 3log(- 2) +--R + +--R 2a x +--R / +--R 2 2 2 2 2 +--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) - 3a +--R Type: Expression Integer +--E + +--S 35 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (7) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 36 +ff:=coshsqrrule ee +--R +--R (8) +--R (- 2cosh(a x)sinh(a x) - cosh(2a x) + 1)log(sinh(a x) - cosh(a x)) +--R + +--R (2a x cosh(a x)coth(a x) + (2log(- 2) - 4a x)cosh(a x))sinh(a x) +--R + +--R (a x cosh(2a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(2a x) - log(- 2) +--R / +--R 2 2 2 +--R 2a cosh(a x)sinh(a x) + a cosh(2a x) - a +--R Type: Expression Integer +--E + +--S 37 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %L sinh(y + x) - %L sinh(y - x) +--I (9) %L cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 38 +gg:=sinhcoshrule ff +--R +--R (10) +--R (- sinh(2a x) - cosh(2a x) + 1)log(sinh(a x) - cosh(a x)) +--R + +--R (a x coth(a x) + log(- 2) - 2a x)sinh(2a x) +--R + +--R (a x cosh(2a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(2a x) - log(- 2) +--R / +--R 2 2 2 +--R a sinh(2a x) + a cosh(2a x) - a +--R Type: Expression Integer +--E + +--S 39 14:622 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R - log(- 1) + log(- 2) +--R (11) --------------------- +--R 2 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.623~~~~~$\displaystyle +\int{\frac{\coth{ax}}{x}}~dx$} +$$\int{\frac{\coth{ax}}{x}}= +-\frac{1}{ax}+\frac{(ax)}{3}-\frac{(ax)^3}{135}+\cdots +\frac{(-1)^{n}2^{2n}B_n(ax)^{2n-1}}{(2n-1)(2n)!}+\cdots +$$ +<<*>>= +)clear all + +--S 40 14:623 Axiom cannot compute this integral +aa:=integrate(coth(a*x)/x,x) +--R +--R +--R x +--I ++ coth(%O a) +--I (1) | ---------- d%O +--I ++ %O +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.624~~~~~$\displaystyle +\int{\frac{dx}{p+q\coth{ax}}}~dx$} +$$\int{\frac{1}{p+q\coth{ax}}}= +\frac{px}{p^2-q^2}-\frac{q}{a(p^2-q^2)}\ln(p\sinh{ax}+q\cosh{ax}) +$$ +<<*>>= +)clear all + +--S 41 +aa:=integrate(1/(p+q*coth(a*x)),x) +--R +--R +--R - 2p sinh(a x) - 2q cosh(a x) +--R q log(-----------------------------) + (- a q - a p)x +--R sinh(a x) - cosh(a x) +--R (1) ----------------------------------------------------- +--R 2 2 +--R a q - a p +--R Type: Union(Expression Integer,...) +--E + +--S 42 +bb:=(p*x)/(p^2-q^2)-q/(a*(p^2-q^2))*log(p*sinh(a*x)+q*cosh(a*x)) +--R +--R q log(p sinh(a x) + q cosh(a x)) - a p x +--R (2) ---------------------------------------- +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 43 +cc:=aa-bb +--R +--R (3) +--R - 2p sinh(a x) - 2q cosh(a x) +--R - q log(p sinh(a x) + q cosh(a x)) + q log(-----------------------------) +--R sinh(a x) - cosh(a x) +--R + +--R - a q x +--R / +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 44 +dd:=expandLog cc +--R +--R (4) +--R - q log(p sinh(a x) + q cosh(a x)) - q log(sinh(a x) - cosh(a x)) +--R + +--R q log(- p sinh(a x) - q cosh(a x)) + q log(2) - a q x +--R / +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 45 14:624 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R q log(2) - 2q log(- 1) +--R (5) ---------------------- +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.625~~~~~$\displaystyle +\int{\coth^n{ax}}~dx$} +$$\int{\coth^n{ax}}= +-\frac{\coth^{n-1}{ax}}{a(n-1)}+\int{\coth^{n-1}{ax}} +$$ +<<*>>= +)clear all + +--S 46 14:625 Axiom cannot compute this integral +aa:=integrate(coth(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | coth(%O a) d%O +--R ++ +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp90-91 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum31.input.pdf b/src/axiom-website/CATS/schaum31.input.pdf new file mode 100644 index 0000000..7778d6b --- /dev/null +++ b/src/axiom-website/CATS/schaum31.input.pdf @@ -0,0 +1,2067 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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00000 n +0000107430 00000 n +0000107826 00000 n +0000108621 00000 n +0000108538 00000 n +0000109530 00000 n +0000109040 00000 n +0000109447 00000 n +0000109914 00000 n +0000109937 00000 n +0000109959 00000 n +trailer +<< +/Size 158 +/Root 2 0 R +/Info 1 0 R +>> +startxref +110080 +%%EOF diff --git a/src/axiom-website/CATS/schaum32.input.pamphlet b/src/axiom-website/CATS/schaum32.input.pamphlet new file mode 100644 index 0000000..e5b2409 --- /dev/null +++ b/src/axiom-website/CATS/schaum32.input.pamphlet @@ -0,0 +1,999 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum32.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.626~~~~~$\displaystyle +\int{{\rm sech~}{ax}}~dx$} +$$\int{{\rm sech~}{ax}}= +\frac{2}{a}\tanh^{-1}{e^{ax}} +$$ +<<*>>= +)spool schaum32.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(sech(a*x),x) +--R +--R +--R 2atan(sinh(a x) + cosh(a x)) +--R (1) ---------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=2/a*atan(%e^(a*x)) +--R +--R a x +--R 2atan(%e ) +--R (2) ------------ +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R a x +--R 2atan(sinh(a x) + cosh(a x)) - 2atan(%e ) +--R (3) ------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 4 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 5 +dd:=atanrule cc +--R +--R a x +--R - %e + %i - sinh(a x) - cosh(a x) + %i +--R %i log(------------) - %i log(----------------------------) +--R a x sinh(a x) + cosh(a x) + %i +--R %e + %i +--R (5) ----------------------------------------------------------- +--R a +--R Type: Expression Complex Integer +--E + +--S 6 +ee:=expandLog dd +--R +--R (6) +--R %i log(sinh(a x) + cosh(a x) + %i) - %i log(sinh(a x) + cosh(a x) - %i) +--R + +--R a x a x +--R - %i log(%e + %i) + %i log(%e - %i) +--R / +--R a +--R Type: Expression Complex Integer +--E + +--S 7 14:626 Schaums and Axiom agree +ff:=complexNormalize ee +--R +--R (7) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.627~~~~~$\displaystyle +\int{{\rm sech}^2~{ax}}~dx$} +$$\int{{\rm sech}^2~{ax}}= +\frac{\tanh{ax}}{a} +$$ +<<*>>= +)clear all + +--S 8 +aa:=integrate(sech(a*x)^2,x) +--R +--R +--R 2 +--R (1) - ------------------------------------------------------- +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Union(Expression Integer,...) +--E + +--S 9 +bb:=tanh(a*x)/a +--R +--R tanh(a x) +--R (2) --------- +--R a +--R Type: Expression Integer +--E + +--S 10 +cc:=aa-bb +--R +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1)tanh(a x) - 2 +--R (3) ------------------------------------------------------------------ +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 11 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 12 +dd:=sinhsqrrule cc +--R +--R 2 +--R (- 4cosh(a x)sinh(a x) - cosh(2a x) - 2cosh(a x) - 1)tanh(a x) - 4 +--R (5) ------------------------------------------------------------------- +--R 2 +--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 13 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 14 +ee:=coshsqrrule dd +--R +--R (- 2cosh(a x)sinh(a x) - cosh(2a x) - 1)tanh(a x) - 2 +--R (7) ----------------------------------------------------- +--R 2a cosh(a x)sinh(a x) + a cosh(2a x) + a +--R Type: Expression Integer +--E + +--S 15 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %L sinh(y + x) - %L sinh(y - x) +--I (8) %L cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 16 +ff:=sinhcoshrule ee +--R +--R (- sinh(2a x) - cosh(2a x) - 1)tanh(a x) - 2 +--R (9) -------------------------------------------- +--R a sinh(2a x) + a cosh(2a x) + a +--R Type: Expression Integer +--E + +--S 17 14:627 Schaums and Axiom differ by a constant +gg:=complexNormalize ff +--R +--R 1 +--R (10) - - +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.628~~~~~$\displaystyle +\int{{\rm sech}^3~{ax}}~dx$} +$$\int{{\rm sech}^3~{ax}}= +\frac{{\rm sech}~{ax}~\tanh{ax}}{2a}+\frac{1}{2a}\tan^{-1}{\rm ~sech~}{ax} +$$ +<<*>>= +)clear all + +--S 18 +aa:=integrate(sech(a*x)^3,x) +--R +--R +--R (1) +--R 4 3 2 2 +--R sinh(a x) + 4cosh(a x)sinh(a x) + (6cosh(a x) + 2)sinh(a x) +--R + +--R 3 4 2 +--R (4cosh(a x) + 4cosh(a x))sinh(a x) + cosh(a x) + 2cosh(a x) + 1 +--R * +--R atan(sinh(a x) + cosh(a x)) +--R + +--R 3 2 2 +--R sinh(a x) + 3cosh(a x)sinh(a x) + (3cosh(a x) - 1)sinh(a x) +--R + +--R 3 +--R cosh(a x) - cosh(a x) +--R / +--R 4 3 2 2 +--R a sinh(a x) + 4a cosh(a x)sinh(a x) + (6a cosh(a x) + 2a)sinh(a x) +--R + +--R 3 4 2 +--R (4a cosh(a x) + 4a cosh(a x))sinh(a x) + a cosh(a x) + 2a cosh(a x) + a +--R Type: Union(Expression Integer,...) +--E + +--S 19 +bb:=(sech(a*x)*tanh(a*x))/(2*a)+1/(2*a)*atan(sinh(a*x)) +--R +--R atan(sinh(a x)) + sech(a x)tanh(a x) +--R (2) ------------------------------------ +--R 2a +--R Type: Expression Integer +--E + +--S 20 14:628 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 4 3 2 2 +--R 2sinh(a x) + 8cosh(a x)sinh(a x) + (12cosh(a x) + 4)sinh(a x) +--R + +--R 3 4 2 +--R (8cosh(a x) + 8cosh(a x))sinh(a x) + 2cosh(a x) + 4cosh(a x) + 2 +--R * +--R atan(sinh(a x) + cosh(a x)) +--R + +--R 4 3 2 2 +--R - sinh(a x) - 4cosh(a x)sinh(a x) + (- 6cosh(a x) - 2)sinh(a x) +--R + +--R 3 4 2 +--R (- 4cosh(a x) - 4cosh(a x))sinh(a x) - cosh(a x) - 2cosh(a x) - 1 +--R * +--R atan(sinh(a x)) +--R + +--R 4 3 +--R - sech(a x)sinh(a x) - 4cosh(a x)sech(a x)sinh(a x) +--R + +--R 2 2 +--R (- 6cosh(a x) - 2)sech(a x)sinh(a x) +--R + +--R 3 +--R (- 4cosh(a x) - 4cosh(a x))sech(a x)sinh(a x) +--R + +--R 4 2 +--R (- cosh(a x) - 2cosh(a x) - 1)sech(a x) +--R * +--R tanh(a x) +--R + +--R 3 2 2 +--R 2sinh(a x) + 6cosh(a x)sinh(a x) + (6cosh(a x) - 2)sinh(a x) +--R + +--R 3 +--R 2cosh(a x) - 2cosh(a x) +--R / +--R 4 3 2 2 +--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) + (12a cosh(a x) + 4a)sinh(a x) +--R + +--R 3 4 2 +--R (8a cosh(a x) + 8a cosh(a x))sinh(a x) + 2a cosh(a x) + 4a cosh(a x) +--R + +--R 2a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.629~~~~~$\displaystyle +\int{{\rm sech}^n~{ax}~{\tanh{ax}}}~dx$} +$$\int{{\rm sech~}^n{ax}~{\tanh{ax}}}= +-\frac{{\rm sech~}^{n}{ax}}{na} +$$ +<<*>>= +)clear all + +--S 21 +aa:=integrate(sech(a*x)^n*tanh(a*x),x) +--R +--R +--R (1) +--R 2sinh(a x) + 2cosh(a x) +--R - sinh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1 +--R + +--R 2sinh(a x) + 2cosh(a x) +--R - cosh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1 +--R / +--R a n +--R Type: Union(Expression Integer,...) +--E + +--S 22 +bb:=-sech(a*x)^n/(n*a) +--R +--R n +--R sech(a x) +--R (2) - ---------- +--R a n +--R Type: Expression Integer +--E + +--S 23 +cc:=aa-bb +--R +--R (3) +--R 2sinh(a x) + 2cosh(a x) +--R - sinh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1 +--R + +--R 2sinh(a x) + 2cosh(a x) +--R - cosh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1 +--R + +--R n +--R sech(a x) +--R / +--R a n +--R Type: Expression Integer +--E + +--S 24 +sechrule:=rule(sech(x) == 1/cosh(x)) +--R +--R 1 +--R (4) sech(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 25 +dd:=sechrule cc +--R +--R (5) +--R 2sinh(a x) + 2cosh(a x) +--R - sinh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1 +--R + +--R 2sinh(a x) + 2cosh(a x) +--R - cosh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1 +--R + +--R 1 n +--R (---------) +--R cosh(a x) +--R / +--R a n +--R Type: Expression Integer +--E + +--S 26 +ee:=expandLog dd +--R +--R (6) +--R sinh +--R 2 2 +--R n log(sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1) +--R + +--R - n log(sinh(a x) + cosh(a x)) - n log(2) +--R + +--R - +--R cosh +--R 2 2 +--R n log(sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1) +--R + +--R - n log(sinh(a x) + cosh(a x)) - n log(2) +--R + +--R 1 n +--R (---------) +--R cosh(a x) +--R / +--R a n +--R Type: Expression Integer +--E + +--S 27 14:629 Schaums and Axiom agree +ff:=complexNormalize ee +--R +--R (7) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.630~~~~~$\displaystyle +\int{\frac{dx}{{\rm sech~}{ax}}}~dx$} +$$\int{\frac{1}{{\rm sech~}{ax}}}= +\frac{{\rm sech}~{ax}}{a} +$$ +<<*>>= +)clear all + +--S 28 +aa:=integrate(1/sech(a*x),x) +--R +--R +--R sinh(a x) +--R (1) --------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 29 +bb:=sinh(a*x)/a +--R +--R sinh(a x) +--R (2) --------- +--R a +--R Type: Expression Integer +--E + +--S 30 14:630 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.631~~~~~$\displaystyle +\int{x{\rm ~sech~}{ax}}~dx$} +$$\int{x{\rm ~sech~}{ax}}= +\frac{1}{a^2}\left\{ +\frac{(ax)^2}{2}-\frac{(ax)^4}{8}+\frac{5(ax)^6}{144}+\cdots +\frac{(-1)^{n}E_n(ax)^{2n+2}}{(2n+2)(2n)!}+\cdots\right\} +$$ +<<*>>= +)clear all + +--S 31 14:631 Axiom cannot compute this integral +aa:=integrate(x*sech(a*x),x) +--R +--R +--R x +--R ++ +--I (1) | %O sech(%O a)d%O +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.632~~~~~$\displaystyle +\int{x~{\rm sech}^2~{ax}}~dx$} +$$\int{x~{\rm sech}^2~{ax}}= +\frac{x\tanh{ax}}{a}-\frac{1}{a^2}\ln\cosh{ax} +$$ +<<*>>= +)clear all + +--S 32 +aa:=integrate(x*sech(a*x)^2,x) +--R +--R +--R (1) +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1) +--R * +--R 2cosh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 2 +--R 2a x sinh(a x) + 4a x cosh(a x)sinh(a x) + 2a x cosh(a x) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Union(Expression Integer,...) +--E + +--S 33 +bb:=(x*tanh(a*x))/a-1/a^2*log(cosh(a*x)) +--R +--R - log(cosh(a x)) + a x tanh(a x) +--R (2) -------------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 34 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1)log(cosh(a x)) +--R + +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1) +--R * +--R 2cosh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 2 +--R (- a x sinh(a x) - 2a x cosh(a x)sinh(a x) - a x cosh(a x) - a x) +--R * +--R tanh(a x) +--R + +--R 2 2 +--R 2a x sinh(a x) + 4a x cosh(a x)sinh(a x) + 2a x cosh(a x) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 35 +dd:=expandLog cc +--R +--R (4) +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1) +--R * +--R log(sinh(a x) - cosh(a x)) +--R + +--R 2 2 +--R (- a x sinh(a x) - 2a x cosh(a x)sinh(a x) - a x cosh(a x) - a x) +--R * +--R tanh(a x) +--R + +--R 2 +--R (- log(- 2) + 2a x)sinh(a x) + (- 2log(- 2) + 4a x)cosh(a x)sinh(a x) +--R + +--R 2 +--R (- log(- 2) + 2a x)cosh(a x) - log(- 2) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 36 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (5) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 37 +ee:=sinhsqrrule dd +--R +--R (6) +--R 2 +--R (4cosh(a x)sinh(a x) + cosh(2a x) + 2cosh(a x) + 1) +--R * +--R log(sinh(a x) - cosh(a x)) +--R + +--R 2 +--R (- 4a x cosh(a x)sinh(a x) - a x cosh(2a x) - 2a x cosh(a x) - a x) +--R * +--R tanh(a x) +--R + +--R (- 4log(- 2) + 8a x)cosh(a x)sinh(a x) + (- log(- 2) + 2a x)cosh(2a x) +--R + +--R 2 +--R (- 2log(- 2) + 4a x)cosh(a x) - log(- 2) - 2a x +--R / +--R 2 2 2 2 2 +--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 38 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (7) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 39 +ff:=coshsqrrule ee +--R +--R (8) +--R (2cosh(a x)sinh(a x) + cosh(2a x) + 1)log(sinh(a x) - cosh(a x)) +--R + +--R (- 2a x cosh(a x)sinh(a x) - a x cosh(2a x) - a x)tanh(a x) +--R + +--R (- 2log(- 2) + 4a x)cosh(a x)sinh(a x) + (- log(- 2) + 2a x)cosh(2a x) +--R + +--R - log(- 2) +--R / +--R 2 2 2 +--R 2a cosh(a x)sinh(a x) + a cosh(2a x) + a +--R Type: Expression Integer +--E + +--S 40 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %P sinh(y + x) - %P sinh(y - x) +--I (9) %P cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 41 +gg:=sinhcoshrule ff +--R +--R (10) +--R (sinh(2a x) + cosh(2a x) + 1)log(sinh(a x) - cosh(a x)) +--R + +--R (- a x sinh(2a x) - a x cosh(2a x) - a x)tanh(a x) +--R + +--R (- log(- 2) + 2a x)sinh(2a x) + (- log(- 2) + 2a x)cosh(2a x) - log(- 2) +--R / +--R 2 2 2 +--R a sinh(2a x) + a cosh(2a x) + a +--R Type: Expression Integer +--E + +--S 42 14:632 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R log(- 1) - log(- 2) +--R (11) ------------------- +--R 2 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.633~~~~~$\displaystyle +\int{\frac{{\rm sech~}{ax}}{x}}~dx$} +$$\int{\frac{{\rm sech~}{ax}}{x}}= +\ln{x}-\frac{(ax)^2}{4}+\frac{5(ax)^4}{96}-\frac{61(ax)^6}{4320}+\cdots +\frac{(-1)^{n}E_n(ax)^{2n}}{2n(2n)!}+\cdots +$$ +<<*>>= +)clear all + +--S 43 14:633 Axiom cannot compute this integral +aa:=integrate(sech(a*x)/x,x) +--R +--R +--R x +--I ++ sech(%O a) +--I (1) | ---------- d%O +--I ++ %O +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.634~~~~~$\displaystyle +\int{\frac{dx}{q+p{\rm ~sech~}{ax}}}~dx$} +$$\int{\frac{1}{q+p{\rm ~sech~}{ax}}}= +\frac{x}{q}-\frac{p}{q}\int{\frac{dx}{p+q\cosh{ax}}} +$$ +<<*>>= +)clear all + +--S 44 +aa:=integrate(1/(q+p*sech(a*x)),x) +--R +--R +--R (1) +--R [ +--R p +--R * +--R log +--R 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) +--R + +--R 2 2 2 2 +--R q cosh(a x) + 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q + 2p q)sinh(a x) + (- 2q + 2p q)cosh(a x) - 2p q + 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R +---------+ +--R | 2 2 +--R a x\|- q + p +--R / +--R +---------+ +--R | 2 2 +--R a q\|- q + p +--R , +--R +-------+ +--R | 2 2 +-------+ +--R (q sinh(a x) + q cosh(a x) + p)\|q - p | 2 2 +--R - 2p atan(-----------------------------------------) + a x\|q - p +--R 2 2 +--R q - p +--R --------------------------------------------------------------------] +--R +-------+ +--R | 2 2 +--R a q\|q - p +--R Type: Union(List Expression Integer,...) +--E + +--S 45 +t1:=integrate(1/(p+q*cosh(a*x)),x) +--R +--R (2) +--R [ +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R / +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R , +--R +-------+ +--R | 2 2 +--R (q sinh(a x) + q cosh(a x) + p)\|q - p +--R 2atan(-----------------------------------------) +--R 2 2 +--R q - p +--R ------------------------------------------------] +--R +-------+ +--R | 2 2 +--R a\|q - p +--R Type: Union(List Expression Integer,...) +--E + +--S 46 +bb1:=x/q-p/q*t1.1 +--R +--R (3) +--R - +--R p +--R * +--R log +--R 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) +--R + +--R 2 2 2 2 +--R q cosh(a x) + 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R +---------+ +--R | 2 2 +--R a x\|- q + p +--R / +--R +---------+ +--R | 2 2 +--R a q\|- q + p +--R Type: Expression Integer +--E + +--S 47 +bb2:=x/q-p/q*t1.2 +--R +--R +-------+ +--R | 2 2 +-------+ +--R (q sinh(a x) + q cosh(a x) + p)\|q - p | 2 2 +--R - 2p atan(-----------------------------------------) + a x\|q - p +--R 2 2 +--R q - p +--R (4) -------------------------------------------------------------------- +--R +-------+ +--R | 2 2 +--R a q\|q - p +--R Type: Expression Integer +--E + +--S 48 +cc1:=aa.1-bb1 +--R +--R (5) +--R p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q + 2p q)sinh(a x) + (- 2q + 2p q)cosh(a x) - 2p q + 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R / +--R +---------+ +--R | 2 2 +--R a q\|- q + p +--R Type: Expression Integer +--E + +--S 49 +cc2:=aa.2-bb1 +--R +--R (6) +--R +-------+ +--R | 2 2 +--R p\|q - p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 (q sinh(a x) + q cosh(a x) + p)\|q - p +--R - 2p\|- q + p atan(-----------------------------------------) +--R 2 2 +--R q - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a q\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 50 +cc3:=aa.1-bb2 +--R +--R (7) +--R +-------+ +--R | 2 2 +--R p\|q - p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q + 2p q)sinh(a x) + (- 2q + 2p q)cosh(a x) - 2p q + 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 (q sinh(a x) + q cosh(a x) + p)\|q - p +--R 2p\|- q + p atan(-----------------------------------------) +--R 2 2 +--R q - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a q\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 51 14:634 Schaums and Axiom agree +cc4:=aa.2-bb2 +--R +--R (8) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.635~~~~~$\displaystyle +\int{{\rm sech}^n~{ax}}~dx$} +$$\int{{\rm sech}^n~{ax}}= +\frac{{\rm sech}^{n-2}~{ax}~\tanh{ax}}{a(n-1)} ++\frac{n-2}{n-1}\int{{\rm sech}^{n-2}~{ax}} +$$ +<<*>>= +)clear all + +--S 52 14:635 Axiom cannot compute this integral +aa:=integrate(sech(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | sech(%O a) d%O +--R ++ +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p91 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum32.input.pdf b/src/axiom-website/CATS/schaum32.input.pdf new file mode 100644 index 0000000..3a29251 --- /dev/null +++ b/src/axiom-website/CATS/schaum32.input.pdf @@ -0,0 +1,2318 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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+0000116022 00000 n +0000116045 00000 n +0000116067 00000 n +trailer +<< +/Size 188 +/Root 2 0 R +/Info 1 0 R +>> +startxref +116188 +%%EOF diff --git a/src/axiom-website/CATS/schaum33.input.pamphlet b/src/axiom-website/CATS/schaum33.input.pamphlet new file mode 100644 index 0000000..61506d7 --- /dev/null +++ b/src/axiom-website/CATS/schaum33.input.pamphlet @@ -0,0 +1,1021 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum33.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.636~~~~~$\displaystyle +\int{{\rm csch~}{ax}}~dx$} +$$\int{{\rm csch~}{ax}}= +\frac{1}{a}\ln\tanh{\frac{ax}{2}} +$$ +<<*>>= +)spool schaum33.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(csch(a*x),x) +--R +--R +--R - log(sinh(a x) + cosh(a x) + 1) + log(sinh(a x) + cosh(a x) - 1) +--R (1) ----------------------------------------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=1/a*log(tanh((a*x)/2)) +--R +--R a x +--R log(tanh(---)) +--R 2 +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R (3) +--R a x +--R - log(tanh(---)) - log(sinh(a x) + cosh(a x) + 1) +--R 2 +--R + +--R log(sinh(a x) + cosh(a x) - 1) +--R / +--R a +--R Type: Expression Integer +--E + +--S 4 14:636 Schaums and Axiom agree +dd:=complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.637~~~~~$\displaystyle +\int{{\rm csch}^2~{ax}}~dx$} +$$\int{{\rm csch}^2~{ax}}= +-\frac{\coth{ax}}{a} +$$ +<<*>>= +)clear all + +--S 5 +aa:=integrate(csch(a*x)^2,x) +--R +--R +--R 2 +--R (1) - ------------------------------------------------------- +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Union(Expression Integer,...) +--E + +--S 6 +bb:=-coth(a*x)/a +--R +--R coth(a x) +--R (2) - --------- +--R a +--R Type: Expression Integer +--E + +--S 7 14:637 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 2 +--R coth(a x)sinh(a x) + 2cosh(a x)coth(a x)sinh(a x) +--R + +--R 2 +--R (cosh(a x) - 1)coth(a x) - 2 +--R / +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.638~~~~~$\displaystyle +\int{{\rm csch}^3~{ax}}~dx$} +$$\int{{\rm csch}^3~{ax}}= +-\frac{{\rm csch~}{ax}\coth{ax}}{2a}-\frac{1}{2a}\ln\tanh\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 8 +aa:=integrate(csch(a*x)^3,x) +--R +--R +--R (1) +--R 4 3 2 2 +--R sinh(a x) + 4cosh(a x)sinh(a x) + (6cosh(a x) - 2)sinh(a x) +--R + +--R 3 4 2 +--R (4cosh(a x) - 4cosh(a x))sinh(a x) + cosh(a x) - 2cosh(a x) + 1 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 4 3 2 2 +--R - sinh(a x) - 4cosh(a x)sinh(a x) + (- 6cosh(a x) + 2)sinh(a x) +--R + +--R 3 4 2 +--R (- 4cosh(a x) + 4cosh(a x))sinh(a x) - cosh(a x) + 2cosh(a x) - 1 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 3 2 2 +--R - 2sinh(a x) - 6cosh(a x)sinh(a x) + (- 6cosh(a x) - 2)sinh(a x) +--R + +--R 3 +--R - 2cosh(a x) - 2cosh(a x) +--R / +--R 4 3 2 2 +--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) + (12a cosh(a x) - 4a)sinh(a x) +--R + +--R 3 4 2 +--R (8a cosh(a x) - 8a cosh(a x))sinh(a x) + 2a cosh(a x) - 4a cosh(a x) +--R + +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 9 +bb:=-(csch(a*x)*coth(a*x))/(2*a)-1/(2*a)*log(tanh((a*x)/2)) +--R +--R a x +--R - log(tanh(---)) - coth(a x)csch(a x) +--R 2 +--R (2) ------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 10 14:638 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 4 3 2 2 +--R sinh(a x) + 4cosh(a x)sinh(a x) + (6cosh(a x) - 2)sinh(a x) +--R + +--R 3 4 2 +--R (4cosh(a x) - 4cosh(a x))sinh(a x) + cosh(a x) - 2cosh(a x) + 1 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 4 3 2 2 +--R sinh(a x) + 4cosh(a x)sinh(a x) + (6cosh(a x) - 2)sinh(a x) +--R + +--R 3 4 2 +--R (4cosh(a x) - 4cosh(a x))sinh(a x) + cosh(a x) - 2cosh(a x) + 1 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 4 3 2 2 +--R - sinh(a x) - 4cosh(a x)sinh(a x) + (- 6cosh(a x) + 2)sinh(a x) +--R + +--R 3 4 2 +--R (- 4cosh(a x) + 4cosh(a x))sinh(a x) - cosh(a x) + 2cosh(a x) - 1 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 4 +--R coth(a x)csch(a x)sinh(a x) +--R + +--R 3 +--R (4cosh(a x)coth(a x)csch(a x) - 2)sinh(a x) +--R + +--R 2 2 +--R ((6cosh(a x) - 2)coth(a x)csch(a x) - 6cosh(a x))sinh(a x) +--R + +--R 3 2 +--R ((4cosh(a x) - 4cosh(a x))coth(a x)csch(a x) - 6cosh(a x) - 2)sinh(a x) +--R + +--R 4 2 3 +--R (cosh(a x) - 2cosh(a x) + 1)coth(a x)csch(a x) - 2cosh(a x) - 2cosh(a x) +--R / +--R 4 3 2 2 +--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) + (12a cosh(a x) - 4a)sinh(a x) +--R + +--R 3 4 2 +--R (8a cosh(a x) - 8a cosh(a x))sinh(a x) + 2a cosh(a x) - 4a cosh(a x) +--R + +--R 2a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.639~~~~~$\displaystyle +\int{{\rm csch}^n~{ax}~{\coth{ax}}}~dx$} +$$\int{{\rm csch~}^n{ax}~{\coth{ax}}}= +-\frac{{\rm csch~}^{n}{ax}}{na} +$$ +<<*>>= +)clear all + +--S 11 +aa:=integrate(csch(a*x)^n*coth(a*x),x) +--R +--R +--R (1) +--R 2sinh(a x) + 2cosh(a x) +--R - sinh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1 +--R + +--R 2sinh(a x) + 2cosh(a x) +--R - cosh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1 +--R / +--R a n +--R Type: Union(Expression Integer,...) +--E + +--S 12 +bb:=-csch(a*x)^n/(n*a) +--R +--R n +--R csch(a x) +--R (2) - ---------- +--R a n +--R Type: Expression Integer +--E + +--S 13 +cc:=aa-bb +--R +--R (3) +--R 2sinh(a x) + 2cosh(a x) +--R - sinh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1 +--R + +--R 2sinh(a x) + 2cosh(a x) +--R - cosh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1 +--R + +--R n +--R csch(a x) +--R / +--R a n +--R Type: Expression Integer +--E + +--S 14 +cschrule:=rule(csch(x) == 1/sinh(x)) +--R +--R 1 +--R (4) csch(x) == ------- +--R sinh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 15 +dd:=cschrule cc +--R +--R (5) +--R 2sinh(a x) + 2cosh(a x) +--R - sinh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1 +--R + +--R 2sinh(a x) + 2cosh(a x) +--R - cosh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1 +--R + +--R 1 n +--R (---------) +--R sinh(a x) +--R / +--R a n +--R Type: Expression Integer +--E + +--S 16 +ee:=expandLog dd +--R +--R (6) +--R sinh +--R 2 2 +--R n log(sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1) +--R + +--R - n log(sinh(a x) + cosh(a x)) - n log(2) +--R + +--R - +--R cosh +--R 2 2 +--R n log(sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1) +--R + +--R - n log(sinh(a x) + cosh(a x)) - n log(2) +--R + +--R 1 n +--R (---------) +--R sinh(a x) +--R / +--R a n +--R Type: Expression Integer +--E + +--S 17 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (7) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 18 +ff:=sinhsqrrule ee +--R +--R (8) +--R sinh +--R 2 +--R 4cosh(a x)sinh(a x) + cosh(2a x) + 2cosh(a x) - 3 +--R n log(--------------------------------------------------) +--R 2 +--R + +--R - n log(sinh(a x) + cosh(a x)) - n log(2) +--R + +--R - +--R cosh +--R 2 +--R 4cosh(a x)sinh(a x) + cosh(2a x) + 2cosh(a x) - 3 +--R n log(--------------------------------------------------) +--R 2 +--R + +--R - n log(sinh(a x) + cosh(a x)) - n log(2) +--R + +--R 1 n +--R (---------) +--R sinh(a x) +--R / +--R a n +--R Type: Expression Integer +--E + +--S 19 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (9) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 20 +gg:=coshsqrrule ff +--R +--R (10) +--R sinh +--R n log(2cosh(a x)sinh(a x) + cosh(2a x) - 1) +--R + +--R - n log(sinh(a x) + cosh(a x)) - n log(2) +--R + +--R - +--R cosh +--R n log(2cosh(a x)sinh(a x) + cosh(2a x) - 1) +--R + +--R - n log(sinh(a x) + cosh(a x)) - n log(2) +--R + +--R 1 n +--R (---------) +--R sinh(a x) +--R / +--R a n +--R Type: Expression Integer +--E + +--S 21 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %O sinh(y + x) - %O sinh(y - x) +--I (11) %O cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 22 +hh:=sinhcoshrule gg +--R +--R (12) +--R sinh +--R n log(sinh(2a x) + cosh(2a x) - 1) - n log(sinh(a x) + cosh(a x)) +--R + +--R - n log(2) +--R + +--R - +--R cosh +--R n log(sinh(2a x) + cosh(2a x) - 1) - n log(sinh(a x) + cosh(a x)) +--R + +--R - n log(2) +--R + +--R 1 n +--R (---------) +--R sinh(a x) +--R / +--R a n +--R Type: Expression Integer +--E + +--S 23 14:639 Schaums and Axiom agree +ii:=complexNormalize hh +--R +--R (13) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.640~~~~~$\displaystyle +\int{\frac{dx}{{\rm csch~}{ax}}}~dx$} +$$\int{\frac{1}{{\rm csch~}{ax}}}= +\frac{1}{a}{\rm cosh}~{ax} +$$ +<<*>>= +)clear all + +--S 24 +aa:=integrate(1/csch(a*x),x) +--R +--R +--R cosh(a x) +--R (1) --------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 25 +bb:=1/a*cosh(a*x) +--R +--R cosh(a x) +--R (2) --------- +--R a +--R Type: Expression Integer +--E + +--S 26 14:640 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.641~~~~~$\displaystyle +\int{x{\rm ~csch~}{ax}}~dx$} +$$\int{x{\rm ~csch~}{ax}}= +\frac{1}{a^2}\left\{ +ax-\frac{(ax)^3}{18}+\frac{7(ax)^5}{1800}+\cdots+ +\frac{2(-1)^n(2^{2n-1}-1)B_n(ax)^{2n+1}}{(2n+1)!}+\cdots\right\} +$$ +<<*>>= +)clear all + +--S 27 14:641 Axiom cannot compute this integral +aa:=integrate(x*csch(a*x),x) +--R +--R +--R x +--R ++ +--I (1) | %O csch(%O a)d%O +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.642~~~~~$\displaystyle +\int{x~{\rm csch}^2~{ax}}~dx$} +$$\int{x~{\rm csch}^2~{ax}}= +-\frac{x\coth{ax}}{a}+\frac{1}{a^2}\ln\sinh{ax} +$$ +<<*>>= +)clear all + +--S 28 +aa:=integrate(x*csch(a*x)^2,x) +--R +--R +--R (1) +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1) +--R * +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 2 +--R - 2a x sinh(a x) - 4a x cosh(a x)sinh(a x) - 2a x cosh(a x) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Union(Expression Integer,...) +--E + +--S 29 +bb:=-(x*coth(a*x))/a+1/a^2*log(sinh(a*x)) +--R +--R log(sinh(a x)) - a x coth(a x) +--R (2) ------------------------------ +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 30 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) + 1)log(sinh(a x)) +--R + +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1) +--R * +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 +--R (a x coth(a x) - 2a x)sinh(a x) +--R + +--R (2a x cosh(a x)coth(a x) - 4a x cosh(a x))sinh(a x) +--R + +--R 2 2 +--R (a x cosh(a x) - a x)coth(a x) - 2a x cosh(a x) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E + +--S 31 +dd:=expandLog cc +--R +--R (4) +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) + 1) +--R * +--R log(sinh(a x) - cosh(a x)) +--R + +--R 2 +--R (a x coth(a x) + log(- 2) - 2a x)sinh(a x) +--R + +--R (2a x cosh(a x)coth(a x) + (2log(- 2) - 4a x)cosh(a x))sinh(a x) +--R + +--R 2 2 +--R (a x cosh(a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(a x) - log(- 2) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E + +--S 32 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (5) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 33 +ee:=sinhsqrrule dd +--R +--R (6) +--R 2 +--R (- 4cosh(a x)sinh(a x) - cosh(2a x) - 2cosh(a x) + 3) +--R * +--R log(sinh(a x) - cosh(a x)) +--R + +--R (4a x cosh(a x)coth(a x) + (4log(- 2) - 8a x)cosh(a x))sinh(a x) +--R + +--R 2 +--R (a x cosh(2a x) + 2a x cosh(a x) - 3a x)coth(a x) +--R + +--R 2 +--R (log(- 2) - 2a x)cosh(2a x) + (2log(- 2) - 4a x)cosh(a x) - 3log(- 2) +--R + +--R 2a x +--R / +--R 2 2 2 2 2 +--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) - 3a +--R Type: Expression Integer +--E + +--S 34 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (7) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 35 +ff:=coshsqrrule ee +--R +--R (8) +--R (- 2cosh(a x)sinh(a x) - cosh(2a x) + 1)log(sinh(a x) - cosh(a x)) +--R + +--R (2a x cosh(a x)coth(a x) + (2log(- 2) - 4a x)cosh(a x))sinh(a x) +--R + +--R (a x cosh(2a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(2a x) - log(- 2) +--R / +--R 2 2 2 +--R 2a cosh(a x)sinh(a x) + a cosh(2a x) - a +--R Type: Expression Integer +--E + +--S 36 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %P sinh(y + x) - %P sinh(y - x) +--I (9) %P cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 37 +gg:=sinhcoshrule ff +--R +--R (10) +--R (- sinh(2a x) - cosh(2a x) + 1)log(sinh(a x) - cosh(a x)) +--R + +--R (a x coth(a x) + log(- 2) - 2a x)sinh(2a x) +--R + +--R (a x cosh(2a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(2a x) - log(- 2) +--R / +--R 2 2 2 +--R a sinh(2a x) + a cosh(2a x) - a +--R Type: Expression Integer +--E + +--S 38 14:642 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R - log(- 1) + log(- 2) +--R (11) --------------------- +--R 2 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.643~~~~~$\displaystyle +\int{\frac{{\rm csch~}{ax}}{x}}~dx$} +$$\int{\frac{{\rm csch~}{ax}}{x}}= +-\frac{1}{ax}-\frac{ax}{6}+\frac{7(ax)^3}{1080}+\cdots +\frac{(-1)^n2(2^{2n-1}-1)B_n(ax)^{2n-1}}{(2n-1)(2n)!}+\cdots +$$ +<<*>>= +)clear all + +--S 39 14:643 Axiom cannot compute this integral +aa:=integrate(csch(a*x)/x,x) +--R +--R +--R x +--I ++ csch(%O a) +--I (1) | ---------- d%O +--I ++ %O +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.644~~~~~$\displaystyle +\int{\frac{dx}{q+p{\rm ~csch~}{ax}}}~dx$} +$$\int{\frac{1}{q+p{\rm ~csch~}{ax}}}= +\frac{x}{q}-\frac{p}{q}\int{\frac{dx}{p+q\sinh{ax}}} +$$ +<<*>>= +)clear all + +--S 40 +aa:=integrate(1/(q+p*csch(a*x)),x) +--R +--R +--R (1) +--R p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (2q + 2p q)sinh(a x) + (2q + 2p q)cosh(a x) + 2p q + 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R + +--R +-------+ +--R | 2 2 +--R a x\|q + p +--R / +--R +-------+ +--R | 2 2 +--R a q\|q + p +--R Type: Union(Expression Integer,...) +--E + +--S 41 +t1:=integrate(1/(p+q*sinh(a*x)),x) +--R +--R (2) +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R / +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Union(Expression Integer,...) +--E + +--S 42 +bb:=x/q-p/q*t1 +--R +--R (3) +--R - +--R p +--R * +--R log +--R 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) +--R + +--R 2 2 2 2 +--R q cosh(a x) + 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R + +--R +-------+ +--R | 2 2 +--R a x\|q + p +--R / +--R +-------+ +--R | 2 2 +--R a q\|q + p +--R Type: Expression Integer +--E + +--S 43 +cc:=aa-bb +--R +--R (4) +--R p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (2q + 2p q)sinh(a x) + (2q + 2p q)cosh(a x) + 2p q + 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R + +--R p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R / +--R +-------+ +--R | 2 2 +--R a q\|q + p +--R Type: Expression Integer +--E + +--S 44 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (5) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 45 +dd:=sinhsqrrule cc +--R +--R (6) +--R p +--R * +--R log +--R 2 2 +--R (4q cosh(a x) + 4p q)sinh(a x) + q cosh(2a x) +--R + +--R 2 2 2 2 +--R 2q cosh(a x) + 4p q cosh(a x) + q + 4p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (4q + 4p q)sinh(a x) + (4q + 4p q)cosh(a x) + 4p q + 4p +--R / +--R 2 +--R (4q cosh(a x) + 4p)sinh(a x) + q cosh(2a x) + 2q cosh(a x) +--R + +--R 4p cosh(a x) - 3q +--R + +--R p +--R * +--R log +--R 2 2 +--R (4q cosh(a x) + 4p q)sinh(a x) + q cosh(2a x) +--R + +--R 2 2 2 2 +--R 2q cosh(a x) + 4p q cosh(a x) + q + 4p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 4q - 4p q)sinh(a x) + (- 4q - 4p q)cosh(a x) - 4p q - 4p +--R / +--R 2 +--R (4q cosh(a x) + 4p)sinh(a x) + q cosh(2a x) + 2q cosh(a x) +--R + +--R 4p cosh(a x) - 3q +--R / +--R +-------+ +--R | 2 2 +--R a q\|q + p +--R Type: Expression Integer +--E + +--S 46 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (7) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 47 +ee:=coshsqrrule dd +--R +--R (8) +--R p +--R * +--R log +--R 2 2 +--R (2q cosh(a x) + 2p q)sinh(a x) + q cosh(2a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (2q + 2p q)sinh(a x) + (2q + 2p q)cosh(a x) + 2p q + 2p +--R / +--R (2q cosh(a x) + 2p)sinh(a x) + q cosh(2a x) + 2p cosh(a x) - q +--R + +--R p +--R * +--R log +--R 2 2 +--R (2q cosh(a x) + 2p q)sinh(a x) + q cosh(2a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R (2q cosh(a x) + 2p)sinh(a x) + q cosh(2a x) + 2p cosh(a x) - q +--R / +--R +-------+ +--R | 2 2 +--R a q\|q + p +--R Type: Expression Integer +--E + +--S 48 14:644 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R 4 2 2 +--R p log(q + p q ) +--R (9) ---------------- +--R +-------+ +--R | 2 2 +--R a q\|q + p +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.645~~~~~$\displaystyle +\int{{\rm csch}^n~{ax}}~dx$} +$$\int{{\rm csch}^n~{ax}}= +\frac{-{\rm csch}^{n-2}~{ax}~\coth{ax}}{a(n-1)} +-\frac{n-2}{n-1}\int{{\rm csch}^{n-2}~{ax}} +$$ +<<*>>= +)clear all + +--S 49 14:645 Axiom cannot compute this integral +aa:=integrate(csch(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | csch(%O a) d%O +--R ++ +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp91-92 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum33.input.pdf b/src/axiom-website/CATS/schaum33.input.pdf new file mode 100644 index 0000000..5ee405c --- /dev/null +++ b/src/axiom-website/CATS/schaum33.input.pdf @@ -0,0 +1,2315 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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+0000114985 00000 n +0000115008 00000 n +0000115030 00000 n +trailer +<< +/Size 188 +/Root 2 0 R +/Info 1 0 R +>> +startxref +115151 +%%EOF diff --git a/src/axiom-website/CATS/schaum34.input.pamphlet b/src/axiom-website/CATS/schaum34.input.pamphlet new file mode 100644 index 0000000..d61b757 --- /dev/null +++ b/src/axiom-website/CATS/schaum34.input.pamphlet @@ -0,0 +1,2492 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum34.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.646~~~~~$\displaystyle +\int{\sinh^{-1}\frac{x}{a}}~dx$} +$$\int{\sinh^{-1}\frac{x}{a}}= +x\sinh^{-1}\frac{x}{a}-\sqrt{x^2+a^2} +$$ +<<*>>= +)spool schaum34.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(asinh(x/a),x) +--R +--R +--R +-------+ +--R +-------+ | 2 2 +-------+ +--R | 2 2 2 \|x + a + x | 2 2 2 2 +--R (x\|x + a - x )log(--------------) + x\|x + a - x - a +--R a +--R (1) ------------------------------------------------------------- +--R +-------+ +--R | 2 2 +--R \|x + a - x +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=x*asinh(x/a)-sqrt(x^2+a^2) +--R +--R +-------+ +--R | 2 2 x +--R (2) - \|x + a + x asinh(-) +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x + a + x x +--R (3) x log(--------------) - x asinh(-) +--R a a +--R Type: Expression Integer +--E + +--S 4 +asinhlogrule:=rule(asinh(x) == log(x+sqrt(x^2+1))) +--R +--R +------+ +--R | 2 +--R (4) asinh(x) == log(\|x + 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 5 +dd:=asinhlogrule cc +--R +--R +-------+ +--R | 2 2 +--R |x + a +--R +-------+ a |------- + x +--R | 2 2 | 2 +--R \|x + a + x \| a +--R (5) x log(--------------) - x log(---------------) +--R a a +--R Type: Expression Integer +--E + +--S 6 +ee:=expandLog dd +--R +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 |x + a +--R (6) x log(\|x + a + x) - x log(a |------- + x) +--R | 2 +--R \| a +--R Type: Expression Integer +--E + +--S 7 14:646 Schaums and Axiom agree +ff:=rootSimp ee +--R +--R (7) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.647~~~~~$\displaystyle +\int{x\sinh^{-1}\frac{x}{a}}~dx$} +$$\int{x\sinh^{-1}\frac{x}{a}}= +\left(\frac{x^2}{2}+\frac{a^2}{4}\right)\sinh^{-1}\frac{x}{a} +-\frac{x\sqrt{x^2+a^2}}{4} +$$ +<<*>>= +)clear all + +--S 8 +aa:=integrate(x*asinh(x/a),x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 3 2 | 2 2 4 2 2 4 \|x + a + x +--R ((4x + 2a x)\|x + a - 4x - 4a x - a )log(--------------) +--R a +--R + +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (2x + a x)\|x + a - 2x - 2a x +--R / +--R +-------+ +--R | 2 2 2 2 +--R 8x\|x + a - 8x - 4a +--R Type: Union(Expression Integer,...) +--E + +--S 9 +bb:=(x^2/2+a^2/4)*asinh(x/a)-(x*sqrt(x^2+a^2))/4 +--R +--R +-------+ +--R | 2 2 2 2 x +--R - x\|x + a + (2x + a )asinh(-) +--R a +--R (2) ---------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 10 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R 2 2 \|x + a + x 2 2 x +--R (2x + a )log(--------------) + (- 2x - a )asinh(-) +--R a a +--R (3) ---------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 11 +asinhlogrule:=rule(asinh(x) == log(x+sqrt(x^2+1))) +--R +--R +------+ +--R | 2 +--R (4) asinh(x) == log(\|x + 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 12 +dd:=asinhlogrule cc +--R +--R +-------+ +--R | 2 2 +--R |x + a +--R +-------+ a |------- + x +--R | 2 2 | 2 +--R 2 2 \|x + a + x 2 2 \| a +--R (2x + a )log(--------------) + (- 2x - a )log(---------------) +--R a a +--R (5) ---------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 13 +ee:=expandLog dd +--R +--R +-------+ +--R +-------+ | 2 2 +--R 2 2 | 2 2 2 2 |x + a +--R (2x + a )log(\|x + a + x) + (- 2x - a )log(a |------- + x) +--R | 2 +--R \| a +--R (6) ---------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 14 14:647 Schaums and Axiom agree +ff:=rootSimp ee +--R +--R (7) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.648~~~~~$\displaystyle +\int{x^2\sinh^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\sinh^{-1}\frac{x}{a}}= +\frac{x^3}{3}\sinh^{-1}\frac{x}{a}+\frac{(2a^2-x^2)\sqrt{x^2+a^2}}{9} +$$ +<<*>>= +)clear all + +--S 15 +aa:=integrate(x^2*asinh(x/a),x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 5 2 3 | 2 2 6 2 4 \|x + a + x +--R ((12x + 3a x )\|x + a - 12x - 9a x )log(--------------) +--R a +--R + +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (4x - 5a x - 6a x)\|x + a - 4x + 3a x + 9a x + 2a +--R / +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (36x + 9a )\|x + a - 36x - 27a x +--R Type: Union(Expression Integer,...) +--E + +--S 16 +bb:=x^3/3*asinh(x/a)+((2*a^2-x^2)*sqrt(x^2+a^2))/9 +--R +--R +-------+ +--R 2 2 | 2 2 3 x +--R (- x + 2a )\|x + a + 3x asinh(-) +--R a +--R (2) ------------------------------------ +--R 9 +--R Type: Expression Integer +--E + +--S 17 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R 3 \|x + a + x 3 x +--R x log(--------------) - x asinh(-) +--R a a +--R (3) ---------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 18 +asinhlogrule:=rule(asinh(x) == log(x+sqrt(x^2+1))) +--R +--R +------+ +--R | 2 +--R (4) asinh(x) == log(\|x + 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 19 +dd:=asinhlogrule cc +--R +--R +-------+ +--R | 2 2 +--R |x + a +--R +-------+ a |------- + x +--R | 2 2 | 2 +--R 3 \|x + a + x 3 \| a +--R x log(--------------) - x log(---------------) +--R a a +--R (5) ---------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 20 +ee:=expandLog dd +--R +--R +-------+ +--R +-------+ | 2 2 +--R 3 | 2 2 3 |x + a +--R x log(\|x + a + x) - x log(a |------- + x) +--R | 2 +--R \| a +--R (6) ---------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 21 14:648 Schaums and Axiom agree +ff:=rootSimp ee +--R +--R (7) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.649~~~~~$\displaystyle +\int{\frac{\sinh^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{\sinh^{-1}(x/a)}{x}}= +\left\{ +\begin{array}{lr} +\displaystyle +\frac{x}{a}-\frac{(x/a)^3}{2\cdot 3\cdot 3} ++\frac{1\cdot 3(x/a)^5}{2\cdot 4\cdot 5\cdot 5} +-\frac{1\cdot 3\cdot 5(x/a)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7}+\cdots& +|x| a\\ +\\ +\displaystyle +-\frac{\ln^2(-2x/a)}{2}+\frac{(a/x)^2}{2\cdot 2\cdot 2} +-\frac{1\cdot 3(a/x)^4}{2\cdot 4\cdot 4\cdot 4} ++\frac{1\cdot 3\cdot 5(a/x)^6}{2\cdot 4\cdot 6\cdot 6\cdot 6}-\cdots& +x<-a\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 22 14:649 Axiom cannot compute this integral +aa:=integrate(asinh(x/a)/x,x) +--R +--R +--I %P +--R x asinh(--) +--R ++ a +--I (1) | --------- d%P +--I ++ %P +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.650~~~~~$\displaystyle +\int{\frac{\sinh^{-1}(x/a)}{x^2}}~dx$} +$$\int{\frac{\sinh^{-1}(x/a)}{x^2}}= +-\frac{\sinh^{-1}(x/a)}{x} +-\frac{1}{a}\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 23 +aa:=integrate(asinh(x/a)/x^2,x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - x log(\|x + a - x + a) + x log(\|x + a - x - a) +--R + +--R +-------+ +--R | 2 2 +--R \|x + a + x +--R - a log(--------------) +--R a +--R / +--R a x +--R Type: Union(Expression Integer,...) +--E + +--S 24 +bb:=-asinh(x/a)/x-1/a*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R | 2 2 +--R \|x + a + a x +--R - x log(--------------) - a asinh(-) +--R x a +--R (2) ------------------------------------ +--R a x +--R Type: Expression Integer +--E + +--S 25 +cc:=aa-bb +--R +--R (3) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - x log(\|x + a - x + a) + x log(\|x + a - x - a) +--R + +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R \|x + a + x \|x + a + a x +--R - a log(--------------) + x log(--------------) + a asinh(-) +--R a x a +--R / +--R a x +--R Type: Expression Integer +--E + +--S 26 +asinhlogrule:=rule(asinh(x) == log(x+sqrt(x^2+1))) +--R +--R +------+ +--R | 2 +--R (4) asinh(x) == log(\|x + 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 27 +dd:=asinhlogrule cc +--R +--R (5) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - x log(\|x + a - x + a) + x log(\|x + a - x - a) +--R + +--R +-------+ +--R | 2 2 +--R |x + a +--R +-------+ +-------+ a |------- + x +--R | 2 2 | 2 2 | 2 +--R \|x + a + x \|x + a + a \| a +--R - a log(--------------) + x log(--------------) + a log(---------------) +--R a x a +--R / +--R a x +--R Type: Expression Integer +--E + +--S 28 +ee:=expandLog dd +--R +--R (6) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - a log(\|x + a + x) + x log(\|x + a + a) +--R + +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - x log(\|x + a - x + a) + x log(\|x + a - x - a) +--R + +--R +-------+ +--R | 2 2 +--R |x + a +--R a log(a |------- + x) - x log(x) +--R | 2 +--R \| a +--R / +--R a x +--R Type: Expression Integer +--E + +--S 29 +ff:=rootSimp ee +--R +--R (7) +--R +-------+ +-------+ +-------+ +--R | 2 2 | 2 2 | 2 2 +--R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x - a) +--R + +--R - log(x) +--R / +--R a +--R Type: Expression Integer +--E + +--S 30 14:650 Schaums and Axiom differ by a constant +gg:=complexNormalize ff +--R +--R log(- 1) +--R (8) - -------- +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.651~~~~~$\displaystyle +\int{\cosh^{-1}\frac{x}{a}}~dx$} +$$\int{\cosh^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +x\cosh^{-1}(x/a)-\sqrt{x^2-a^2},\quad\cosh^{-1}\frac{x}{a} > 0\\ +\\ +\displaystyle +x\cosh^{-1}(x/a)+\sqrt{x^2-a^2},\quad\cosh^{-1}\frac{x}{a} < 0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 31 +aa:=integrate(acosh(x/a),x) +--R +--R +--R +-------+ +--R +-------+ | 2 2 +-------+ +--R | 2 2 2 \|x - a + x | 2 2 2 2 +--R (x\|x - a - x )log(--------------) + x\|x - a - x + a +--R a +--R (1) ------------------------------------------------------------- +--R +-------+ +--R | 2 2 +--R \|x - a - x +--R Type: Union(Expression Integer,...) +--E + +--S 32 +bb1:=x*acosh(x/a)-sqrt(x^2-a^2) +--R +--R +-------+ +--R | 2 2 x +--R (2) - \|x - a + x acosh(-) +--R a +--R Type: Expression Integer +--E + +--S 33 +bb2:=x*acosh(x/a)+sqrt(x^2-a^2) +--R +--R +-------+ +--R | 2 2 x +--R (3) \|x - a + x acosh(-) +--R a +--R Type: Expression Integer +--E + +--S 34 +cc1:=aa-bb1 +--R +--R +-------+ +--R | 2 2 +--R \|x - a + x x +--R (4) x log(--------------) - x acosh(-) +--R a a +--R Type: Expression Integer +--E + +--S 35 +cc2:=aa-bb2 +--R +--R (5) +--R +-------+ +--R +-------+ | 2 2 +-------+ +--R | 2 2 2 \|x - a + x x | 2 2 +--R (x\|x - a - x )log(--------------) + (- x acosh(-) + 2x)\|x - a +--R a a +--R + +--R 2 x 2 2 +--R x acosh(-) - 2x + 2a +--R a +--R / +--R +-------+ +--R | 2 2 +--R \|x - a - x +--R Type: Expression Integer +--E + +--S 36 +acoshlogrule:=rule(acosh(x) == log(x+sqrt(x^2-1))) +--R +--R +------+ +--R | 2 +--R (6) acosh(x) == log(\|x - 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 37 +dd1:=acoshlogrule cc1 +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R +-------+ a |------- + x +--R | 2 2 | 2 +--R \|x - a + x \| a +--R (7) x log(--------------) - x log(---------------) +--R a a +--R Type: Expression Integer +--E + +--S 38 +ee1:=expandLog dd1 +--R +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 |x - a +--R (8) x log(\|x - a + x) - x log(a |------- + x) +--R | 2 +--R \| a +--R Type: Expression Integer +--E + +--S 39 14:651 Schaums and Axiom agree +ff1:=rootSimp ee1 +--R +--R (9) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.652~~~~~$\displaystyle +\int{x\cosh^{-1}\frac{x}{a}}~dx$} +$$\int{x\cosh^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{4}(2x^2-a^2)\cosh^{-1}(x/a)-\frac{1}{4}x\sqrt{x^2-a^2}, +\quad\cosh^{-1}(x/a)>0\\ +\\ +\displaystyle +\frac{1}{4}(2x^2-a^2)\cosh^{-1}(x/a)+\frac{1}{4}x\sqrt{x^2-a^2}, +\quad\cosh^{-1}(x/a)<0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 40 +aa:=integrate(x*acosh(x/a),x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 3 2 | 2 2 4 2 2 4 \|x - a + x +--R ((4x - 2a x)\|x - a - 4x + 4a x - a )log(--------------) +--R a +--R + +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (2x - a x)\|x - a - 2x + 2a x +--R / +--R +-------+ +--R | 2 2 2 2 +--R 8x\|x - a - 8x + 4a +--R Type: Union(Expression Integer,...) +--E + +--S 41 +bb1:=1/4*(2*x^2-a^2)*acosh(x/a)-1/4*x*sqrt(x^2-a^2) +--R +--R +-------+ +--R | 2 2 2 2 x +--R - x\|x - a + (2x - a )acosh(-) +--R a +--R (2) ---------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 42 +bb2:=1/4*(2*x^2-a^2)*acosh(x/a)+1/4*x*sqrt(x^2-a^2) +--R +--R +-------+ +--R | 2 2 2 2 x +--R x\|x - a + (2x - a )acosh(-) +--R a +--R (3) -------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 43 +cc1:=aa-bb1 +--R +--R +-------+ +--R | 2 2 +--R 2 2 \|x - a + x 2 2 x +--R (2x - a )log(--------------) + (- 2x + a )acosh(-) +--R a a +--R (4) ---------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 44 +cc2:=aa-bb2 +--R +--R (5) +--R +-------+ +--R +-------+ | 2 2 +--R 3 2 | 2 2 4 2 2 4 \|x - a + x +--R ((4x - 2a x)\|x - a - 4x + 4a x - a )log(--------------) +--R a +--R + +--R +-------+ +--R 3 2 x 3 2 | 2 2 +--R ((- 4x + 2a x)acosh(-) + 4x - 2a x)\|x - a +--R a +--R + +--R 4 2 2 4 x 4 2 2 +--R (4x - 4a x + a )acosh(-) - 4x + 4a x +--R a +--R / +--R +-------+ +--R | 2 2 2 2 +--R 8x\|x - a - 8x + 4a +--R Type: Expression Integer +--E + +--S 45 +acoshlogrule:=rule(acosh(x) == log(x+sqrt(x^2-1))) +--R +--R +------+ +--R | 2 +--R (6) acosh(x) == log(\|x - 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 46 +dd1:=acoshlogrule cc1 +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R +-------+ a |------- + x +--R | 2 2 | 2 +--R 2 2 \|x - a + x 2 2 \| a +--R (2x - a )log(--------------) + (- 2x + a )log(---------------) +--R a a +--R (7) ---------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 47 +ee1:=expandLog dd1 +--R +--R +-------+ +--R +-------+ | 2 2 +--R 2 2 | 2 2 2 2 |x - a +--R (2x - a )log(\|x - a + x) + (- 2x + a )log(a |------- + x) +--R | 2 +--R \| a +--R (8) ---------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 48 14:652 Schaums and Axiom agree +ff1:=rootSimp ee1 +--R +--R (9) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.653~~~~~$\displaystyle +\int{x^2\cosh^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\cosh^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{3}x^3\cosh^{-1}(x/a)-\frac{1}{9}(x^2+2a^2)\sqrt{x^2-a^2}, +\quad\cosh^{-1}(x/a)>0\\ +\\ +\displaystyle +\frac{1}{3}x^3\cosh^{-1}(x/a)+\frac{1}{9}(x^2+2a^2)\sqrt{x^2-a^2}, +\quad\cosh^{-1}(x/a)<0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 49 +aa:=integrate(x^2*acosh(x/a),x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 5 2 3 | 2 2 6 2 4 \|x - a + x +--R ((12x - 3a x )\|x - a - 12x + 9a x )log(--------------) +--R a +--R + +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (4x + 5a x - 6a x)\|x - a - 4x - 3a x + 9a x - 2a +--R / +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (36x - 9a )\|x - a - 36x + 27a x +--R Type: Union(Expression Integer,...) +--E + +--S 50 +bb1:=1/3*x^3*acosh(x/a)-1/9*(x^2+2*a^2)*sqrt(x^2-a^2) +--R +--R +-------+ +--R 2 2 | 2 2 3 x +--R (- x - 2a )\|x - a + 3x acosh(-) +--R a +--R (2) ------------------------------------ +--R 9 +--R Type: Expression Integer +--E + +--S 51 +bb2:=1/3*x^3*acosh(x/a)+1/9*(x^2+2*a^2)*sqrt(x^2-a^2) +--R +--R +-------+ +--R 2 2 | 2 2 3 x +--R (x + 2a )\|x - a + 3x acosh(-) +--R a +--R (3) ---------------------------------- +--R 9 +--R Type: Expression Integer +--E + +--S 52 +cc1:=aa-bb1 +--R +--R +-------+ +--R | 2 2 +--R 3 \|x - a + x 3 x +--R x log(--------------) - x acosh(-) +--R a a +--R (4) ---------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 53 +cc2:=aa-bb2 +--R +--R (5) +--R +-------+ +--R +-------+ | 2 2 +--R 5 2 3 | 2 2 6 2 4 \|x - a + x +--R ((12x - 3a x )\|x - a - 12x + 9a x )log(--------------) +--R a +--R + +--R +-------+ +--R 5 2 3 x 5 2 3 4 | 2 2 +--R ((- 12x + 3a x )acosh(-) + 8x + 10a x - 12a x)\|x - a +--R a +--R + +--R 6 2 4 x 6 2 4 4 2 6 +--R (12x - 9a x )acosh(-) - 8x - 6a x + 18a x - 4a +--R a +--R / +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (36x - 9a )\|x - a - 36x + 27a x +--R Type: Expression Integer +--E + +--S 54 +acoshlogrule:=rule(acosh(x) == log(x+sqrt(x^2-1))) +--R +--R +------+ +--R | 2 +--R (6) acosh(x) == log(\|x - 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 55 +dd1:=acoshlogrule cc1 +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R +-------+ a |------- + x +--R | 2 2 | 2 +--R 3 \|x - a + x 3 \| a +--R x log(--------------) - x log(---------------) +--R a a +--R (7) ---------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 56 +ee1:=expandLog dd1 +--R +--R +-------+ +--R +-------+ | 2 2 +--R 3 | 2 2 3 |x - a +--R x log(\|x - a + x) - x log(a |------- + x) +--R | 2 +--R \| a +--R (8) ---------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 57 14:653 Schaums and Axiom agree +ff1:=rootSimp ee1 +--R +--R (9) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.654~~~~~$\displaystyle +\int{\frac{\cosh^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{\cosh^{-1}(x/a)}{x}}= +\begin{array}{l} +\displaystyle +\pm\left[\frac{1}{2}\ln^2(2x/a)+\frac{(a/x)^2}{2\cdot 2\cdot 2} ++\frac{1\cdot 3(a/x)^4}{2\cdot 4\cdot 4\cdot 4} ++\frac{1\cdot 3\cdot 5(a/x)^6}{2\cdot 4\cdot 6\cdot 6\cdot 6}+\cdots\right]\\ +\\ +\displaystyle +\hbox{\hskip 2cm}+{\rm if\ }\cosh^{-1}(x/a)>0, +\quad -{\rm if\ }\cosh^{-1}(x/a)<0, +\end{array} +$$ +<<*>>= +)clear all + +--S 58 14:654 Axiom cannot compute this integral +aa:=integrate(acosh(x/a)/x,x) +--R +--R +--I %P +--R x acosh(--) +--R ++ a +--I (1) | --------- d%P +--I ++ %P +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.655~~~~~$\displaystyle +\int{\frac{\cosh^{-1}(x/a)}{x^2}}~dx$} +$$\int{\frac{\cosh^{-1}(x/a)}{x^2}}= +\begin{array}{l} +\displaystyle +-\frac{\cosh^{-1}(x/a)}{x} +\mp\frac{1}{a}\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right)\\ +\\ +\displaystyle +\hbox{\hskip 1cm}-{\rm if\ }\cosh^{-1}(x/a)>0, +\quad +{\rm if\ }\cosh^{-1}(x/a)<0, +\end{array} +$$ +<<*>>= +)clear all + +--S 59 +aa:=integrate(acosh(x/a)/x^2,x) +--R +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R \|x - a + x \|x - a - x +--R - a log(--------------) + 2x atan(--------------) +--R a a +--R (1) ------------------------------------------------- +--R a x +--R Type: Union(Expression Integer,...) +--E + +--S 60 +bb1:=-acosh(x/a)/x-1/a*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R | 2 2 +--R \|x + a + a x +--R - x log(--------------) - a acosh(-) +--R x a +--R (2) ------------------------------------ +--R a x +--R Type: Expression Integer +--E + +--S 61 +bb2:=-acosh(x/a)/x+1/a*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R | 2 2 +--R \|x + a + a x +--R x log(--------------) - a acosh(-) +--R x a +--R (3) ---------------------------------- +--R a x +--R Type: Expression Integer +--E + +--S 62 +cc1:=aa-bb1 +--R +--R (4) +--R +-------+ +-------+ +-------+ +--R | 2 2 | 2 2 | 2 2 +--R \|x + a + a \|x - a + x \|x - a - x +--R x log(--------------) - a log(--------------) + 2x atan(--------------) +--R x a a +--R + +--R x +--R a acosh(-) +--R a +--R / +--R a x +--R Type: Expression Integer +--E + +--S 63 14:655 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +-------+ +-------+ +-------+ +--R | 2 2 | 2 2 | 2 2 +--R \|x + a + a \|x - a + x \|x - a - x +--R - x log(--------------) - a log(--------------) + 2x atan(--------------) +--R x a a +--R + +--R x +--R a acosh(-) +--R a +--R / +--R a x +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.656~~~~~$\displaystyle +\int{\tanh^{-1}\frac{x}{a}}~dx$} +$$\int{\tanh^{-1}\frac{x}{a}}= +x\tanh^{-1}\frac{x}{a}+\frac{a}{2}\ln(a^2-x^2) +$$ +<<*>>= +)clear all + +--S 64 +aa:=integrate(atanh(x/a),x) +--R +--R +--R 2 2 - x - a +--R a log(x - a ) + x log(-------) +--R x - a +--R (1) ------------------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E + +--S 65 +bb:=x*atanh(x/a)+a/2*log(a^2-x^2) +--R +--R 2 2 x +--R a log(- x + a ) + 2x atanh(-) +--R a +--R (2) ------------------------------ +--R 2 +--R Type: Expression Integer +--E + +--S 66 +cc:=aa-bb +--R +--R 2 2 - x - a 2 2 x +--R a log(x - a ) + x log(-------) - a log(- x + a ) - 2x atanh(-) +--R x - a a +--R (3) ---------------------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 67 +atanhrule:=rule(atanh(x) == 1/2*log((1+x)/(1-x))) +--R +--R - x - 1 +--R log(-------) +--R x - 1 +--R (4) atanh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 68 +dd:=atanhrule cc +--R +--R 2 2 2 2 +--R a log(x - a ) - a log(- x + a ) +--R (5) --------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 69 14:656 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R a log(- 1) +--R (6) ---------- +--R 2 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.657~~~~~$\displaystyle +\int{x*\tanh^{-1}\frac{x}{a}}~dx$} +$$\int{x*\tanh^{-1}\frac{x}{a}}= +\frac{ax}{2}+\frac{1}{2}(x^2-a^2)\tanh^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 70 +aa:=integrate(x*atanh(x/a),x) +--R +--R +--R 2 2 - x - a +--R (x - a )log(-------) + 2a x +--R x - a +--R (1) ---------------------------- +--R 4 +--R Type: Union(Expression Integer,...) +--E + +--S 71 +bb:=(a*x)/2+1/2*(x^2-a^2)*atanh(x/a) +--R +--R 2 2 x +--R (x - a )atanh(-) + a x +--R a +--R (2) ----------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 72 +cc:=aa-bb +--R +--R 2 2 - x - a 2 2 x +--R (x - a )log(-------) + (- 2x + 2a )atanh(-) +--R x - a a +--R (3) --------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 73 +atanhrule:=rule(atanh(x) == 1/2*log((1+x)/(1-x))) +--R +--R - x - 1 +--R log(-------) +--R x - 1 +--R (4) atanh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 74 14:657 Schaums and Axiom agree +dd:=atanhrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.658~~~~~$\displaystyle +\int{x^2\tanh^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\tanh^{-1}\frac{x}{a}}= +\frac{ax^2}{6}+\frac{x^3}{3}\tanh^{-1}\frac{x}{a} ++\frac{a^3}{6}\ln(a^2-x^2) +$$ +<<*>>= +)clear all + +--S 75 +aa:=integrate(x^2*atanh(x/a),x) +--R +--R +--R 3 2 2 3 - x - a 2 +--R a log(x - a ) + x log(-------) + a x +--R x - a +--R (1) -------------------------------------- +--R 6 +--R Type: Union(Expression Integer,...) +--E + +--S 76 +bb:=(a*x^2)/6+x^3/3*atanh(x/a)+a^3/6*log(a^2-x^2) +--R +--R 3 2 2 3 x 2 +--R a log(- x + a ) + 2x atanh(-) + a x +--R a +--R (2) ------------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 77 +cc:=aa-bb +--R +--R 3 2 2 3 - x - a 3 2 2 3 x +--R a log(x - a ) + x log(-------) - a log(- x + a ) - 2x atanh(-) +--R x - a a +--R (3) ---------------------------------------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 78 +atanhrule:=rule(atanh(x) == 1/2*log((1+x)/(1-x))) +--R +--R - x - 1 +--R log(-------) +--R x - 1 +--R (4) atanh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 79 +dd:=atanhrule cc +--R +--R 3 2 2 3 2 2 +--R a log(x - a ) - a log(- x + a ) +--R (5) --------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 80 14:658 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R 3 +--R a log(- 1) +--R (6) ---------- +--R 6 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.659~~~~~$\displaystyle +\int{\frac{\tanh^{-1}(x/a)}{a}}~dx$} +$$\int{\frac{\tanh^{-1}(x/a)}{a}}= +\frac{x}{a}+\frac{(x/a)^3}{3^2}+\frac{(x/a)^5}{5^2}+\cdots +$$ +<<*>>= +)clear all + +--S 81 14:659 Axiom cannot compute this integral +aa:=integrate(atanh(x/a)/x,x) +--R +--R +--I %P +--R x atanh(--) +--R ++ a +--I (1) | --------- d%P +--I ++ %P +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.660~~~~~$\displaystyle +\int{\frac{tanh^{-1}(x/a)}{x^2}}~dx$} +$$\int{\frac{tanh^{-1}(x/a)}{x^2}}= +-\frac{\tanh^{-1}(x/a)}{x}+\frac{1}{2a}\ln\left(\frac{x^2}{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 82 +aa:=integrate(atanh(x/a)/x^2,x) +--R +--R +--R 2 2 - x - a +--R - x log(x - a ) + 2x log(x) - a log(-------) +--R x - a +--R (1) --------------------------------------------- +--R 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 83 +bb:=-atanh(x/a)/x+1/(2*a)*log(x^2/(a^2-x^2)) +--R +--R 2 +--R x x +--R x log(- -------) - 2a atanh(-) +--R 2 2 a +--R x - a +--R (2) ------------------------------ +--R 2a x +--R Type: Expression Integer +--E + +--S 84 +cc:=aa-bb +--R +--R (3) +--R 2 +--R 2 2 x - x - a +--R - x log(x - a ) + 2x log(x) - x log(- -------) - a log(-------) +--R 2 2 x - a +--R x - a +--R + +--R x +--R 2a atanh(-) +--R a +--R / +--R 2a x +--R Type: Expression Integer +--E + +--S 85 +atanhrule:=rule(atanh(x) == 1/2*log((1+x)/(1-x))) +--R +--R - x - 1 +--R log(-------) +--R x - 1 +--R (4) atanh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 86 +dd:=atanhrule cc +--R +--R 2 +--R 2 2 x +--R - log(x - a ) + 2log(x) - log(- -------) +--R 2 2 +--R x - a +--R (5) ----------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 87 14:660 Schaums and Axiom agree +ee:=expandLog dd +--R +--R log(- 1) +--R (6) - -------- +--R 2a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.661~~~~~$\displaystyle +\int{\coth^{-1}\frac{x}{a}}~dx$} +$$\int{\coth^{-1}\frac{x}{a}}= +x\coth^{-1}{x}+\frac{a}{2}\ln(x^2-a^2) +$$ + +Note that it appears there is a typo in Schaums (1968 printing 4). +$$\int{\coth^{-1}\frac{x}{a}}= +x\coth^{-1}{x/a}+\frac{a}{2}\ln(x^2-a^2) +$$ +<<*>>= +)clear all + +--S 88 +aa:=integrate(acoth(x/a),x) +--R +--R +--R 2 2 x + a +--R a log(x - a ) + x log(-----) +--R x - a +--R (1) ----------------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E + +--S 89 +bb:=x*acoth(x/a)+a/2*log(x^2-a^2) +--R +--R 2 2 x +--R a log(x - a ) + 2x acoth(-) +--R a +--R (2) ---------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 90 +cc:=aa-bb +--R +--R x + a x +--R x log(-----) - 2x acoth(-) +--R x - a a +--R (3) -------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 91 +acothrule:=rule(acoth(x) == 1/2*log((x+1)/(x-1))) +--R +--R x + 1 +--R log(-----) +--R x - 1 +--R (4) acoth(x) == ---------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 92 14:661 Schaums and Axiom agree +dd:=acothrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.662~~~~~$\displaystyle +\int{x\coth^{-1}\frac{x}{a}}~dx$} +$$\int{x\coth^{-1}\frac{x}{a}}= +\frac{ax}{2}+\frac{1}{2}(x^2-a^2)\coth^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 93 +aa:=integrate(x*acoth(x/a),x) +--R +--R +--R 2 2 x + a +--R (x - a )log(-----) + 2a x +--R x - a +--R (1) -------------------------- +--R 4 +--R Type: Union(Expression Integer,...) +--E + +--S 94 +bb:=(a*x)/2+1/2*(x^2-a^2)*acoth(x/a) +--R +--R 2 2 x +--R (x - a )acoth(-) + a x +--R a +--R (2) ----------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 95 +cc:=aa-bb +--R +--R 2 2 x + a 2 2 x +--R (x - a )log(-----) + (- 2x + 2a )acoth(-) +--R x - a a +--R (3) ------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 96 +acothrule:=rule(acoth(x) == 1/2*log((x+1)/(x-1))) +--R +--R x + 1 +--R log(-----) +--R x - 1 +--R (4) acoth(x) == ---------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 97 14:662 Schaums and Axiom agree +dd:=acothrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.663~~~~~$\displaystyle +\int{x^2\coth^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\coth^{-1}\frac{x}{a}}= +\frac{ax^2}{6}+\frac{x^3}{3}\coth^{-1}\frac{x}{a} ++\frac{a^3}{6}\ln(x^2-a^2) +$$ +<<*>>= +)clear all + +--S 98 +aa:=integrate(x^2*acoth(x/a),x) +--R +--R +--R 3 2 2 3 x + a 2 +--R a log(x - a ) + x log(-----) + a x +--R x - a +--R (1) ------------------------------------ +--R 6 +--R Type: Union(Expression Integer,...) +--E + +--S 99 +bb:=(a*x^2)/6+x^3/3*acoth(x/a)+a^3/6*log(x^2-a^2) +--R +--R 3 2 2 3 x 2 +--R a log(x - a ) + 2x acoth(-) + a x +--R a +--R (2) ----------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 100 +cc:=aa-bb +--R +--R 3 x + a 3 x +--R x log(-----) - 2x acoth(-) +--R x - a a +--R (3) -------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 101 +acothrule:=rule(acoth(x) == 1/2*log((x+1)/(x-1))) +--R +--R x + 1 +--R log(-----) +--R x - 1 +--R (4) acoth(x) == ---------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 102 14:663 Schaums and Axiom agree +dd:=acothrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.664~~~~~$\displaystyle +\int{\frac{\coth^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{\coth^{-1}(x/a)}{x}}= +-\left(\frac{a}{x}+\frac{(a/x)^3}{3^2}+\frac{(a/x)^5}{5^2}+\cdots\right) +$$ +<<*>>= +)clear all + +--S 103 14:664 Axiom cannot compute this integral +aa:=integrate(acoth(x/a)/x,x) +--R +--R +--I %P +--R x acoth(--) +--R ++ a +--I (1) | --------- d%P +--I ++ %P +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.665~~~~~$\displaystyle +\int{\frac{\coth^{-1}(x/a)}{x^2}}~dx$} +$$\int{\frac{\coth^{-1}(x/a)}{x^2}}= +-\frac{\coth^{-1}(x/a)}{x}+\frac{1}{2a}\ln\left(\frac{x^2}{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 104 +aa:=integrate(acoth(x/a)/x^2,x) +--R +--R +--R 2 2 x + a +--R - x log(x - a ) + 2x log(x) - a log(-----) +--R x - a +--R (1) ------------------------------------------- +--R 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 105 +bb:=-acoth(x/a)/x+1/(2*a)*log(x^2/(x^2-a^2)) +--R +--R 2 +--R x x +--R x log(-------) - 2a acoth(-) +--R 2 2 a +--R x - a +--R (2) ---------------------------- +--R 2a x +--R Type: Expression Integer +--E + +--S 106 +cc:=aa-bb +--R +--R (3) +--R 2 +--R 2 2 x + a x x +--R - x log(x - a ) + 2x log(x) - a log(-----) - x log(-------) + 2a acoth(-) +--R x - a 2 2 a +--R x - a +--R -------------------------------------------------------------------------- +--R 2a x +--R Type: Expression Integer +--E + +--S 107 +acothrule:=rule(acoth(x) == 1/2*log((x+1)/(x-1))) +--R +--R x + 1 +--R log(-----) +--R x - 1 +--R (4) acoth(x) == ---------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 108 +dd:=acothrule cc +--R +--R 2 +--R 2 2 x +--R - log(x - a ) + 2log(x) - log(-------) +--R 2 2 +--R x - a +--R (5) --------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 109 14:665 Schaums and Axiom agree +ee:=expandLog dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.666~~~~~$\displaystyle +\int{{\rm sech}^{-1}\frac{x}{a}}~dx$} +$$\int{{\rm sech}^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +x{\rm ~sech}^{-1}(x/a)+a\sin^{-1}(x/a),\quad{\rm sech}^{-1}(x/a)>0\\ +\\ +\displaystyle +x{\rm ~sech}^{-1}(x/a)-a\sin^{-1}(x/a),\quad{\rm sech}^{-1}(x/a)<0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 110 +aa:=integrate(asech(x/a),x) +--R +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R (1) x log(----------------) - 2a atan(----------------) +--R x x +--R Type: Union(Expression Integer,...) +--E + +--S 111 +bb1:=x*asech(x/a)+a*asin(x/a) +--R +--R x x +--R (2) a asin(-) + x asech(-) +--R a a +--R Type: Expression Integer +--E + +--S 112 +bb2:=x*asech(x/a)-a*asin(x/a) +--R +--R x x +--R (3) - a asin(-) + x asech(-) +--R a a +--R Type: Expression Integer +--E + +--S 113 +cc1:=aa-bb1 +--R +--R (4) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a x x +--R x log(----------------) - 2a atan(----------------) - a asin(-) - x asech(-) +--R x x a a +--R Type: Expression Integer +--E + +--S 114 +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a x x +--R x log(----------------) - 2a atan(----------------) + a asin(-) - x asech(-) +--R x x a a +--R Type: Expression Integer +--E + +--S 115 +asechrule:=rule(asech(x) == log(1/x+sqrt(1/x^2-1))) +--R +--R +--------+ +--R | 2 +--R |- x + 1 +--R x |-------- + 1 +--R | 2 +--R \| x +--R (6) asech(x) == log(----------------) +--R x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 116 +dd1:=asechrule cc1 +--R +--R (7) +--R +---------+ +--R | 2 2 +--R |- x + a +--R x |--------- + a +---------+ +--R | 2 | 2 2 +--R \| x \|- x + a + a +--R - x log(-----------------) + x log(----------------) +--R x x +--R + +--R +---------+ +--R | 2 2 +--R \|- x + a - a x +--R - 2a atan(----------------) - a asin(-) +--R x a +--R Type: Expression Integer +--E + +--S 117 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (8) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 118 +ee1:=asinrule dd1 +--R +--R (9) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R |- x + a |- x + a +--R x |--------- + a a |--------- - %i x +--R | 2 | 2 +--R \| x \| a +--R - x log(-----------------) - %i a log(--------------------) +--R x a +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R x log(----------------) - 2a atan(----------------) +--R x x +--R Type: Expression Complex Integer +--E + +--S 119 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (10) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 120 +ff1:=atanrule ee1 +--R +--R (11) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R |- x + a |- x + a +--R x |--------- + a a |--------- - %i x +--R | 2 | 2 +--R \| x \| a +--R - x log(-----------------) - %i a log(--------------------) +--R x a +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a - \|- x + a + %i x + a +--R x log(----------------) + %i a log(-------------------------) +--R x +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 121 +gg1:=expandLog ff1 +--R +--R (12) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R |- x + a |- x + a +--R - x log(x |--------- + a) - %i a log(a |--------- - %i x) +--R | 2 | 2 +--R \| x \| a +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R - %i a log(\|- x + a + %i x - a) + x log(\|- x + a + a) +--R + +--R +---------+ +--R | 2 2 +--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1) +--R Type: Expression Complex Integer +--E + +--S 122 +hh1:=rootSimp gg1 +--R +--R (13) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R | 2 2 +--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1) +--R Type: Expression Complex Integer +--E + +--S 123 14:666 Schaums and Axiom agree +ii1:=complexNormalize hh1 +--R +--R (14) 0 +--R Type: Expression Complex Integer +--E + +@ +Note that Axiom has a built-in assumption about the sign of asech(x/a). +We can see this if we simplify the cc2 value and show that it differs +by a complex value of x. +<<*>>= +--S 124 +dd2:=asechrule cc2 +--R +--R (15) +--R +---------+ +--R | 2 2 +--R |- x + a +--R x |--------- + a +---------+ +--R | 2 | 2 2 +--R \| x \|- x + a + a +--R - x log(-----------------) + x log(----------------) +--R x x +--R + +--R +---------+ +--R | 2 2 +--R \|- x + a - a x +--R - 2a atan(----------------) + a asin(-) +--R x a +--R Type: Expression Integer +--E + +--S 125 +ee2:=asinrule dd2 +--R +--R (16) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R |- x + a |- x + a +--R x |--------- + a a |--------- - %i x +--R | 2 | 2 +--R \| x \| a +--R - x log(-----------------) + %i a log(--------------------) +--R x a +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R x log(----------------) - 2a atan(----------------) +--R x x +--R Type: Expression Complex Integer +--E + +--S 126 +ff2:=atanrule ee2 +--R +--R (17) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R |- x + a |- x + a +--R x |--------- + a a |--------- - %i x +--R | 2 | 2 +--R \| x \| a +--R - x log(-----------------) + %i a log(--------------------) +--R x a +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a - \|- x + a + %i x + a +--R x log(----------------) + %i a log(-------------------------) +--R x +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 127 +gg2:=expandLog ff2 +--R +--R (18) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R |- x + a |- x + a +--R - x log(x |--------- + a) + %i a log(a |--------- - %i x) +--R | 2 | 2 +--R \| x \| a +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R - %i a log(\|- x + a + %i x - a) + x log(\|- x + a + a) +--R + +--R +---------+ +--R | 2 2 +--R %i a log(\|- x + a - %i x - a) - %i a log(a) + %i a log(- 1) +--R Type: Expression Complex Integer +--E + +--S 128 +hh2:=rootSimp gg2 +--R +--R (19) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) + %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R | 2 2 +--R %i a log(%i\|x - a - %i x - a) - %i a log(a) + %i a log(- 1) +--R Type: Expression Complex Integer +--E + +--S 129 +ii2:=complexNormalize hh2 +--R +--R +-------+ +--R | 2 2 +--R (20) 2%i a log(%i\|x - a - %i x) - 2%i a log(a) +--R Type: Expression Complex Integer +--E + +@ +Thus we can conjecture that solutions that show up with x in only the +imaginary part do so when the assumption of the sign of an inverse +function differs. + +\section{\cite{1}:14.667~~~~~$\displaystyle +\int{x{\rm ~sech}^{-1}\frac{x}{a}}~dx$} +$$\int{x{\rm ~sech}^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{2}x^2{\rm ~sech}^{-1}(x/a)-\frac{1}{2}a\sqrt{a^2-x^2}, +\quad{\rm sech}^{-1}(x/a)>0\\ +\\ +\displaystyle +\frac{1}{2}x^2{\rm ~sech}^{-1}(x/a)+\frac{1}{2}a\sqrt{a^2-x^2}, +\quad{\rm sech}^{-1}(x/a)<0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 130 +aa:=integrate(x*asech(x/a),x) +--R +--R +--R +---------+ +--R +---------+ | 2 2 +--R 2 | 2 2 2 \|- x + a + a 2 +--R (x \|- x + a - a x )log(----------------) + a x +--R x +--R (1) --------------------------------------------------- +--R +---------+ +--R | 2 2 +--R 2\|- x + a - 2a +--R Type: Union(Expression Integer,...) +--E + +--S 131 +bb1:=1/2*x^2*asech(x/a)-1/2*a*sqrt(a^2-x^2) +--R +--R +---------+ +--R | 2 2 2 x +--R - a\|- x + a + x asech(-) +--R a +--R (2) ---------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 132 +bb2:=1/2*x^2*asech(x/a)+1/2*a*sqrt(a^2-x^2) +--R +--R +---------+ +--R | 2 2 2 x +--R a\|- x + a + x asech(-) +--R a +--R (3) -------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 133 +cc1:=aa-bb1 +--R +--R +---------+ +--R | 2 2 +--R 2 \|- x + a + a 2 x 2 +--R x log(----------------) - x asech(-) - a +--R x a +--R (4) ----------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 134 +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +--R +---------+ | 2 2 +--R 2 | 2 2 2 \|- x + a + a +--R (x \|- x + a - a x )log(----------------) +--R x +--R + +--R +---------+ +--R 2 x 2 | 2 2 2 x 2 3 +--R (- x asech(-) + a )\|- x + a + a x asech(-) + 2a x - a +--R a a +--R / +--R +---------+ +--R | 2 2 +--R 2\|- x + a - 2a +--R Type: Expression Integer +--E + +--S 135 +asechrule:=rule(asech(x) == log(1/x+sqrt(1/x^2-1))) +--R +--R +--------+ +--R | 2 +--R |- x + 1 +--R x |-------- + 1 +--R | 2 +--R \| x +--R (6) asech(x) == log(----------------) +--R x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 136 +dd1:=asechrule cc1 +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R x |--------- + a +---------+ +--R | 2 | 2 2 +--R 2 \| x 2 \|- x + a + a 2 +--R - x log(-----------------) + x log(----------------) - a +--R x x +--R (7) --------------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 137 +ee1:=expandLog dd1 +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 |- x + a 2 | 2 2 2 +--R - x log(x |--------- + a) + x log(\|- x + a + a) - a +--R | 2 +--R \| x +--R (8) --------------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 138 14:667 Schaums and Axiom differ by a constant +ff1:=rootSimp ee1 +--R +--R 2 +--R a +--R (9) - -- +--R 2 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.668~~~~~$\displaystyle +\int{\frac{{\rm sech}^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{{\rm sech}^{-1}(x/a)}{x}}= +\left\{ +\begin{array}{l} +\displaystyle +-\frac{1}{2}\ln(a/x)\ln(4a/x)-\frac{(x/a)^2}{2\cdot 2\cdot 2} +-\frac{1\cdot 3(x/a)^4}{2\cdot 4\cdot 4\cdot 4} +-\cdots,\quad{\rm sech}^{-1}(x/a)>0\\ +\\ +\displaystyle +\frac{1}{2}\ln(a/x)\ln(4a/x)+\frac{(x/a)^2}{2\cdot 2\cdot 2} ++\frac{1\cdot 3(x/a)^4}{2\cdot 4\cdot 4\cdot 4} ++\cdots,\quad{\rm sech}^{-1}(x/a)<0\\ +\end{array} +\right. +$$ + +This is a interesting result since Axiom gives a closed form +solution to the problem but Schaums gives a series solution. +<<*>>= +)clear all + +--S 139 14:668 Axiom cannot compute this integral +aa:=integrate(asech(x/a)/x,x) +--R +--R +--I %P +--R x asech(--) +--R ++ a +--I (1) | --------- d%P +--I ++ %P +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.669~~~~~$\displaystyle +\int{{\rm csch}^{-1}\frac{x}{a}}~dx$} +$$\int{{\rm csch}^{-1}\frac{x}{a}}= +x{\rm ~csch}^{-1}\frac{x}{a}\pm a\sinh^{-1}\frac{x}{a} +\quad +{\rm if\ }x>0, -{\rm if\ }x<0 +$$ +<<*>>= +)clear all + +--S 140 +aa:=integrate(acsch(x/a),x) +--R +--R +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 \|x + a + a +--R (1) - a log(\|x + a - x) + x log(--------------) +--R x +--R Type: Union(Expression Integer,...) +--E + +--S 141 +bb1:=x*acsch(x/a)+a*asinh(x/a) +--R +--R x x +--R (2) a asinh(-) + x acsch(-) +--R a a +--R Type: Expression Integer +--E + +--S 142 +bb2:=x*acsch(x/a)-a*asinh(x/a) +--R +--R x x +--R (3) - a asinh(-) + x acsch(-) +--R a a +--R Type: Expression Integer +--E + +--S 143 +cc1:=aa-bb1 +--R +--R (4) +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 \|x + a + a x x +--R - a log(\|x + a - x) + x log(--------------) - a asinh(-) - x acsch(-) +--R x a a +--R Type: Expression Integer +--E + +--S 144 14:669 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 \|x + a + a x x +--R - a log(\|x + a - x) + x log(--------------) + a asinh(-) - x acsch(-) +--R x a a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.670~~~~~$\displaystyle +\int{x{\rm ~csch}^{-1}\frac{x}{a}}~dx$} +$$\int{x{\rm ~csch}^{-1}\frac{x}{a}}= +\frac{x^2}{2}{\rm ~csch}^{-1}\frac{x}{a}\pm \frac{a\sqrt{x^2+a^2}}{2} +\quad +{\rm if\ }x>0, -{\rm if\ }x<0 +$$ +<<*>>= +)clear all + +--S 145 +aa:=integrate(x*acsch(x/a),x) +--R +--R +--R +-------+ +--R +-------+ | 2 2 +-------+ +--R 2 | 2 2 3 \|x + a + a | 2 2 2 3 +--R (x \|x + a - x )log(--------------) - a x\|x + a + a x + a +--R x +--R (1) ------------------------------------------------------------------ +--R +-------+ +--R | 2 2 +--R 2\|x + a - 2x +--R Type: Union(Expression Integer,...) +--E + +--S 146 +bb1:=x^2/2*acsch(x/a)+(a*sqrt(x^2+a^2))/2 +--R +--R +-------+ +--R | 2 2 2 x +--R a\|x + a + x acsch(-) +--R a +--R (2) ------------------------ +--R 2 +--R Type: Expression Integer +--E + +--S 147 +bb2:=x^2/2*acsch(x/a)-(a*sqrt(x^2+a^2))/2 +--R +--R +-------+ +--R | 2 2 2 x +--R - a\|x + a + x acsch(-) +--R a +--R (3) -------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 148 +cc1:=aa-bb1 +--R +--R +-------+ +--R | 2 2 +--R 2 \|x + a + a 2 x +--R x log(--------------) - x acsch(-) +--R x a +--R (4) ---------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 149 14:670 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +-------+ +--R +-------+ | 2 2 +-------+ +--R 2 | 2 2 3 \|x + a + a 2 x | 2 2 +--R (x \|x + a - x )log(--------------) + (- x acsch(-) - 2a x)\|x + a +--R x a +--R + +--R 3 x 2 3 +--R x acsch(-) + 2a x + 2a +--R a +--R / +--R +-------+ +--R | 2 2 +--R 2\|x + a - 2x +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.671~~~~~$\displaystyle +\int{\frac{{\rm csch}^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{{\rm csch}^{-1}(x/a)}{x}}= +\left\{ +\begin{array}{lr} +\displaystyle +\frac{1}{2}\ln(x/a)\ln(4a/x)+\frac{1(x/a)^2}{2\cdot 2\cdot 2} +-\frac{1\cdot 3(x/a)^4}{2\cdot 4\cdot 4\cdot 4}+\cdots& +0a +\end{array} +\right. +$$ + +Schaums gives 3 different series expansions for this integral +but Axiom has computed a closed form. +<<*>>= +)clear all + +--S 150 14:671 Axiom cannot compute this integral +aa:=integrate(acsch(x/a)/x,x) +--R +--R +--I %P +--R x acsch(--) +--R ++ a +--I (1) | --------- d%P +--I ++ %P +--R Type: Union(Expression Integer,...) +--E + +@ + +\section{\cite{1}:14.672~~~~~$\displaystyle +\int{x^m\sinh^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\sinh^{-1}\frac{x}{a}}= +\frac{x^{m+1}}{m+1}\sinh^{-1}\frac{x}{a} +-\frac{1}{m+1}\int{\frac{x^{m+1}}{\sqrt{x^2+a^2}}} +$$ +<<*>>= +)clear all + +--S 151 14:672 Axiom cannot compute this integral +aa:=integrate(x^m*asinh(x/a),x) +--R +--R +--R x +--I ++ %P m +--I (1) | asinh(--)%P d%P +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.673~~~~~$\displaystyle +\int{x^m\cosh^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\cosh^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{lr} +\displaystyle +\frac{x^{m+1}}{m+1}\cosh^{-1}\frac{x}{a} +-\frac{1}{m+1}\int{\frac{x^{m+1}}{\sqrt{x^2-a^2}}},& +\quad\cosh^{-1}(x/a)>0\\ +\\ +\displaystyle +\frac{x^{m+1}}{m+1}\cosh^{-1}\frac{x}{a} ++\frac{1}{m+1}\int{\frac{x^{m+1}}{\sqrt{x^2-a^2}}},& +\quad\cosh^{-1}(x/a)<0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 152 14:673 Axiom cannot compute this integral +aa:=integrate(x^m*acosh(x/a),x) +--R +--R +--R x +--I ++ %P m +--I (1) | acosh(--)%P d%P +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.674~~~~~$\displaystyle +\int{x^m\tanh^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\tanh^{-1}\frac{x}{a}}= +\frac{x^{m+1}}{m+1}\tanh^{-1}\frac{x}{a} +-\frac{a}{m+1}\int{\frac{x^{m+1}}{a^2-x^2}} +$$ +<<*>>= +)clear all + +--S 153 14:674 Axiom cannot compute this integral +aa:=integrate(x^m*atanh(x/a),x) +--R +--R +--R x +--I ++ %P m +--I (1) | atanh(--)%P d%P +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.675~~~~~$\displaystyle +\int{x^m\coth^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\coth^{-1}\frac{x}{a}}= +\frac{x^{m+1}}{m+1}\coth^{-1}\frac{x}{a} +-\frac{a}{m+1}\int{\frac{x^{m+1}}{a^2-x^2}} +$$ +<<*>>= +)clear all + +--S 154 14:675 Axiom cannot compute this integral +aa:=integrate(x^m*acoth(x/a),x) +--R +--R +--R x +--I ++ %P m +--I (1) | acoth(--)%P d%P +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.676~~~~~$\displaystyle +\int{x^m{\rm ~sech}^{-1}\frac{x}{a}}~dx$} +$$\int{x^m{\rm ~sech}^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{lr} +\displaystyle +\frac{x^{m+1}}{m+1}{\rm ~sech}^{-1}\frac{x}{a} ++\frac{a}{m+1}\int{\frac{x^m}{\sqrt{a^2-x^2}}}& +{\rm sech}^{-1}(x/a)>0\\ +\\ +\displaystyle +\frac{x^{m+1}}{m+1}{\rm ~sech}^{-1}\frac{x}{a} +-\frac{a}{m+1}\int{\frac{x^m}{\sqrt{a^2-x^2}}}& +{\rm sech}^{-1}(x/a)<0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 155 14:676 Axiom cannot compute this integral +aa:=integrate(x^m*asech(x/a),x) +--R +--R +--R x +--I ++ %P m +--I (1) | asech(--)%P d%P +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.677~~~~~$\displaystyle +\int{x^m{\rm ~csch}^{-1}\frac{x}{a}}~dx$} +$$\int{x^m{\rm ~csch}^{-1}\frac{x}{a}}= +\frac{x^{m+1}}{m+1}{\rm ~csch}^{-1}\frac{x}{a} +\pm\frac{a}{m+1}\int{\frac{x^m}{\sqrt{x^2+a^2}}} +\quad+{\rm if\ }x>0 +~-{\rm if\ }x<0 +$$ +<<*>>= +)clear all + +--S 156 14:677 Axiom cannot compute this integral +aa:=integrate(x^m*acsch(x/a),x) +--R +--R +--R x +--I ++ %P m +--I (1) | acsch(--)%P d%P +--R ++ a +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp92-93 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum34.input.pdf b/src/axiom-website/CATS/schaum34.input.pdf new file mode 100644 index 0000000..1048462 --- /dev/null +++ b/src/axiom-website/CATS/schaum34.input.pdf @@ -0,0 +1,3928 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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+\frac{2(apx+3aq-2bp)}{3a^2}\sqrt{ax+b}$$ +<<*>>= +)spool schaum4.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate((p*x+q)/sqrt(a*x+b),x) +--R +--R +--R +-------+ +--R (2a p x + 6a q - 4b p)\|a x + b +--R (1) -------------------------------- +--R 2 +--R 3a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=(2*(a*p*x+3*a*q-2*b*p))/(3*a^2)*sqrt(a*x+b) +--R +--R +-------+ +--R (2a p x + 6a q - 4b p)\|a x + b +--R (2) -------------------------------- +--R 2 +--R 3a +--R Type: Expression Integer +--E + +--S 3 14:113 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.114~~~~~$\displaystyle +\int{\frac{dx}{(px+q)\sqrt{ax+b}}}$} +$$\int{\frac{1}{(px+q)\sqrt{ax+b}}}= +\left\{ +\begin{array}{l} +\frac{1}{\sqrt{bp-aq}\sqrt{p}}\ln\left( +\frac{\sqrt{p(ax+b)}-\sqrt{bp-aq}}{\sqrt{p(ax+b)}+\sqrt{bp-aq}}\right)\\ +\frac{2}{\sqrt{aq-bp}\sqrt{p}}\tan^{-1}\sqrt{\frac{p(ax+b)}{aq-bp}} +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 4 +aa:=integrate(1/((p*x+q)*sqrt(a*x+b)),x) +--R +--R +--R (1) +--R +--------------+ +--R 2 +-------+ | 2 +--R (2a p q - 2b p )\|a x + b + (a p x - a q + 2b p)\|- a p q + b p +--R log(------------------------------------------------------------------) +--R p x + q +--R [-----------------------------------------------------------------------, +--R +--------------+ +--R | 2 +--R \|- a p q + b p +--R +------------+ +--R | 2 +-------+ +--R \|a p q - b p \|a x + b +--R 2atan(-------------------------) +--R a q - b p +--R --------------------------------] +--R +------------+ +--R | 2 +--R \|a p q - b p +--R Type: Union(List Expression Integer,...) +--E + +--S 5 +aa1:=aa.1 +--R +--R (2) +--R +--------------+ +--R 2 +-------+ | 2 +--R (2a p q - 2b p )\|a x + b + (a p x - a q + 2b p)\|- a p q + b p +--R log(------------------------------------------------------------------) +--R p x + q +--R ----------------------------------------------------------------------- +--R +--------------+ +--R | 2 +--R \|- a p q + b p +--R Type: Expression Integer +--E + +--S 6 +aa2:=aa.2 +--R +--R +------------+ +--R | 2 +-------+ +--R \|a p q - b p \|a x + b +--R 2atan(-------------------------) +--R a q - b p +--R (3) -------------------------------- +--R +------------+ +--R | 2 +--R \|a p q - b p +--R Type: Expression Integer +--E + +--S 7 +bb1:=1/sqrt(b*p-a*q)*log((sqrt(p*(a*x+b))-sqrt(b*p-a*q))/(sqrt(p*(a*x+b))+sqrt(b*p-a*q))) +--R +--R +-----------+ +-----------+ +--R \|a p x + b p - \|- a q + b p +--R log(-------------------------------) +--R +-----------+ +-----------+ +--R \|a p x + b p + \|- a q + b p +--R (4) ------------------------------------ +--R +-----------+ +--R \|- a q + b p +--R Type: Expression Integer +--E + +--S 8 +bb2:=2/(sqrt(a*q-b*p)*sqrt(p))*atan(sqrt((p*(a*x+b))/(a*q-b*p))) +--R +--R +-----------+ +--R |a p x + b p +--R 2atan( |----------- ) +--R \| a q - b p +--R (5) --------------------- +--R +-+ +---------+ +--R \|p \|a q - b p +--R Type: Expression Integer +--E + +--S 9 +cc1:=aa1-bb1 +--R +--R (6) +--R +-----------+ +--R \|- a q + b p +--R * +--R +--------------+ +--R 2 +-------+ | 2 +--R (2a p q - 2b p )\|a x + b + (a p x - a q + 2b p)\|- a p q + b p +--R log(------------------------------------------------------------------) +--R p x + q +--R + +--R +--------------+ +-----------+ +-----------+ +--R | 2 \|a p x + b p - \|- a q + b p +--R - \|- a p q + b p log(-------------------------------) +--R +-----------+ +-----------+ +--R \|a p x + b p + \|- a q + b p +--R / +--R +--------------+ +--R | 2 +-----------+ +--R \|- a p q + b p \|- a q + b p +--R Type: Expression Integer +--E + +--S 10 +cc2:=aa1-bb2 +--R +--R (7) +--R +-+ +---------+ +--R \|p \|a q - b p +--R * +--R +--------------+ +--R 2 +-------+ | 2 +--R (2a p q - 2b p )\|a x + b + (a p x - a q + 2b p)\|- a p q + b p +--R log(------------------------------------------------------------------) +--R p x + q +--R + +--R +--------------+ +-----------+ +--R | 2 |a p x + b p +--R - 2\|- a p q + b p atan( |----------- ) +--R \| a q - b p +--R / +--R +--------------+ +--R | 2 +-+ +---------+ +--R \|- a p q + b p \|p \|a q - b p +--R Type: Expression Integer +--E + +--S 11 +cc3:=aa2-bb1 +--R +--R (8) +--R +------------+ +-----------+ +-----------+ +--R | 2 \|a p x + b p - \|- a q + b p +--R - \|a p q - b p log(-------------------------------) +--R +-----------+ +-----------+ +--R \|a p x + b p + \|- a q + b p +--R + +--R +------------+ +--R | 2 +-------+ +--R +-----------+ \|a p q - b p \|a x + b +--R 2\|- a q + b p atan(-------------------------) +--R a q - b p +--R / +--R +------------+ +--R +-----------+ | 2 +--R \|- a q + b p \|a p q - b p +--R Type: Expression Integer +--E + +--S 12 14:114 Axiom cannot simplify these answers +cc4:=aa2-bb2 +--R +--R (9) +--R +------------+ +--R | 2 +-------+ +--R +-+ +---------+ \|a p q - b p \|a x + b +--R 2\|p \|a q - b p atan(-------------------------) +--R a q - b p +--R + +--R +------------+ +-----------+ +--R | 2 |a p x + b p +--R - 2\|a p q - b p atan( |----------- ) +--R \| a q - b p +--R / +--R +------------+ +--R +-+ +---------+ | 2 +--R \|p \|a q - b p \|a p q - b p +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.115~~~~~$\displaystyle\int{\frac{\sqrt{ax+b}}{px+q}}~dx$} +$$\int{\frac{\sqrt{ax+b}}{px+q}}= +\left\{ +\begin{array}{l} +\frac{2\sqrt{ax+b}}{p}+\frac{\sqrt{bp-aq}}{p\sqrt{p}}\ln\left( +\frac{\sqrt{p(ax+b)}-\sqrt{bp-aq}}{\sqrt{p(ax+b)}+\sqrt{bp-aq}}\right)\\ +\frac{2\sqrt{ax+b}}{p}-\frac{2\sqrt{aq-bp}}{p\sqrt{p}} +\tan^{-1}\sqrt{\frac{p(ax+b)}{aq-bp}} +\end{array} +\right.$$ +<<*>>= +)clear all + +--S 13 +aa:=integrate(sqrt(a*x+b)/(p*x+q),x) +--R +--R +--R (1) +--R [ +--R +-----------+ +--R |- a q + b p +-------+ +--R +-----------+ - 2p |----------- \|a x + b + a p x - a q + 2b p +--R |- a q + b p \| p +--R |----------- log(-------------------------------------------------) +--R \| p p x + q +--R + +--R +-------+ +--R 2\|a x + b +--R / +--R p +--R , +--R +---------+ +-------+ +--R |a q - b p \|a x + b +-------+ +--R - 2 |--------- atan(------------ + 2\|a x + b +--R \| p +---------+ +--R |a q - b p +--R |--------- +--R \| p +--R -----------------------------------------------] +--R p +--R Type: Union(List Expression Integer,...) +--E + +--S 14 +aa1:=aa.1 +--R +--R (2) +--R +-----------+ +--R |- a q + b p +-------+ +--R +-----------+ - 2p |----------- \|a x + b + a p x - a q + 2b p +--R |- a q + b p \| p +--R |----------- log(-------------------------------------------------) +--R \| p p x + q +--R + +--R +-------+ +--R 2\|a x + b +--R / +--R p +--R Type: Expression Integer +--E + +--S 15 +aa2:=aa.2 +--R +--R +---------+ +-------+ +--R |a q - b p \|a x + b +-------+ +--R - 2 |--------- atan(------------ + 2\|a x + b +--R \| p +---------+ +--R |a q - b p +--R |--------- +--R \| p +--R (3) ----------------------------------------------- +--R p +--R Type: Expression Integer +--E + +--S 16 +bb1:=(2*sqrt(a*x+b))/p+sqrt(b*p-a*q)/(p*sqrt(p))*log((sqrt(p*(a*x+b))-sqrt(b*p-a*q))/(sqrt(p*(a*x+b))+sqrt(b*p-a*q))) +--R +--R +-----------+ +-----------+ +--R +-----------+ \|a p x + b p - \|- a q + b p +-+ +-------+ +--R \|- a q + b p log(-------------------------------) + 2\|p \|a x + b +--R +-----------+ +-----------+ +--R \|a p x + b p + \|- a q + b p +--R (4) -------------------------------------------------------------------- +--R +-+ +--R p\|p +--R Type: Expression Integer +--E + +--S 17 +bb2:=(2*sqrt(a*x+b))/p-(2*sqrt(a*q-b*p))/(p*sqrt(p))*atan(sqrt((p*(a*x+b))/(a*q-b*p))) +--R +--R +-----------+ +--R +---------+ |a p x + b p +-+ +-------+ +--R - 2\|a q - b p atan( |----------- ) + 2\|p \|a x + b +--R \| a q - b p +--R (5) ----------------------------------------------------- +--R +-+ +--R p\|p +--R Type: Expression Integer +--E + +--S 18 +cc1:=aa1-bb1 +--R +--R (6) +--R +-----------+ +-----------+ +--R +-----------+ \|a p x + b p - \|- a q + b p +--R - \|- a q + b p log(-------------------------------) +--R +-----------+ +-----------+ +--R \|a p x + b p + \|- a q + b p +--R + +--R +-----------+ +--R |- a q + b p +-------+ +--R +-----------+ - 2p |----------- \|a x + b + a p x - a q + 2b p +--R |- a q + b p +-+ \| p +--R |----------- \|p log(-------------------------------------------------) +--R \| p p x + q +--R / +--R +-+ +--R p\|p +--R Type: Expression Integer +--E + +--S 19 +cc2:=aa1-bb2 +--R +--R (7) +--R +-----------+ +--R |- a q + b p +-------+ +--R +-----------+ - 2p |----------- \|a x + b + a p x - a q + 2b p +--R |- a q + b p +-+ \| p +--R |----------- \|p log(-------------------------------------------------) +--R \| p p x + q +--R + +--R +-----------+ +--R +---------+ |a p x + b p +--R 2\|a q - b p atan( |----------- ) +--R \| a q - b p +--R / +--R +-+ +--R p\|p +--R Type: Expression Integer +--E + +--S 20 +cc3:=aa2-bb1 +--R +--R (8) +--R +-----------+ +-----------+ +--R +-----------+ \|a p x + b p - \|- a q + b p +--R - \|- a q + b p log(-------------------------------) +--R +-----------+ +-----------+ +--R \|a p x + b p + \|- a q + b p +--R + +--R +---------+ +-------+ +--R +-+ |a q - b p \|a x + b +--R - 2\|p |--------- atan(------------) +--R \| p +---------+ +--R |a q - b p +--R |--------- +--R \| p +--R / +--R +-+ +--R p\|p +--R Type: Expression Integer +--E + +--S 21 14:115 Axiom cannot simplify these answers +cc4:=aa2-bb2 +--R +--R (9) +--R +---------+ +-------+ +-----------+ +--R +-+ |a q - b p \|a x + b +---------+ |a p x + b p +--R - 2\|p |--------- atan(------------) + 2\|a q - b p atan( |----------- ) +--R \| p +---------+ \| a q - b p +--R |a q - b p +--R |--------- +--R \| p +--R ------------------------------------------------------------------------- +--R +-+ +--R p\|p +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.116~~~~~$\displaystyle\int{(px+b)^n\sqrt{ax+b}}~dx$} +$$\int{(px+b)^n\sqrt{ax+b}}= +\frac{2(px+q)^{n+1}\sqrt{ax+b}}{(2n+3)p}+\frac{bp-aq}{(2n+3)p} +\int{\frac{(px+q)^n}{\sqrt{ax+b}}}$$ + +<<*>>= +)clear all + +--S 22 14:116 Axiom cannot compute this integral +aa:=integrate((p*x+q)^n*sqrt(a*x+b),x) +--R +--R +--R x +--R ++ n +--------+ +--I (1) | (q + %L p) \|b + %L a d%L +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.117~~~~~$\displaystyle +\int{\frac{dx}{(px+b)^n\sqrt{ax+b}}}$} +$$\int{\frac{1}{(px+b)^n\sqrt{ax+b}}}= +\frac{\sqrt{ax+b}}{(n-1)(aq-bp)(px+q)^{n-1}}+ +\frac{(2n-3)a}{2(n-1)(aq-bp)} +\int{\frac{1}{(px+q)^{n-1}\sqrt{ax+b}}}$$ + +<<*>>= +)clear all + +--S 23 14:117 Axiom cannot compute this integral +aa:=integrate(1/((p*x+q)^n*sqrt(a*x+b)),x) +--R +--R +--R x +--R ++ 1 +--I (1) | ---------------------- d%L +--R ++ n +--------+ +--I (q + %L p) \|b + %L a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.118~~~~~$\displaystyle +\int{\frac{(px+q)^n}{\sqrt{ax+b}}}~dx$} +$$\int{\frac{(px+q)^n}{\sqrt{ax+b}}}= +\frac{2(px+q)^n\sqrt{ax+b}}{(2n+1)a}+ +\frac{2n(aq-bp)}{(2n+1)a} +\int{\frac{(px+q)^{n-1}}{\sqrt{ax+b}}}$$ +<<*>>= +)clear all + +--S 24 14:118 Axiom cannot compute this integral +aa:=integrate((p*x+q)^n/sqrt(a*x+b),x) +--R +--R +--R x n +--I ++ (q + %L p) +--I (1) | ----------- d%L +--R ++ +--------+ +--I \|b + %L a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.119~~~~~$\displaystyle +\int{\frac{\sqrt{ax+b}}{(px+q)^n}}~dx$} +$$\int{\frac{\sqrt{ax+b}}{(px+q)^n}}= +\frac{-\sqrt{ax+b}}{(n-1)p(px+q)^{n-1}}+ +\frac{a}{2(n-1)p}\int{\frac{1}{(px+q)^{n-1}\sqrt{ax+b}}}$$ +<<*>>= +)clear all + +--S 25 14:119 Axiom cannot compute this integral +aa:=integrate(sqrt(a*x+b)/(p*x+q)^n,x) +--R +--R +--R x +--------+ +--I ++ \|b + %L a +--I (1) | ----------- d%L +--R ++ n +--I (q + %L p) +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p63 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum4.input.pdf b/src/axiom-website/CATS/schaum4.input.pdf new file mode 100644 index 0000000..a9291fc --- /dev/null +++ b/src/axiom-website/CATS/schaum4.input.pdf @@ -0,0 +1,1765 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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+\begin{array}{l} +\frac{2}{\sqrt{ap}}\ln\left(\sqrt{a(px+q)}+\sqrt{p(ax+b)}\right)\\ +\frac{2}{\sqrt{-ap}}\tan^{-1}\sqrt{\frac{-p(ax+b)}{a(px+b)}} +\end{array} +\right.$$ +<<*>>= +)spool schaum5.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(1/sqrt((a*x+b)*(p*x+q)),x) +--R +--R +--R (1) +--R [ +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R / +--R +---+ +--R \|a p +--R , +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R 2atan(-------------------------------------------------------) +--R a p x +--R --------------------------------------------------------------] +--R +-----+ +--R \|- a p +--R Type: Union(List Expression Integer,...) +--E + +--S 2 +aa1:=aa.1 +--R +--R (2) +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R / +--R +---+ +--R \|a p +--R Type: Expression Integer +--E + +--S 3 +aa2:=aa.2 +--R +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R 2atan(-------------------------------------------------------) +--R a p x +--R (3) -------------------------------------------------------------- +--R +-----+ +--R \|- a p +--R Type: Expression Integer +--E + +--S 4 +bb1:=2/sqrt(a*p)*log(sqrt(a*(p*x+q))+sqrt(p*(a*x+b))) +--R +--R +-----------+ +-----------+ +--R 2log(\|a p x + a q + \|a p x + b p ) +--R (4) ------------------------------------- +--R +---+ +--R \|a p +--R Type: Expression Integer +--E + +--S 5 +bb2:=2/sqrt(-a*p)*atan(sqrt((-p*(a*x+b))/(a*(p*x+q)))) +--R +--R +-------------+ +--R |- a p x - b p +--R 2atan( |------------- ) +--R \| a p x + a q +--R (5) ----------------------- +--R +-----+ +--R \|- a p +--R Type: Expression Integer +--E + +--S 6 +cc1:=aa1-bb1 +--R +--R (6) +--R +-----------+ +-----------+ +--R - 2log(\|a p x + a q + \|a p x + b p ) +--R + +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R / +--R +---+ +--R \|a p +--R Type: Expression Integer +--E + +--S 7 +cc2:=aa1-bb2 +--R +--R (7) +--R +-----+ +--R \|- a p +--R * +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R + +--R +-------------+ +--R +---+ |- a p x - b p +--R - 2\|a p atan( |------------- ) +--R \| a p x + a q +--R / +--R +-----+ +---+ +--R \|- a p \|a p +--R Type: Expression Integer +--E + +--S 8 +cc3:=aa2-bb1 +--R +--R (8) +--R +-----+ +-----------+ +-----------+ +--R - 2\|- a p log(\|a p x + a q + \|a p x + b p ) +--R + +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R +---+ \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R 2\|a p atan(-------------------------------------------------------) +--R a p x +--R / +--R +-----+ +---+ +--R \|- a p \|a p +--R Type: Expression Integer +--E + +--S 9 14:120 Axiom cannot simplify these answers +cc4:=aa2-bb2 +--R +--R (9) +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R 2atan(-------------------------------------------------------) +--R a p x +--R + +--R +-------------+ +--R |- a p x - b p +--R - 2atan( |------------- ) +--R \| a p x + a q +--R / +--R +-----+ +--R \|- a p +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.121~~~~~$\displaystyle +\int{\frac{x~dx}{\sqrt{(ax+b)(px+q)}}}$} +$$\int{\frac{x}{\sqrt{(ax+b)(px+q)}}}= +\frac{\sqrt{(ax+b)(px+q)}}{ap}-\frac{bp+aq}{2ap} +\int{\frac{1}{\sqrt{(ax+b)(px+q)}}} +$$ +<<*>>= +)clear all + +--S 10 +aa:=integrate(x/sqrt((a*x+b)*(p*x+q)),x) +--R +--R +--R (1) +--R [ +--R +---------------------------+ +--R +---+ | 2 +--R (2a q + 2b p)\|b q \|a p x + (a q + b p)x + b q +--R + +--R 2 2 2 2 2 2 +--R (- a q - 2a b p q - b p )x - 2a b q - 2b p q +--R * +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q + 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R - 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R + +--R +---------------------------+ +--R +---+ | 2 +--R (- 2a q - 2b p)x\|a p \|a p x + (a q + b p)x + b q +--R + +--R 2 +---+ +---+ +--R (4a p x + (2a q + 2b p)x)\|a p \|b q +--R / +--R +---------------------------+ +--R +---+ +---+ | 2 +--R 4a p\|a p \|b q \|a p x + (a q + b p)x + b q +--R + +--R 2 2 +---+ +--R ((- 2a p q - 2a b p )x - 4a b p q)\|a p +--R , +--R +--R +---------------------------+ +--R +---+ | 2 +--R (- 2a q - 2b p)\|b q \|a p x + (a q + b p)x + b q +--R + +--R 2 2 2 2 2 2 +--R (a q + 2a b p q + b p )x + 2a b q + 2b p q +--R * +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R atan(-------------------------------------------------------) +--R a p x +--R + +--R +---------------------------+ +--R +-----+ | 2 +--R (- a q - b p)x\|- a p \|a p x + (a q + b p)x + b q +--R + +--R 2 +-----+ +---+ +--R (2a p x + (a q + b p)x)\|- a p \|b q +--R / +--R +---------------------------+ +--R +-----+ +---+ | 2 +--R 2a p\|- a p \|b q \|a p x + (a q + b p)x + b q +--R + +--R 2 2 +-----+ +--R ((- a p q - a b p )x - 2a b p q)\|- a p +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 11 +bb1:=integrate(1/(sqrt(a*x+b)*(p*x+q)),x) +--R +--R (2) +--R +--------------+ +--R 2 +-------+ | 2 +--R (2a p q - 2b p )\|a x + b + (a p x - a q + 2b p)\|- a p q + b p +--R log(------------------------------------------------------------------) +--R p x + q +--R [-----------------------------------------------------------------------, +--R +--------------+ +--R | 2 +--R \|- a p q + b p +--R +------------+ +--R | 2 +-------+ +--R \|a p q - b p \|a x + b +--R 2atan(-------------------------) +--R a q - b p +--R --------------------------------] +--R +------------+ +--R | 2 +--R \|a p q - b p +--R Type: Union(List Expression Integer,...) +--E + +--S 12 +bb2:=sqrt((a*x+b)*(p*x+q))/(a*p)-(b*p+a*q)/(2*a*p) +--R +--R +---------------------------+ +--R | 2 +--R 2\|a p x + (a q + b p)x + b q - a q - b p +--R (3) ------------------------------------------- +--R 2a p +--R Type: Expression Integer +--E + +--S 13 +bb:=bb2*bb1 +--R +--R (4) +--R [ +--R +---------------------------+ +--R | 2 +--R (2\|a p x + (a q + b p)x + b q - a q - b p) +--R * +--R +--------------+ +--R 2 +-------+ | 2 +--R (2a p q - 2b p )\|a x + b + (a p x - a q + 2b p)\|- a p q + b p +--R log(------------------------------------------------------------------) +--R p x + q +--R / +--R +--------------+ +--R | 2 +--R 2a p\|- a p q + b p +--R , +--R +------------+ +--R +---------------------------+ | 2 +-------+ +--R | 2 \|a p q - b p \|a x + b +--R (2\|a p x + (a q + b p)x + b q - a q - b p)atan(-------------------------) +--R a q - b p +--R ---------------------------------------------------------------------------- +--R +------------+ +--R | 2 +--R a p\|a p q - b p +--R ] +--R Type: Vector Expression Integer +--E + +--S 14 14:121 Axiom cannot simplify this answer +cc:=aa-bb +--R +--R (5) +--R [ +--R +---+ +---+ +---+ +--R ((2a q + 2b p)\|a p \|b q + ((2a q + 2b p)x + 4b q)\|a p ) +--R * +--R +---------------------------+ +--R | 2 +--R \|a p x + (a q + b p)x + b q +--R + +--R 2 +---+ +---+ +--R (- 4a p x + (- 4a q - 4b p)x - 4b q)\|a p \|b q +--R + +--R 2 2 2 2 2 2 +---+ +--R ((- a q - 2a b p q - b p )x - 2a b q - 2b p q)\|a p +--R * +--R +--------------+ +--R 2 +-------+ | 2 +--R (2a p q - 2b p )\|a x + b + (a p x - a q + 2b p)\|- a p q + b p +--R log(------------------------------------------------------------------) +--R p x + q +--R + +--R +--------------+ +---------------------------+ +--R | 2 +---+ | 2 +--R (2a q + 2b p)\|- a p q + b p \|b q \|a p x + (a q + b p)x + b q +--R + +--R +--------------+ +--R 2 2 2 2 2 2 | 2 +--R ((- a q - 2a b p q - b p )x - 2a b q - 2b p q)\|- a p q + b p +--R * +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q + 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R - 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R + +--R +--------------+ +---------------------------+ +--R | 2 +---+ | 2 +--R (- 2a q - 2b p)x\|- a p q + b p \|a p \|a p x + (a q + b p)x + b q +--R + +--R +--------------+ +--R 2 | 2 +---+ +---+ +--R (4a p x + (2a q + 2b p)x)\|- a p q + b p \|a p \|b q +--R / +--R +--------------+ +---------------------------+ +--R | 2 +---+ +---+ | 2 +--R 4a p\|- a p q + b p \|a p \|b q \|a p x + (a q + b p)x + b q +--R + +--R +--------------+ +--R 2 2 | 2 +---+ +--R ((- 2a p q - 2a b p )x - 4a b p q)\|- a p q + b p \|a p +--R , +--R +--R +------------+ +---------------------------+ +--R +---+ | 2 | 2 +--R (- 2a q - 2b p)\|b q \|a p q - b p \|a p x + (a q + b p)x + b q +--R + +--R +------------+ +--R 2 2 2 2 2 2 | 2 +--R ((a q + 2a b p q + b p )x + 2a b q + 2b p q)\|a p q - b p +--R * +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R atan(-------------------------------------------------------) +--R a p x +--R + +--R +-----+ +---+ +-----+ +--R ((2a q + 2b p)\|- a p \|b q + ((2a q + 2b p)x + 4b q)\|- a p ) +--R * +--R +---------------------------+ +--R | 2 +--R \|a p x + (a q + b p)x + b q +--R + +--R 2 +-----+ +---+ +--R (- 4a p x + (- 4a q - 4b p)x - 4b q)\|- a p \|b q +--R + +--R 2 2 2 2 2 2 +-----+ +--R ((- a q - 2a b p q - b p )x - 2a b q - 2b p q)\|- a p +--R * +--R +------------+ +--R | 2 +-------+ +--R \|a p q - b p \|a x + b +--R atan(-------------------------) +--R a q - b p +--R + +--R +------------+ +---------------------------+ +--R +-----+ | 2 | 2 +--R (- a q - b p)x\|- a p \|a p q - b p \|a p x + (a q + b p)x + b q +--R + +--R +------------+ +--R 2 +-----+ +---+ | 2 +--R (2a p x + (a q + b p)x)\|- a p \|b q \|a p q - b p +--R / +--R +------------+ +---------------------------+ +--R +-----+ +---+ | 2 | 2 +--R 2a p\|- a p \|b q \|a p q - b p \|a p x + (a q + b p)x + b q +--R + +--R +------------+ +--R 2 2 +-----+ | 2 +--R ((- a p q - a b p )x - 2a b p q)\|- a p \|a p q - b p +--R ] +--R Type: Vector Expression Integer +--E +@ + +\section{\cite{1}:14.122~~~~~$\displaystyle\int{\sqrt{(ax+b)(px+q)}}~dx$} +$$\int{\sqrt{(ax+b)(px+q)}}= +\frac{2apx+bp+aq}{4ap}\sqrt{(ax+b)(px+q)}- +\frac{(bp-aq)^2}{8ap}\int{\frac{1}{\sqrt{(ax+b)(px+q)}}} +$$ +<<*>>= +)clear all + +--S 15 +aa:=integrate(sqrt((a*x+b)*(p*x+q)),x) +--R +--R +--R (1) +--R [ +--R 3 3 2 2 2 2 3 3 2 3 2 2 +--R (4a q - 4a b p q - 4a b p q + 4b p )x + 8a b q - 16a b p q +--R + +--R 3 2 +--R 8b p q +--R * +--R +---------------------------+ +--R +---+ | 2 +--R \|b q \|a p x + (a q + b p)x + b q +--R + +--R 4 4 3 3 2 2 2 2 3 3 4 4 2 +--R (- a q - 4a b p q + 10a b p q - 4a b p q - b p )x +--R + +--R 3 4 2 2 3 3 2 2 4 3 2 2 4 +--R (- 8a b q + 8a b p q + 8a b p q - 8b p q)x - 8a b q +--R + +--R 3 3 4 2 2 +--R 16a b p q - 8b p q +--R * +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q + 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R - 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R + +--R 3 2 2 2 2 3 3 +--R (- 4a p q - 24a b p q - 4a b p )x +--R + +--R 3 3 2 2 2 2 3 3 2 +--R (- 2a q - 46a b p q - 46a b p q - 2b p )x +--R + +--R 2 3 2 2 3 2 +--R (- 8a b q - 48a b p q - 8b p q)x +--R * +--R +---------------------------+ +--R +---+ | 2 +--R \|a p \|a p x + (a q + b p)x + b q +--R + +--R 3 2 2 3 4 3 2 2 2 2 3 3 +--R (16a p q + 16a b p )x + (24a p q + 80a b p q + 24a b p )x +--R + +--R 3 3 2 2 2 2 3 3 2 +--R (6a q + 74a b p q + 74a b p q + 6b p )x +--R + +--R 2 3 2 2 3 2 +--R (8a b q + 48a b p q + 8b p q)x +--R * +--R +---+ +---+ +--R \|a p \|b q +--R / +--R 2 2 +---+ +---+ +--R ((32a p q + 32a b p )x + 64a b p q)\|a p \|b q +--R * +--R +---------------------------+ +--R | 2 +--R \|a p x + (a q + b p)x + b q +--R + +--R 3 2 2 2 2 3 2 2 2 2 2 +--R (- 8a p q - 48a b p q - 8a b p )x + (- 64a b p q - 64a b p q)x +--R + +--R 2 2 +--R - 64a b p q +--R * +--R +---+ +--R \|a p +--R , +--R +--R 3 3 2 2 2 2 3 3 2 3 +--R (- 4a q + 4a b p q + 4a b p q - 4b p )x - 8a b q +--R + +--R 2 2 3 2 +--R 16a b p q - 8b p q +--R * +--R +---------------------------+ +--R +---+ | 2 +--R \|b q \|a p x + (a q + b p)x + b q +--R + +--R 4 4 3 3 2 2 2 2 3 3 4 4 2 +--R (a q + 4a b p q - 10a b p q + 4a b p q + b p )x +--R + +--R 3 4 2 2 3 3 2 2 4 3 2 2 4 3 3 +--R (8a b q - 8a b p q - 8a b p q + 8b p q)x + 8a b q - 16a b p q +--R + +--R 4 2 2 +--R 8b p q +--R * +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R atan(-------------------------------------------------------) +--R a p x +--R + +--R 3 2 2 2 2 3 3 +--R (- 2a p q - 12a b p q - 2a b p )x +--R + +--R 3 3 2 2 2 2 3 3 2 +--R (- a q - 23a b p q - 23a b p q - b p )x +--R + +--R 2 3 2 2 3 2 +--R (- 4a b q - 24a b p q - 4b p q)x +--R * +--R +---------------------------+ +--R +-----+ | 2 +--R \|- a p \|a p x + (a q + b p)x + b q +--R + +--R 3 2 2 3 4 3 2 2 2 2 3 3 +--R (8a p q + 8a b p )x + (12a p q + 40a b p q + 12a b p )x +--R + +--R 3 3 2 2 2 2 3 3 2 +--R (3a q + 37a b p q + 37a b p q + 3b p )x +--R + +--R 2 3 2 2 3 2 +--R (4a b q + 24a b p q + 4b p q)x +--R * +--R +-----+ +---+ +--R \|- a p \|b q +--R / +--R 2 2 +-----+ +---+ +--R ((16a p q + 16a b p )x + 32a b p q)\|- a p \|b q +--R * +--R +---------------------------+ +--R | 2 +--R \|a p x + (a q + b p)x + b q +--R + +--R 3 2 2 2 2 3 2 2 2 2 2 +--R (- 4a p q - 24a b p q - 4a b p )x + (- 32a b p q - 32a b p q)x +--R + +--R 2 2 +--R - 32a b p q +--R * +--R +-----+ +--R \|- a p +--R ] +--R Type: Union(List Expression Integer,...) +--E +@ +Since there are two parts to the aa variable we split them: +<<*>>= +--S 16 +aa1:=aa.1 +--R +--R (2) +--R 3 3 2 2 2 2 3 3 2 3 2 2 +--R (4a q - 4a b p q - 4a b p q + 4b p )x + 8a b q - 16a b p q +--R + +--R 3 2 +--R 8b p q +--R * +--R +---------------------------+ +--R +---+ | 2 +--R \|b q \|a p x + (a q + b p)x + b q +--R + +--R 4 4 3 3 2 2 2 2 3 3 4 4 2 +--R (- a q - 4a b p q + 10a b p q - 4a b p q - b p )x +--R + +--R 3 4 2 2 3 3 2 2 4 3 2 2 4 3 3 +--R (- 8a b q + 8a b p q + 8a b p q - 8b p q)x - 8a b q + 16a b p q +--R + +--R 4 2 2 +--R - 8b p q +--R * +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q + 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R - 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R + +--R 3 2 2 2 2 3 3 +--R (- 4a p q - 24a b p q - 4a b p )x +--R + +--R 3 3 2 2 2 2 3 3 2 +--R (- 2a q - 46a b p q - 46a b p q - 2b p )x +--R + +--R 2 3 2 2 3 2 +--R (- 8a b q - 48a b p q - 8b p q)x +--R * +--R +---------------------------+ +--R +---+ | 2 +--R \|a p \|a p x + (a q + b p)x + b q +--R + +--R 3 2 2 3 4 3 2 2 2 2 3 3 +--R (16a p q + 16a b p )x + (24a p q + 80a b p q + 24a b p )x +--R + +--R 3 3 2 2 2 2 3 3 2 +--R (6a q + 74a b p q + 74a b p q + 6b p )x +--R + +--R 2 3 2 2 3 2 +--R (8a b q + 48a b p q + 8b p q)x +--R * +--R +---+ +---+ +--R \|a p \|b q +--R / +--R 2 2 +---+ +---+ +--R ((32a p q + 32a b p )x + 64a b p q)\|a p \|b q +--R * +--R +---------------------------+ +--R | 2 +--R \|a p x + (a q + b p)x + b q +--R + +--R 3 2 2 2 2 3 2 2 2 2 2 +--R (- 8a p q - 48a b p q - 8a b p )x + (- 64a b p q - 64a b p q)x +--R + +--R 2 2 +--R - 64a b p q +--R * +--R +---+ +--R \|a p +--R Type: Expression Integer +--E + +--S 17 +aa2:=aa.2 +--R +--R (3) +--R 3 3 2 2 2 2 3 3 2 3 2 2 +--R (- 4a q + 4a b p q + 4a b p q - 4b p )x - 8a b q + 16a b p q +--R + +--R 3 2 +--R - 8b p q +--R * +--R +---------------------------+ +--R +---+ | 2 +--R \|b q \|a p x + (a q + b p)x + b q +--R + +--R 4 4 3 3 2 2 2 2 3 3 4 4 2 +--R (a q + 4a b p q - 10a b p q + 4a b p q + b p )x +--R + +--R 3 4 2 2 3 3 2 2 4 3 2 2 4 3 3 +--R (8a b q - 8a b p q - 8a b p q + 8b p q)x + 8a b q - 16a b p q +--R + +--R 4 2 2 +--R 8b p q +--R * +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R atan(-------------------------------------------------------) +--R a p x +--R + +--R 3 2 2 2 2 3 3 +--R (- 2a p q - 12a b p q - 2a b p )x +--R + +--R 3 3 2 2 2 2 3 3 2 +--R (- a q - 23a b p q - 23a b p q - b p )x +--R + +--R 2 3 2 2 3 2 +--R (- 4a b q - 24a b p q - 4b p q)x +--R * +--R +---------------------------+ +--R +-----+ | 2 +--R \|- a p \|a p x + (a q + b p)x + b q +--R + +--R 3 2 2 3 4 3 2 2 2 2 3 3 +--R (8a p q + 8a b p )x + (12a p q + 40a b p q + 12a b p )x +--R + +--R 3 3 2 2 2 2 3 3 2 +--R (3a q + 37a b p q + 37a b p q + 3b p )x +--R + +--R 2 3 2 2 3 2 +--R (4a b q + 24a b p q + 4b p q)x +--R * +--R +-----+ +---+ +--R \|- a p \|b q +--R / +--R 2 2 +-----+ +---+ +--R ((16a p q + 16a b p )x + 32a b p q)\|- a p \|b q +--R * +--R +---------------------------+ +--R | 2 +--R \|a p x + (a q + b p)x + b q +--R + +--R 3 2 2 2 2 3 2 2 2 2 2 +--R (- 4a p q - 24a b p q - 4a b p )x + (- 32a b p q - 32a b p q)x +--R + +--R 2 2 +--R - 32a b p q +--R * +--R +-----+ +--R \|- a p +--R Type: Expression Integer +--E +@ +We break the books answer into 3 parts, the first term, the coefficient +of the second term, and the integral. +<<*>>= +--S 18 +bba:=((2*a*p*x+b*p+a*q)/(4*a*p))*sqrt((a*x+b)*(p*x+q)) +--R +--R +---------------------------+ +--R | 2 +--R (2a p x + a q + b p)\|a p x + (a q + b p)x + b q +--R (4) -------------------------------------------------- +--R 4a p +--R Type: Expression Integer +--E + +--S 19 +bbb:=-(b*p-a*q)^2/(8*a*p) +--R +--R 2 2 2 2 +--R - a q + 2a b p q - b p +--R (5) ------------------------ +--R 8a p +--R Type: Fraction Polynomial Integer +--E + +--S 20 +bbc:=integrate(1/sqrt((a*x+b)*(p*x+q)),x) +--R +--R (6) +--R [ +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R / +--R +---+ +--R \|a p +--R , +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R 2atan(-------------------------------------------------------) +--R a p x +--R --------------------------------------------------------------] +--R +-----+ +--R \|- a p +--R Type: Union(List Expression Integer,...) +--E +@ +Since the integral has two parts, we break them apart +<<*>>= +--S 21 +bbc1:=bbc.1 +--R +--R (7) +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R / +--R +---+ +--R \|a p +--R Type: Expression Integer +--E + +--S 22 +bbc2:=bbc.2 +--R +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R 2atan(-------------------------------------------------------) +--R a p x +--R (8) -------------------------------------------------------------- +--R +-----+ +--R \|- a p +--R Type: Expression Integer +--E +@ +And now we construct the two bb answers based on the integral parts +<<*>>= +--S 23 +bb1:=bba+bbb*bbc1 +--R +--R (9) +--R 2 2 2 2 +--R (- a q + 2a b p q - b p ) +--R * +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R + +--R +---------------------------+ +--R +---+ | 2 +--R (4a p x + 2a q + 2b p)\|a p \|a p x + (a q + b p)x + b q +--R / +--R +---+ +--R 8a p\|a p +--R Type: Expression Integer +--E + +--S 24 +bb2:=bba+bbb*bbc2 +--R +--R (10) +--R 2 2 2 2 +--R (- a q + 2a b p q - b p ) +--R * +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R atan(-------------------------------------------------------) +--R a p x +--R + +--R +---------------------------+ +--R +-----+ | 2 +--R (2a p x + a q + b p)\|- a p \|a p x + (a q + b p)x + b q +--R / +--R +-----+ +--R 4a p\|- a p +--R Type: Expression Integer +--E +@ +So there are 4 possible combinations that might yield an answer. +We construct all four. +<<*>>= +--S 25 +cc1:=aa1-bb1 +--R +--R (11) +--R 3 3 2 2 2 2 3 3 2 3 2 2 +--R (4a q - 4a b p q - 4a b p q + 4b p )x + 8a b q - 16a b p q +--R + +--R 3 2 +--R 8b p q +--R * +--R +---------------------------+ +--R +---+ | 2 +--R \|b q \|a p x + (a q + b p)x + b q +--R + +--R 4 4 3 3 2 2 2 2 3 3 4 4 2 +--R (- a q - 4a b p q + 10a b p q - 4a b p q - b p )x +--R + +--R 3 4 2 2 3 3 2 2 4 3 2 2 4 3 3 +--R (- 8a b q + 8a b p q + 8a b p q - 8b p q)x - 8a b q + 16a b p q +--R + +--R 4 2 2 +--R - 8b p q +--R * +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q + 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R - 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R + +--R 3 3 2 2 2 2 3 3 2 3 2 2 +--R (4a q - 4a b p q - 4a b p q + 4b p )x + 8a b q - 16a b p q +--R + +--R 3 2 +--R 8b p q +--R * +--R +---------------------------+ +--R +---+ | 2 +--R \|b q \|a p x + (a q + b p)x + b q +--R + +--R 4 4 3 3 2 2 2 2 3 3 4 4 2 +--R (- a q - 4a b p q + 10a b p q - 4a b p q - b p )x +--R + +--R 3 4 2 2 3 3 2 2 4 3 2 2 4 3 3 +--R (- 8a b q + 8a b p q + 8a b p q - 8b p q)x - 8a b q + 16a b p q +--R + +--R 4 2 2 +--R - 8b p q +--R * +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R + +--R 2 3 2 2 3 2 2 3 3 2 +---+ +--R ((8a b q + 16a b p q + 8b p q)x + 16a b q + 16b p q )\|a p +--R * +--R +---------------------------+ +--R | 2 +--R \|a p x + (a q + b p)x + b q +--R + +--R 3 3 2 2 2 2 3 3 2 +--R (- 2a q - 14a b p q - 14a b p q - 2b p )x +--R + +--R 2 3 2 2 3 2 2 3 3 2 +--R (- 16a b q - 32a b p q - 16b p q)x - 16a b q - 16b p q +--R * +--R +---+ +---+ +--R \|a p \|b q +--R / +--R 2 2 +---+ +---+ +--R ((32a p q + 32a b p )x + 64a b p q)\|a p \|b q +--R * +--R +---------------------------+ +--R | 2 +--R \|a p x + (a q + b p)x + b q +--R + +--R 3 2 2 2 2 3 2 2 2 2 2 +--R (- 8a p q - 48a b p q - 8a b p )x + (- 64a b p q - 64a b p q)x +--R + +--R 2 2 +--R - 64a b p q +--R * +--R +---+ +--R \|a p +--R Type: Expression Integer +--E + +--S 26 +cc2:=aa1-bb2 +--R +--R (12) +--R 3 3 2 2 2 2 3 3 2 3 2 2 +--R (4a q - 4a b p q - 4a b p q + 4b p )x + 8a b q - 16a b p q +--R + +--R 3 2 +--R 8b p q +--R * +--R +---------------------------+ +--R +-----+ +---+ | 2 +--R \|- a p \|b q \|a p x + (a q + b p)x + b q +--R + +--R 4 4 3 3 2 2 2 2 3 3 4 4 2 +--R (- a q - 4a b p q + 10a b p q - 4a b p q - b p )x +--R + +--R 3 4 2 2 3 3 2 2 4 3 2 2 4 +--R (- 8a b q + 8a b p q + 8a b p q - 8b p q)x - 8a b q +--R + +--R 3 3 4 2 2 +--R 16a b p q - 8b p q +--R * +--R +-----+ +--R \|- a p +--R * +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q + 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R - 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R + +--R 3 3 2 2 2 2 3 3 2 3 2 2 +--R (8a q - 8a b p q - 8a b p q + 8b p )x + 16a b q - 32a b p q +--R + +--R 3 2 +--R 16b p q +--R * +--R +---------------------------+ +--R +---+ +---+ | 2 +--R \|a p \|b q \|a p x + (a q + b p)x + b q +--R + +--R 4 4 3 3 2 2 2 2 3 3 4 4 2 +--R (- 2a q - 8a b p q + 20a b p q - 8a b p q - 2b p )x +--R + +--R 3 4 2 2 3 3 2 2 4 3 2 2 4 +--R (- 16a b q + 16a b p q + 16a b p q - 16b p q)x - 16a b q +--R + +--R 3 3 4 2 2 +--R 32a b p q - 16b p q +--R * +--R +---+ +--R \|a p +--R * +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R atan(-------------------------------------------------------) +--R a p x +--R + +--R 2 3 2 2 3 2 2 3 3 2 +-----+ +---+ +--R ((8a b q + 16a b p q + 8b p q)x + 16a b q + 16b p q )\|- a p \|a p +--R * +--R +---------------------------+ +--R | 2 +--R \|a p x + (a q + b p)x + b q +--R + +--R 3 3 2 2 2 2 3 3 2 +--R (- 2a q - 14a b p q - 14a b p q - 2b p )x +--R + +--R 2 3 2 2 3 2 2 3 3 2 +--R (- 16a b q - 32a b p q - 16b p q)x - 16a b q - 16b p q +--R * +--R +-----+ +---+ +---+ +--R \|- a p \|a p \|b q +--R / +--R 2 2 +-----+ +---+ +---+ +--R ((32a p q + 32a b p )x + 64a b p q)\|- a p \|a p \|b q +--R * +--R +---------------------------+ +--R | 2 +--R \|a p x + (a q + b p)x + b q +--R + +--R 3 2 2 2 2 3 2 2 2 2 2 +--R (- 8a p q - 48a b p q - 8a b p )x + (- 64a b p q - 64a b p q)x +--R + +--R 2 2 +--R - 64a b p q +--R * +--R +-----+ +---+ +--R \|- a p \|a p +--R Type: Expression Integer +--E + +--S 27 +cc3:=aa1-bb1 +--R +--R (13) +--R 3 3 2 2 2 2 3 3 2 3 2 2 +--R (4a q - 4a b p q - 4a b p q + 4b p )x + 8a b q - 16a b p q +--R + +--R 3 2 +--R 8b p q +--R * +--R +---------------------------+ +--R +---+ | 2 +--R \|b q \|a p x + (a q + b p)x + b q +--R + +--R 4 4 3 3 2 2 2 2 3 3 4 4 2 +--R (- a q - 4a b p q + 10a b p q - 4a b p q - b p )x +--R + +--R 3 4 2 2 3 3 2 2 4 3 2 2 4 3 3 +--R (- 8a b q + 8a b p q + 8a b p q - 8b p q)x - 8a b q + 16a b p q +--R + +--R 4 2 2 +--R - 8b p q +--R * +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q + 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R - 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R + +--R 3 3 2 2 2 2 3 3 2 3 2 2 +--R (4a q - 4a b p q - 4a b p q + 4b p )x + 8a b q - 16a b p q +--R + +--R 3 2 +--R 8b p q +--R * +--R +---------------------------+ +--R +---+ | 2 +--R \|b q \|a p x + (a q + b p)x + b q +--R + +--R 4 4 3 3 2 2 2 2 3 3 4 4 2 +--R (- a q - 4a b p q + 10a b p q - 4a b p q - b p )x +--R + +--R 3 4 2 2 3 3 2 2 4 3 2 2 4 3 3 +--R (- 8a b q + 8a b p q + 8a b p q - 8b p q)x - 8a b q + 16a b p q +--R + +--R 4 2 2 +--R - 8b p q +--R * +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R + +--R 2 3 2 2 3 2 2 3 3 2 +---+ +--R ((8a b q + 16a b p q + 8b p q)x + 16a b q + 16b p q )\|a p +--R * +--R +---------------------------+ +--R | 2 +--R \|a p x + (a q + b p)x + b q +--R + +--R 3 3 2 2 2 2 3 3 2 +--R (- 2a q - 14a b p q - 14a b p q - 2b p )x +--R + +--R 2 3 2 2 3 2 2 3 3 2 +--R (- 16a b q - 32a b p q - 16b p q)x - 16a b q - 16b p q +--R * +--R +---+ +---+ +--R \|a p \|b q +--R / +--R 2 2 +---+ +---+ +--R ((32a p q + 32a b p )x + 64a b p q)\|a p \|b q +--R * +--R +---------------------------+ +--R | 2 +--R \|a p x + (a q + b p)x + b q +--R + +--R 3 2 2 2 2 3 2 2 2 2 2 +--R (- 8a p q - 48a b p q - 8a b p )x + (- 64a b p q - 64a b p q)x +--R + +--R 2 2 +--R - 64a b p q +--R * +--R +---+ +--R \|a p +--R Type: Expression Integer +--E + +--S 28 14:122 Axiom cannot simplify this answer +cc4:=aa2-bb2 +--R +--R (14) +--R 2 3 2 2 3 2 2 3 3 2 +--R ((4a b q + 8a b p q + 4b p q)x + 8a b q + 8b p q ) +--R * +--R +---------------------------+ +--R | 2 +--R \|a p x + (a q + b p)x + b q +--R + +--R 3 3 2 2 2 2 3 3 2 +--R (- a q - 7a b p q - 7a b p q - b p )x +--R + +--R 2 3 2 2 3 2 2 3 3 2 +--R (- 8a b q - 16a b p q - 8b p q)x - 8a b q - 8b p q +--R * +--R +---+ +--R \|b q +--R / +--R +---------------------------+ +--R 2 2 +---+ | 2 +--R ((16a p q + 16a b p )x + 32a b p q)\|b q \|a p x + (a q + b p)x + b q +--R + +--R 3 2 2 2 2 3 2 2 2 2 2 +--R (- 4a p q - 24a b p q - 4a b p )x + (- 32a b p q - 32a b p q)x +--R + +--R 2 2 +--R - 32a b p q +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.123~~~~~$\displaystyle\int{\sqrt{\frac{px+q}{ax+b}}}~dx$} +$$\int{\sqrt{\frac{px+q}{ax+b}}}= +\frac{\sqrt{(ax+b)(px+q)}}{a}+\frac{aq-bp}{2a} +\int{\frac{1}{\sqrt{(ax+b)(px+q)}}} +$$ +<<*>>= +)clear all + +--S 29 +aa:=integrate(sqrt((p*x+q)/(a*x+b)),x) +--R +--R +--R (1) +--R [ +--R (a q - b p) +--R * +--R +-------+ +--R +---+ 2 |p x + q +--R log((2a p x + a q + b p)\|a p + (2a p x + 2a b p) |------- ) +--R \|a x + b +--R + +--R +-------+ +--R |p x + q +---+ +--R (2a x + 2b) |------- \|a p +--R \|a x + b +--R / +--R +---+ +--R 2a\|a p +--R , +--R +-------+ +--R +-----+ |p x + q +--R \|- a p |------- +-------+ +--R \|a x + b +-----+ |p x + q +--R (a q - b p)atan(------------------) + (a x + b)\|- a p |------- +--R p \|a x + b +--R -----------------------------------------------------------------] +--R +-----+ +--R a\|- a p +--R Type: Union(List Expression Integer,...) +--E + +--S 30 +aa1:=aa.1 +--R +--R (2) +--R +-------+ +--R +---+ 2 |p x + q +--R (a q - b p)log((2a p x + a q + b p)\|a p + (2a p x + 2a b p) |------- ) +--R \|a x + b +--R + +--R +-------+ +--R |p x + q +---+ +--R (2a x + 2b) |------- \|a p +--R \|a x + b +--R / +--R +---+ +--R 2a\|a p +--R Type: Expression Integer +--E + +--S 31 +aa2:=aa.2 +--R +--R +-------+ +--R +-----+ |p x + q +--R \|- a p |------- +-------+ +--R \|a x + b +-----+ |p x + q +--R (a q - b p)atan(------------------) + (a x + b)\|- a p |------- +--R p \|a x + b +--R (3) ----------------------------------------------------------------- +--R +-----+ +--R a\|- a p +--R Type: Expression Integer +--E + +--S 32 +bba:=sqrt((a*x+b)*(p*x+q))/a +--R +--R +---------------------------+ +--R | 2 +--R \|a p x + (a q + b p)x + b q +--R (4) ------------------------------ +--R a +--R Type: Expression Integer +--E + +--S 33 +bbb:=(a*q-b*p)/(2*a) +--R +--R a q - b p +--R (5) --------- +--R 2a +--R Type: Fraction Polynomial Integer +--E + +--S 34 +bbc:=integrate(1/(sqrt((a*x+b)*(p*x+q))),x) +--R +--R (6) +--R [ +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R / +--R +---+ +--R \|a p +--R , +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R 2atan(-------------------------------------------------------) +--R a p x +--R --------------------------------------------------------------] +--R +-----+ +--R \|- a p +--R Type: Union(List Expression Integer,...) +--E + +--S 35 +bbc1:=bbc.1 +--R +--R (7) +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R / +--R +---+ +--R \|a p +--R Type: Expression Integer +--E + +--S 36 +bbc2:=bbc.2 +--R +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R 2atan(-------------------------------------------------------) +--R a p x +--R (8) -------------------------------------------------------------- +--R +-----+ +--R \|- a p +--R Type: Expression Integer +--E + +--S 37 +bb1:=bba+bbb*bbc1 +--R +--R (9) +--R (a q - b p) +--R * +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R + +--R +---------------------------+ +--R +---+ | 2 +--R 2\|a p \|a p x + (a q + b p)x + b q +--R / +--R +---+ +--R 2a\|a p +--R Type: Expression Integer +--E + +--S 38 +bb2:=bba+bbb*bbc2 +--R +--R (10) +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R (a q - b p)atan(-------------------------------------------------------) +--R a p x +--R + +--R +---------------------------+ +--R +-----+ | 2 +--R \|- a p \|a p x + (a q + b p)x + b q +--R / +--R +-----+ +--R a\|- a p +--R Type: Expression Integer +--E + +--S 39 +cc1:=aa1-bb1 +--R +--R (11) +--R +-------+ +--R +---+ 2 |p x + q +--R (a q - b p)log((2a p x + a q + b p)\|a p + (2a p x + 2a b p) |------- ) +--R \|a x + b +--R + +--R (- a q + b p) +--R * +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R + +--R +---------------------------+ +-------+ +--R +---+ | 2 |p x + q +---+ +--R - 2\|a p \|a p x + (a q + b p)x + b q + (2a x + 2b) |------- \|a p +--R \|a x + b +--R / +--R +---+ +--R 2a\|a p +--R Type: Expression Integer +--E + +--S 40 +cc2:=aa1-bb2 +--R +--R (12) +--R +-----+ +--R (a q - b p)\|- a p +--R * +--R +-------+ +--R +---+ 2 |p x + q +--R log((2a p x + a q + b p)\|a p + (2a p x + 2a b p) |------- ) +--R \|a x + b +--R + +--R +---+ +--R (- 2a q + 2b p)\|a p +--R * +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R atan(-------------------------------------------------------) +--R a p x +--R + +--R +---------------------------+ +--R +-----+ +---+ | 2 +--R - 2\|- a p \|a p \|a p x + (a q + b p)x + b q +--R + +--R +-------+ +--R +-----+ |p x + q +---+ +--R (2a x + 2b)\|- a p |------- \|a p +--R \|a x + b +--R / +--R +-----+ +---+ +--R 2a\|- a p \|a p +--R Type: Expression Integer +--E + +--S 41 +cc3:=aa2-bb1 +--R +--R (13) +--R +-----+ +--R (- a q + b p)\|- a p +--R * +--R log +--R +---------------------------+ +--R +---+ +---+ | 2 +--R (2\|a p \|b q - 2a p x)\|a p x + (a q + b p)x + b q +--R + +--R +---+ 2 +---+ +--R 2a p x\|b q + (- 2a p x + (- a q - b p)x - 2b q)\|a p +--R / +--R +---------------------------+ +--R +---+ | 2 +--R 2\|b q \|a p x + (a q + b p)x + b q + (- a q - b p)x - 2b q +--R + +--R +-------+ +--R +-----+ |p x + q +--R \|- a p |------- +--R +---+ \|a x + b +--R (2a q - 2b p)\|a p atan(------------------) +--R p +--R + +--R +---------------------------+ +--R +-----+ +---+ | 2 +--R - 2\|- a p \|a p \|a p x + (a q + b p)x + b q +--R + +--R +-------+ +--R +-----+ |p x + q +---+ +--R (2a x + 2b)\|- a p |------- \|a p +--R \|a x + b +--R / +--R +-----+ +---+ +--R 2a\|- a p \|a p +--R Type: Expression Integer +--E + +--S 42 14:123 Axiom cannot simplify these results +cc4:=aa2-bb2 +--R +--R (14) +--R (- a q + b p) +--R * +--R +---------------------------+ +--R +-----+ | 2 +-----+ +---+ +--R \|- a p \|a p x + (a q + b p)x + b q - \|- a p \|b q +--R atan(-------------------------------------------------------) +--R a p x +--R + +--R +-------+ +--R +-----+ |p x + q +--R \|- a p |------- +--R \|a x + b +--R (a q - b p)atan(------------------) +--R p +--R + +--R +---------------------------+ +-------+ +--R +-----+ | 2 +-----+ |p x + q +--R - \|- a p \|a p x + (a q + b p)x + b q + (a x + b)\|- a p |------- +--R \|a x + b +--R / +--R +-----+ +--R a\|- a p +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.124~~~~~$\displaystyle +\int{\frac{dx}{(px+q)\sqrt{(ax+b)(px+q)}}}~dx$} +$$\int{\frac{1}{(px+q)\sqrt{(ax+b)(px+q)}}}= +\frac{2\sqrt{ax+b}}{(aq-bp)\sqrt{px+q}} +$$ +<<*>>= +)clear all + +--S 43 +aa:=integrate(1/((p*x+q)*sqrt((a*x+b)*(p*x+q))),x) +--R +--R +--R 2x +--R (1) --------------------------------------------------- +--R +---------------------------+ +--R | 2 +---+ +--R q\|a p x + (a q + b p)x + b q + (- p x - q)\|b q +--R Type: Union(Expression Integer,...) +--E + +--S 44 +bb:=(2*sqrt(a*x+b))/((a*q-b*p)*sqrt(p*x+q)) +--R +--R +-------+ +--R 2\|a x + b +--R (2) --------------------- +--R +-------+ +--R (a q - b p)\|p x + q +--R Type: Expression Integer +--E + +--S 45 14:124 Axiom cannot simplify this result +cc:=aa-bb +--R +--R (3) +--R +---------------------------+ +--R +-------+ | 2 +-------+ +--R - 2q\|a x + b \|a p x + (a q + b p)x + b q + (2a q - 2b p)x\|p x + q +--R + +--R +---+ +-------+ +--R (2p x + 2q)\|b q \|a x + b +--R / +--R +---------------------------+ +--R 2 +-------+ | 2 +--R (a q - b p q)\|p x + q \|a p x + (a q + b p)x + b q +--R + +--R 2 2 +---+ +-------+ +--R ((- a p q + b p )x - a q + b p q)\|b q \|p x + q +--R Type: Expression Integer +--E + + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp63-64 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum5.input.pdf b/src/axiom-website/CATS/schaum5.input.pdf new file mode 100644 index 0000000..8bc22b5 --- /dev/null +++ b/src/axiom-website/CATS/schaum5.input.pdf @@ -0,0 +1,2930 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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+\begin{document} +\title{\$SPAD/input schaum6.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.125~~~~~$\displaystyle\int{\frac{dx}{x^2+a^2}}$} +$$\int{\frac{1}{x^2+a^2}}=\frac{1}{a}\tan^{-1}\frac{x}{a}$$ +<<*>>= +)spool schaum6.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(1/(x^2+a^2),x) +--R +--R +--R x +--R atan(-) +--R a +--R (1) ------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=(1/a)*atan(x/a) +--R +--R x +--R atan(-) +--R a +--R (2) ------- +--R a +--R Type: Expression Integer +--E + +--S 3 14:125 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.126~~~~~$\displaystyle\int{\frac{x~dx}{x^2+a^2}}$} +$$\int{\frac{x}{x^2+a^2}}=\frac{1}{2}\ln(x^2+a^2)$$ +<<*>>= +)clear all + +--S 4 +aa:=integrate(x/(x^2+a^2),x) +--R +--R +--R 2 2 +--R log(x + a ) +--R (1) ------------ +--R 2 +--R Type: Union(Expression Integer,...) +--E + +--S 5 +bb:=(1/2)*log(x^2+a^2) +--R +--R 2 2 +--R log(x + a ) +--R (2) ------------ +--R 2 +--R Type: Expression Integer +--E + +--S 6 14:126 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.127~~~~~$\displaystyle\int{\frac{x^2~dx}{x^2+a^2}}$} +$$\int{\frac{x^2}{x^2+a^2}}=x-a\tan^{-1}\frac{x}{a}$$ +<<*>>= +)clear all + +--S 7 +aa:=integrate(x^2/(x^2+a^2),x) +--R +--R +--R x +--R (1) - a atan(-) + x +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 8 +bb:=x-a*atan(x/a) +--R +--R x +--R (2) - a atan(-) + x +--R a +--R Type: Expression Integer +--E + +--S 9 14:127 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.128~~~~~$\displaystyle\int{\frac{x^3~dx}{x^2+a^2}}$} +$$\int{\frac{x^3}{x^2+a^2}}=\frac{x^2}{2}-\frac{a^2}{2}\ln(x^2+a^2)$$ + +<<*>>= +)clear all + +--S 10 +aa:=integrate(x^3/(x^2+a^2),x) +--R +--R +--R 2 2 2 2 +--R - a log(x + a ) + x +--R (1) --------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E + +--S 11 +bb:=x^2/2-a^2/2*log(x^2+a^2) +--R +--R 2 2 2 2 +--R - a log(x + a ) + x +--R (2) --------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 12 14:128 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.129~~~~~$\displaystyle\int{\frac{dx}{x(x^2+a^2)}}$} +$$\int{\frac{1}{x(x^2+a^2)}}= +\frac{1}{2a^2}\ln\left(\frac{x^2}{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 13 +aa:=integrate(1/(x*(x^2+a^2)),x) +--R +--R +--R 2 2 +--R - log(x + a ) + 2log(x) +--R (1) ------------------------ +--R 2 +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 14 +bb:=1/(2*a^2)*log(x^2/(x^2+a^2)) +--R +--R 2 +--R x +--R log(-------) +--R 2 2 +--R x + a +--R (2) ------------ +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 15 +cc:=aa-bb +--R +--R 2 +--R 2 2 x +--R - log(x + a ) + 2log(x) - log(-------) +--R 2 2 +--R x + a +--R (3) --------------------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 16 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 17 +dd:=divlog cc +--R +--R 2 +--R - log(x ) + 2log(x) +--R (5) ------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 18 +logpow:=rule(log(a^n) == n*log(a)) +--R +--R n +--R (6) log(a ) == n log(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 19 14:129 Schaums and Axiom agree +ee:=logpow dd +--R +--R (7) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.130~~~~~$\displaystyle\int{\frac{dx}{x^2(x^2+a^2)}}$} +$$\int{\frac{1}{x^2(x^2+a^2)}}= +-\frac{1}{a^2x}-\frac{1}{a^3}\tan^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 20 +aa:=integrate(1/(x^2*(x^2+a^2)),x) +--R +--R +--R x +--R - x atan(-) - a +--R a +--R (1) --------------- +--R 3 +--R a x +--R Type: Union(Expression Integer,...) +--E + +--S 21 +bb:=-1/(a^2*x)-1/a^3*atan(x/a) +--R +--R x +--R - x atan(-) - a +--R a +--R (2) --------------- +--R 3 +--R a x +--R Type: Expression Integer +--E + +--S 22 14:130 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.131~~~~~$\displaystyle\int{\frac{dx}{x^3(x^2+a^2)}}$} +$$\int{\frac{1}{x^3(x^2+a^2)}}= +-\frac{1}{2a^2x^2}-\frac{1}{2a^4}\ln\left(\frac{x^2}{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 23 +aa:=integrate(1/(x^3*(x^2+a^2)),x) +--R +--R +--R 2 2 2 2 2 +--R x log(x + a ) - 2x log(x) - a +--R (1) ------------------------------- +--R 4 2 +--R 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 24 +bb:=-1/(2*a^2*x^2)-1/(2*a^4)*log(x^2/(x^2+a^2)) +--R +--R 2 +--R 2 x 2 +--R - x log(-------) - a +--R 2 2 +--R x + a +--R (2) --------------------- +--R 4 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 25 +cc:=aa-bb +--R +--R 2 +--R 2 2 x +--R log(x + a ) - 2log(x) + log(-------) +--R 2 2 +--R x + a +--R (3) ------------------------------------- +--R 4 +--R 2a +--R Type: Expression Integer +--E + +--S 26 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 27 +dd:=divlog cc +--R +--R 2 +--R log(x ) - 2log(x) +--R (5) ----------------- +--R 4 +--R 2a +--R Type: Expression Integer +--E + +--S 28 +logpow:=rule(log(a^n) == n*log(a)) +--R +--R n +--R (6) log(a ) == n log(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 29 14:131 Schaums and Axiom agree +ee:=logpow dd +--R +--R (7) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.132~~~~~$\displaystyle\int{\frac{dx}{(x^2+a^2)^2}}$} +$$\int{\frac{1}{(x^2+a^2)^2}}= +\frac{x}{2a^2(x^2+a^2)}+\frac{1}{2a^3}\tan^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 30 +aa:=integrate(1/((x^2+a^2)^2),x) +--R +--R +--R 2 2 x +--R (x + a )atan(-) + a x +--R a +--R (1) ---------------------- +--R 3 2 5 +--R 2a x + 2a +--R Type: Union(Expression Integer,...) +--E + +--S 31 +bb:=x/(2*a^2*(x^2+a^2))+1/(2*a^3)*atan(x/a) +--R +--R 2 2 x +--R (x + a )atan(-) + a x +--R a +--R (2) ---------------------- +--R 3 2 5 +--R 2a x + 2a +--R Type: Expression Integer +--E + +--S 32 14:132 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.133~~~~~$\displaystyle\int{\frac{x~dx}{(x^2+a^2)^2}}$} +$$\int{\frac{x}{(x^2+a^2)^2}}= +\frac{-1}{2(x^2+a^2)} +$$ +<<*>>= +)clear all + +--S 33 +aa:=integrate(x/((x^2+a^2)^2),x) +--R +--R +--R 1 +--R (1) - --------- +--R 2 2 +--R 2x + 2a +--R Type: Union(Expression Integer,...) +--E + +--S 34 +bb:=-1/(2*(x^2+a^2)) +--R +--R 1 +--R (2) - --------- +--R 2 2 +--R 2x + 2a +--R Type: Fraction Polynomial Integer +--E + +--S 35 14:133 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.134~~~~~$\displaystyle\int{\frac{x^2dx}{(x^2+a^2)^2}}$} +$$\int{\frac{x^2}{(x^2+a^2)^2}}= +\frac{-x}{2(x^2+a^2)}+\frac{1}{2a}\tan^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 36 +aa:=integrate(x^2/((x^2+a^2)^2),x) +--R +--R +--R 2 2 x +--R (x + a )atan(-) - a x +--R a +--R (1) ---------------------- +--R 2 3 +--R 2a x + 2a +--R Type: Union(Expression Integer,...) +--E + +--S 37 +bb:=-x/(2*(x^2+a^2))+1/(2*a)*atan(x/a) +--R +--R 2 2 x +--R (x + a )atan(-) - a x +--R a +--R (2) ---------------------- +--R 2 3 +--R 2a x + 2a +--R Type: Expression Integer +--E + +--S 38 14:134 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.135~~~~~$\displaystyle\int{\frac{x^3dx}{(x^2+a^2)^2}}$} +$$\int{\frac{x^3}{(x^2+a^2)^2}}= +\frac{a^2}{2(x^2+a^2)}+\frac{1}{2}\ln(x^2+a^2) +$$ +<<*>>= +)clear all + +--S 39 +aa:=integrate(x^3/((x^2+a^2)^2),x) +--R +--R +--R 2 2 2 2 2 +--R (x + a )log(x + a ) + a +--R (1) -------------------------- +--R 2 2 +--R 2x + 2a +--R Type: Union(Expression Integer,...) +--E + +--S 40 +bb:=a^2/(2*(x^2+a^2))+1/2*log(x^2+a^2) +--R +--R 2 2 2 2 2 +--R (x + a )log(x + a ) + a +--R (2) -------------------------- +--R 2 2 +--R 2x + 2a +--R Type: Expression Integer +--E + +--S 41 14:135 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.136~~~~~$\displaystyle\int{\frac{dx}{x(x^2+a^2)^2}}$} +$$\int{\frac{1}{x(x^2+a^2)^2}}= +\frac{1}{2a^2(x^2+a^2)}+\frac{1}{2a^4}\ln\left(\frac{x^2}{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 42 +aa:=integrate(1/(x*(x^2+a^2)^2),x) +--R +--R +--R 2 2 2 2 2 2 2 +--R (- x - a )log(x + a ) + (2x + 2a )log(x) + a +--R (1) ------------------------------------------------ +--R 4 2 6 +--R 2a x + 2a +--R Type: Union(Expression Integer,...) +--E + +--S 43 +bb:=1/(2*a^2*(x^2+a^2))+1/(2*a^4)*log(x^2/(x^2+a^2)) +--R +--R 2 +--R 2 2 x 2 +--R (x + a )log(-------) + a +--R 2 2 +--R x + a +--R (2) -------------------------- +--R 4 2 6 +--R 2a x + 2a +--R Type: Expression Integer +--E + +--S 44 +cc:=aa-bb +--R +--R 2 +--R 2 2 x +--R - log(x + a ) + 2log(x) - log(-------) +--R 2 2 +--R x + a +--R (3) --------------------------------------- +--R 4 +--R 2a +--R Type: Expression Integer +--E + +--S 45 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 46 +dd:=divlog cc +--R +--R 2 +--R - log(x ) + 2log(x) +--R (5) ------------------- +--R 4 +--R 2a +--R Type: Expression Integer +--E + +--S 47 +logpow:=rule(log(a^n) == n*log(a)) +--R +--R n +--R (6) log(a ) == n log(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 48 14:136 Schaums and Axiom agree +ee:=logpow dd +--R +--R (7) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.137~~~~~$\displaystyle\int{\frac{dx}{x^2(x^2+a^2)^2}}$} +$$\int{\frac{1}{x^2(x^2+a^2)^2}}= +-\frac{1}{a^4x}-\frac{x}{2a^4(x^2+a^2)}-\frac{3}{2a^5}\tan^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 49 +aa:=integrate(1/(x^2*(x^2+a^2)^2),x) +--R +--R 3 2 x 2 3 +--R (- 3x - 3a x)atan(-) - 3a x - 2a +--R a +--R (1) ----------------------------------- +--R 5 3 7 +--R 2a x + 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 50 +bb:=-1/(a^4*x)-x/(2*a^4*(x^2+a^2))-3/(2*a^5)*atan(x/a) +--R +--R 3 2 x 2 3 +--R (- 3x - 3a x)atan(-) - 3a x - 2a +--R a +--R (2) ----------------------------------- +--R 5 3 7 +--R 2a x + 2a x +--R Type: Expression Integer +--E + +--S 51 14:137 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.138~~~~~$\displaystyle\int{\frac{dx}{x^3(x^2+a^2)^2}}$} +$$\int{\frac{1}{x^3(x^2+a^2)^2}}= +-\frac{1}{2a^4x^2}-\frac{1}{2a^4(x^2+a^2)}- +\frac{1}{a^6}\ln\left(\frac{x^2}{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 52 +aa:=integrate(1/(x^3*(x^2+a^2)^2),x) +--R +--R +--R 4 2 2 2 2 4 2 2 2 2 4 +--R (2x + 2a x )log(x + a ) + (- 4x - 4a x )log(x) - 2a x - a +--R (1) -------------------------------------------------------------- +--R 6 4 8 2 +--R 2a x + 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 53 +bb:=-1/(2*a^4*x^2)-1/(2*a^4*(x^2+a^2))-1/a^6*log(x^2/(x^2+a^2)) +--R +--R 2 +--R 4 2 2 x 2 2 4 +--R (- 2x - 2a x )log(-------) - 2a x - a +--R 2 2 +--R x + a +--R (2) ---------------------------------------- +--R 6 4 8 2 +--R 2a x + 2a x +--R Type: Expression Integer +--E + +--S 54 +cc:=aa-bb +--R +--R 2 +--R 2 2 x +--R log(x + a ) - 2log(x) + log(-------) +--R 2 2 +--R x + a +--R (3) ------------------------------------- +--R 6 +--R a +--R Type: Expression Integer +--E + +--S 55 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 56 +dd:=divlog cc +--R +--R 2 +--R log(x ) - 2log(x) +--R (5) ----------------- +--R 6 +--R a +--R Type: Expression Integer +--E + +--S 57 +logpow:=rule(log(a^n) == n*log(a)) +--R +--R n +--R (6) log(a ) == n log(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 58 14:138 Schaums and Axiom agree +ee:=logpow dd +--R +--R (7) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.139~~~~~$\displaystyle\int{\frac{dx}{(x^2+a^2)^n}}$} +$$\int{\frac{1}{(x^2+a^2)^n}}= +\frac{x}{2(n-1)a^2(x^2+a^2)^{n-1}}+\frac{2n-3}{(2n-2)a^2} +\int{\frac{1}{(x^2+a^2)^{n-1}}} +$$ +<<*>>= +)clear all + +--S 59 14:139 Axiom cannot do this integral +aa:=integrate(1/((x^2+a^2)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | ----------- d%L +--R ++ 2 2 n +--I (a + %L ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.140~~~~~$\displaystyle\int{\frac{x~dx}{(x^2+a^2)^n}}$} +$$\int{\frac{x}{(x^2+a^2)^n}}= +\frac{-1}{2(n-1)(x^2+a^2)^{n-1}} +$$ +<<*>>= +)clear all + +--S 60 +aa:=integrate(x/((x^2+a^2)^n),x) +--R +--R +--R 2 2 +--R - x - a +--R (1) ------------------------ +--R 2 2 +--R n log(x + a ) +--R (2n - 2)%e +--R Type: Union(Expression Integer,...) +--E + +--S 61 +bb:=-1/(2*(n-1)*(x^2+a^2)^(n-1)) +--R +--R 1 +--R (2) - ---------------------- +--R 2 2 n - 1 +--R (2n - 2)(x + a ) +--R Type: Expression Integer +--E + +--S 62 +cc:=aa-bb +--R +--R 2 2 +--R n log(x + a ) 2 2 2 2 n - 1 +--R %e + (- x - a )(x + a ) +--R (3) -------------------------------------------- +--R 2 2 +--R 2 2 n - 1 n log(x + a ) +--R (2n - 2)(x + a ) %e +--R Type: Expression Integer +--E + +--S 63 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 64 +dd:=explog cc +--R +--R 2 2 n 2 2 2 2 n - 1 +--R (x + a ) + (- x - a )(x + a ) +--R (5) -------------------------------------- +--R 2 2 n - 1 2 2 n +--R (2n - 2)(x + a ) (x + a ) +--R Type: Expression Integer +--E + +--S 65 14:140 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.141~~~~~$\displaystyle\int{\frac{dx}{x(x^2+a^2)^n}}$} +$$\int{\frac{1}{x(x^2+a^2)^n}}= +\frac{1}{2(n-1)a^2(x^2+a^2)^{n-1}}+\frac{1}{a^2} +\int{\frac{1}{x(x^2+a^2)^{n-1}}} +$$ +<<*>>= +)clear all + +--S 66 14:141 Axiom cannot do this integral +aa:=integrate(1/(x*(x^2+a^2)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | -------------- d%L +--R ++ 2 2 n +--I %L (a + %L ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.142~~~~~$\displaystyle\int{\frac{x^mdx}{(x^2+a^2)^n}}$} +$$\int{\frac{x^m}{(x^2+a^2)^n}}= +\int{\frac{x^{m-2}}{(x^2+a^2)^{n-1}}} - +a^2\int{\frac{x^{m-2}}{(x^2+a^2)^n}} +$$ +<<*>>= +)clear all + +--S 67 14:142 Axiom cannot do this integral +aa:=integrate(x^m/((x^2+a^2)^n),x) +--R +--R +--R x m +--I ++ %L +--I (1) | ----------- d%L +--R ++ 2 2 n +--I (a + %L ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.143~~~~~$\displaystyle\int{\frac{dx}{x^m(x^2+a^2)^n}}$} +$$\int{\frac{1}{x^m(x^2+a^2)^n}}= +\frac{1}{a^2}\int{\frac{1}{x^m(x^2+a^2)^{n-1}}}- +\frac{1}{a^2}\int{\frac{1}{x^{m-2}(x^2+a^2)^n}} +$$ +<<*>>= +)clear all + +--S 68 14:143 Axiom cannot do this integral +aa:=integrate(1/(x^m*(x^2+a^2)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | -------------- d%L +--R ++ m 2 2 n +--I %L (a + %L ) +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p64 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum6.input.pdf b/src/axiom-website/CATS/schaum6.input.pdf new file mode 100644 index 0000000..0309298 --- /dev/null +++ b/src/axiom-website/CATS/schaum6.input.pdf @@ -0,0 +1,2336 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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- log(x + a) + log(x - a) +--R (1) ------------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=1/(2*a)*log((x-a)/(x+a)) +--R +--R x - a +--R log(-----) +--R x + a +--R (2) ---------- +--R 2a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R x - a +--R - log(x + a) + log(x - a) - log(-----) +--R x + a +--R (3) -------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 4 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 5 14:144 Schaums and Axiom agree +dd:=divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.145~~~~~$\displaystyle\int{\frac{x~dx}{x^2-a^2}}$} +$$\int{\frac{x}{x^2-a^2}}=\frac{1}{2}\ln(x^2-a^2)$$ +<<*>>= +)clear all + +--S 6 +aa:=integrate(x/(x^2-a^2),x) +--R +--R +--R 2 2 +--R log(x - a ) +--R (1) ------------ +--R 2 +--R Type: Union(Expression Integer,...) +--E + +--S 7 +bb:=1/2*log(x^2-a^2) +--R +--R 2 2 +--R log(x - a ) +--R (2) ------------ +--R 2 +--R Type: Expression Integer +--E + +--S 8 14:145 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.146~~~~~$\displaystyle\int{\frac{x^2~dx}{x^2-a^2}}$} +$$\int{\frac{x^2}{x^2-a^2}}=x+\frac{a}{2}\ln\left(\frac{x-a}{x+a}\right)$$ +<<*>>= +)clear all + +--S 9 +aa:=integrate(x^2/(x^2-a^2),x) +--R +--R +--R - a log(x + a) + a log(x - a) + 2x +--R (1) ---------------------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E + +--S 10 +bb:=x+a/2*log((x-a)/(x+a)) +--R +--R x - a +--R a log(-----) + 2x +--R x + a +--R (2) ----------------- +--R 2 +--R Type: Expression Integer +--E + +--S 11 +cc:=aa-bb +--R +--R x - a +--R - a log(x + a) + a log(x - a) - a log(-----) +--R x + a +--R (3) -------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 12 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 13 14:146 Schaums and Axiom agree +dd:=divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.147~~~~~$\displaystyle\int{\frac{x^3~dx}{x^2-a^2}}$} +$$\int{\frac{x^3}{x^2-a^2}}=\frac{x^2}{2}+\frac{a^2}{2}\ln(x^2-a^2)$$ + +<<*>>= +)clear all + +--S 14 +aa:=integrate(x^3/(x^2-a^2),x) +--R +--R +--R 2 2 2 2 +--R a log(x - a ) + x +--R (1) ------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E + +--S 15 +bb:=x^2/2+a^2/2*log(x^2-a^2) +--R +--R 2 2 2 2 +--R a log(x - a ) + x +--R (2) ------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 16 14:147 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.148~~~~~$\displaystyle\int{\frac{dx}{x(x^2-a^2)}}$} +$$\int{\frac{1}{x(x^2-a^2)}}= +\frac{1}{2a^2}\ln\left(\frac{x^2-a^2}{x^2}\right) +$$ +<<*>>= +)clear all + +--S 17 +aa:=integrate(1/(x*(x^2-a^2)),x) +--R +--R +--R 2 2 +--R log(x - a ) - 2log(x) +--R (1) ---------------------- +--R 2 +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 18 +bb:=1/(2*a^2)*log((x^2-a^2)/x^2) +--R +--R 2 2 +--R x - a +--R log(-------) +--R 2 +--R x +--R (2) ------------ +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 19 +cc:=aa-bb +--R +--R 2 2 +--R 2 2 x - a +--R log(x - a ) - 2log(x) - log(-------) +--R 2 +--R x +--R (3) ------------------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 20 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 21 +dd:=divlog cc +--R +--R 2 +--R log(x ) - 2log(x) +--R (5) ----------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 22 +logpow:=rule(log(a^n) == n*log(a)) +--R +--R n +--R (6) log(a ) == n log(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 23 14:148 Schaums and Axiom agree +ee:=logpow dd +--R +--R (7) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.149~~~~~$\displaystyle\int{\frac{dx}{x^2(x^2-a^2)}}$} +$$\int{\frac{1}{x^2(x^2-a^2)}}= +\frac{1}{a^2x}+\frac{1}{2a^3}\ln\left(\frac{x-a}{x+a}\right) +$$ +<<*>>= +)clear all + +--S 24 +aa:=integrate(1/(x^2*(x^2-a^2)),x) +--R +--R +--R - x log(x + a) + x log(x - a) + 2a +--R (1) ---------------------------------- +--R 3 +--R 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 25 +bb:=1/(a^2*x)+1/(2*a^3)*log((x-a)/(x+a)) +--R +--R x - a +--R x log(-----) + 2a +--R x + a +--R (2) ----------------- +--R 3 +--R 2a x +--R Type: Expression Integer +--E + +--S 26 +cc:=aa-bb +--R +--R x - a +--R - log(x + a) + log(x - a) - log(-----) +--R x + a +--R (3) -------------------------------------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 27 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 28 14:149 Schaums and Axiom agree +dd:=divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.150~~~~~$\displaystyle\int{\frac{dx}{x^3(x^2-a^2)}}$} +$$\int{\frac{1}{x^3(x^2-a^2)}}= +\frac{1}{2a^2x^2}-\frac{1}{2a^4}\ln\left(\frac{x^2}{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 29 +aa:=integrate(1/(x^3*(x^2-a^2)),x) +--R +--R +--R 2 2 2 2 2 +--R x log(x - a ) - 2x log(x) + a +--R (1) ------------------------------- +--R 4 2 +--R 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 30 +bb:=1/(2*a^2*x^2)-1/(2*a^4)*log(x^2/(x^2-a^2)) +--R +--R 2 +--R 2 x 2 +--R - x log(-------) + a +--R 2 2 +--R x - a +--R (2) --------------------- +--R 4 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 31 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (3) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 32 +t1:=divlog bb +--R +--R 2 2 2 2 2 2 +--R - x log(x ) + x log(x - a ) + a +--R (4) --------------------------------- +--R 4 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 33 +logpow:=rule(log(a^n) == n*log(a)) +--R +--R n +--R (5) log(a ) == n log(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 34 +t2:=logpow t1 +--R +--R 2 2 2 2 2 +--R x log(x - a ) - 2x log(x) + a +--R (6) ------------------------------- +--R 4 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 35 14:150 Schaums and Axiom agree +cc:=aa-t2 +--R +--R (7) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.151~~~~~$\displaystyle\int{\frac{dx}{(x^2-a^2)^2}}$} +$$\int{\frac{1}{(x^2-a^2)^2}}= +\frac{-x}{2a^2(x^2-a^2)}-\frac{1}{4a^3}\ln\left(\frac{x-a}{x+a}\right) +$$ +<<*>>= +)clear all + +--S 36 +aa:=integrate(1/((x^2-a^2)^2),x) +--R +--R +--R 2 2 2 2 +--R (x - a )log(x + a) + (- x + a )log(x - a) - 2a x +--R (1) -------------------------------------------------- +--R 3 2 5 +--R 4a x - 4a +--R Type: Union(Expression Integer,...) +--E + +--S 37 +bb:=-x/(2*a^2*(x^2-a^2))-1/(4*a^3)*log((x-a)/(x+a)) +--R +--R 2 2 x - a +--R (- x + a )log(-----) - 2a x +--R x + a +--R (2) ---------------------------- +--R 3 2 5 +--R 4a x - 4a +--R Type: Expression Integer +--E + +--S 38 +cc:=aa-bb +--R +--R x - a +--R log(x + a) - log(x - a) + log(-----) +--R x + a +--R (3) ------------------------------------ +--R 3 +--R 4a +--R Type: Expression Integer +--E + +--S 39 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 40 14:151 Schaums and Axiom agree +dd:=divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.152~~~~~$\displaystyle\int{\frac{x~dx}{(x^2-a^2)^2}}$} +$$\int{\frac{x}{(x^2-a^2)^2}}= +\frac{-1}{2(x^2-a^2)} +$$ +<<*>>= +)clear all + +--S 41 +aa:=integrate(x/((x^2-a^2)^2),x) +--R +--R +--R 1 +--R (1) - --------- +--R 2 2 +--R 2x - 2a +--R Type: Union(Expression Integer,...) +--E + +--S 42 +bb:=-1/(2*(x^2-a^2)) +--R +--R 1 +--R (2) - --------- +--R 2 2 +--R 2x - 2a +--R Type: Fraction Polynomial Integer +--E + +--S 43 14:152 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.153~~~~~$\displaystyle\int{\frac{x^2dx}{(x^2-a^2)^2}}$} +$$\int{\frac{x^2}{(x^2-a^2)^2}}= +\frac{-x}{2(x^2-a^2)}+\frac{1}{4a}\ln\left(\frac{x-a}{x+a}\right) +$$ +<<*>>= +)clear all + +--S 44 +aa:=integrate(x^2/((x^2-a^2)^2),x) +--R +--R +--R 2 2 2 2 +--R (- x + a )log(x + a) + (x - a )log(x - a) - 2a x +--R (1) -------------------------------------------------- +--R 2 3 +--R 4a x - 4a +--R Type: Union(Expression Integer,...) +--E + +--S 45 +bb:=-x/(2*(x^2-a^2))+1/(4*a)*log((x-a)/(x+a)) +--R +--R 2 2 x - a +--R (x - a )log(-----) - 2a x +--R x + a +--R (2) -------------------------- +--R 2 3 +--R 4a x - 4a +--R Type: Expression Integer +--E + +--S 46 +cc:=aa-bb +--R +--R x - a +--R - log(x + a) + log(x - a) - log(-----) +--R x + a +--R (3) -------------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 47 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 48 14:153 Schaums and Axiom agree +dd:=divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.154~~~~~$\displaystyle\int{\frac{x^3dx}{(x^2-a^2)^2}}$} +$$\int{\frac{x^3}{(x^2-a^2)^2}}= +\frac{-a^2}{2(x^2-a^2)}+\frac{1}{2}\ln(x^2-a^2) +$$ +<<*>>= +)clear all + +--S 49 +aa:=integrate(x^3/((x^2-a^2)^2),x) +--R +--R +--R 2 2 2 2 2 +--R (x - a )log(x - a ) - a +--R (1) -------------------------- +--R 2 2 +--R 2x - 2a +--R Type: Union(Expression Integer,...) +--E + +--S 50 +bb:=-a^2/(2*(x^2-a^2))+1/2*log(x^2-a^2) +--R +--R 2 2 2 2 2 +--R (x - a )log(x - a ) - a +--R (2) -------------------------- +--R 2 2 +--R 2x - 2a +--R Type: Expression Integer +--E + +--S 51 14:154 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.155~~~~~$\displaystyle\int{\frac{dx}{x(x^2-a^2)^2}}$} +$$\int{\frac{1}{x(x^2-a^2)^2}}= +\frac{-1}{2a^2(x^2-a^2)}+\frac{1}{2a^4}\ln\left(\frac{x^2}{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 52 +aa:=integrate(1/(x*(x^2-a^2)^2),x) +--R +--R +--R 2 2 2 2 2 2 2 +--R (- x + a )log(x - a ) + (2x - 2a )log(x) - a +--R (1) ------------------------------------------------ +--R 4 2 6 +--R 2a x - 2a +--R Type: Union(Expression Integer,...) +--E + +--S 53 +bb:=-1/(2*a^2*(x^2-a^2))+1/(2*a^4)*log(x^2/(x^2-a^2)) +--R +--R 2 +--R 2 2 x 2 +--R (x - a )log(-------) - a +--R 2 2 +--R x - a +--R (2) -------------------------- +--R 4 2 6 +--R 2a x - 2a +--R Type: Expression Integer +--E + +--S 54 +cc:=aa-bb +--R +--R 2 +--R 2 2 x +--R - log(x - a ) + 2log(x) - log(-------) +--R 2 2 +--R x - a +--R (3) --------------------------------------- +--R 4 +--R 2a +--R Type: Expression Integer +--E + +--S 55 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 56 +dd:=divlog cc +--R +--R 2 +--R - log(x ) + 2log(x) +--R (5) ------------------- +--R 4 +--R 2a +--R Type: Expression Integer +--E + +--S 57 +logpow:=rule(log(a^n) == n*log(a)) +--R +--R n +--R (6) log(a ) == n log(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 58 14:155 Schaums and Axiom agree +ee:=logpow dd +--R +--R (7) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.156~~~~~$\displaystyle\int{\frac{dx}{x^2(x^2-a^2)^2}}$} +$$\int{\frac{1}{x^2(x^2-a^2)^2}}= +-\frac{1}{a^4x}-\frac{x}{2a^4(x^2-a^2)}- +\frac{3}{4a^5}\ln\left(\frac{x-a}{x+a}\right) +$$ +<<*>>= +)clear all + +--S 59 +aa:=integrate(1/(x^2*(x^2-a^2)^2),x) +--R +--R 3 2 3 2 2 3 +--R (3x - 3a x)log(x + a) + (- 3x + 3a x)log(x - a) - 6a x + 4a +--R (1) --------------------------------------------------------------- +--R 5 3 7 +--R 4a x - 4a x +--R Type: Union(Expression Integer,...) +--E + +--S 60 +bb:=-1/(a^4*x)-x/(2*a^4*(x^2-a^2))-3/(4*a^5)*log((x-a)/(x+a)) +--R +--R 3 2 x - a 2 3 +--R (- 3x + 3a x)log(-----) - 6a x + 4a +--R x + a +--R (2) -------------------------------------- +--R 5 3 7 +--R 4a x - 4a x +--R Type: Expression Integer +--E + +--S 61 +cc:=aa-bb +--R +--R x - a +--R 3log(x + a) - 3log(x - a) + 3log(-----) +--R x + a +--R (3) --------------------------------------- +--R 5 +--R 4a +--R Type: Expression Integer +--E + +--S 62 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 63 14:156 Schaums and Axiom agree +dd:=divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.157~~~~~$\displaystyle\int{\frac{dx}{x^3(x^2-a^2)^2}}$} +$$\int{\frac{1}{x^3(x^2-a^2)^2}}= +-\frac{1}{2a^4x^2}-\frac{1}{2a^4(x^2-a^2)}+ +\frac{1}{a^6}\ln\left(\frac{x^2}{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 64 +aa:=integrate(1/(x^3*(x^2-a^2)^2),x) +--R +--R +--R 4 2 2 2 2 4 2 2 2 2 4 +--R (- 2x + 2a x )log(x - a ) + (4x - 4a x )log(x) - 2a x + a +--R (1) -------------------------------------------------------------- +--R 6 4 8 2 +--R 2a x - 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 65 +bb:=-1/(2*a^4*x^2)-1/(2*a^4*(x^2-a^2))+1/a^6*log(x^2/(x^2-a^2)) +--R +--R 2 +--R 4 2 2 x 2 2 4 +--R (2x - 2a x )log(-------) - 2a x + a +--R 2 2 +--R x - a +--R (2) -------------------------------------- +--R 6 4 8 2 +--R 2a x - 2a x +--R Type: Expression Integer +--E + +--S 66 +cc:=aa-bb +--R +--R 2 +--R 2 2 x +--R - log(x - a ) + 2log(x) - log(-------) +--R 2 2 +--R x - a +--R (3) --------------------------------------- +--R 6 +--R a +--R Type: Expression Integer +--E + +--S 67 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 68 +dd:=divlog cc +--R +--R 2 +--R - log(x ) + 2log(x) +--R (5) ------------------- +--R 6 +--R a +--R Type: Expression Integer +--E + +--S 69 +logpow:=rule(log(a^n) == n*log(a)) +--R +--R n +--R (6) log(a ) == n log(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 70 14:157 Schaums and Axiom agree +ee:=logpow dd +--R +--R (7) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.158~~~~~$\displaystyle\int{\frac{dx}{(x^2-a^2)^n}}$} +$$\int{\frac{1}{(x^2-a^2)^n}}= +\frac{-x}{2(n-1)a^2(x^2-a^2)^{n-1}}- +\frac{2n-3}{(2n-2)a^2}\int{\frac{1}{(x^2-a^2)^{n-1}}} +$$ +<<*>>= +)clear all + +--S 71 14:158 Axiom cannot do this integral +aa:=integrate(1/((x^2-a^2)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | ------------- d%L +--R ++ 2 2 n +--I (- a + %L ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.159~~~~~$\displaystyle\int{\frac{x~dx}{(x^2-a^2)^n}}$} +$$\int{\frac{x}{(x^2-a^2)^n}}= +\frac{-1}{2(n-1)(x^2-a^2)^{n-1}} +$$ +<<*>>= +)clear all + +--S 72 +aa:=integrate(x/((x^2-a^2)^n),x) +--R +--R +--R 2 2 +--R - x + a +--R (1) ------------------------ +--R 2 2 +--R n log(x - a ) +--R (2n - 2)%e +--R Type: Union(Expression Integer,...) +--E + +--S 73 +bb:=-1/(2*(n-1)*(x^2-a^2)^(n-1)) +--R +--R 1 +--R (2) - ---------------------- +--R 2 2 n - 1 +--R (2n - 2)(x - a ) +--R Type: Expression Integer +--E + +--S 74 +cc:=aa-bb +--R +--R 2 2 +--R n log(x - a ) 2 2 2 2 n - 1 +--R %e + (- x + a )(x - a ) +--R (3) -------------------------------------------- +--R 2 2 +--R 2 2 n - 1 n log(x - a ) +--R (2n - 2)(x - a ) %e +--R Type: Expression Integer +--E + +--S 75 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 76 +dd:=explog cc +--R +--R 2 2 n 2 2 2 2 n - 1 +--R (x - a ) + (- x + a )(x - a ) +--R (5) -------------------------------------- +--R 2 2 n - 1 2 2 n +--R (2n - 2)(x - a ) (x - a ) +--R Type: Expression Integer +--E + +--S 77 14:159 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.160~~~~~$\displaystyle\int{\frac{dx}{x(x^2-a^2)^n}}$} +$$\int{\frac{1}{x(x^2-a^2)^n}}= +\frac{-1}{2(n-1)a^2(x^2-a^2)^{n-1}}- +\frac{1}{a^2}\int{\frac{1}{x(x^2-a^2)^{n-1}}} +$$ +<<*>>= +)clear all + +--S 78 14:160 Axiom cannot compute this integral +aa:=integrate(1/(x*(x^2-a^2)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | ---------------- d%L +--R ++ 2 2 n +--I %L (- a + %L ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.161~~~~~$\displaystyle\int{\frac{x^mdx}{(x^2-a^2)^n}}$} +$$\int{\frac{x^m}{(x^2-a^2)^n}}= +\int{\frac{x^{m-2}}{(x^2-a^2)^{n-1}}}+ +a^2\int\frac{x^{m-2}}{(x^2-a^2)^n} +$$ +<<*>>= +)clear all + +--S 79 14:161 Axiom cannot compute this integral +aa:=integrate(x^m/((x^2-a^2)^n),x) +--R +--R +--R x m +--I ++ %L +--I (1) | ------------- d%L +--R ++ 2 2 n +--I (- a + %L ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.162~~~~~$\displaystyle\int{\frac{dx}{x^m(x^2-a^2)^n}}$} +$$\int{\frac{1}{x^m(x^2-a^2)^n}}= +\frac{1}{a^2}\int{\frac{1}{x^{m-2}(x^2-a^2)^n}}- +\frac{1}{a^2}\int{\frac{1}{x^m(x^2-a^2)^{n-1}}} +$$ +<<*>>= +)clear all + +--S 80 14:162 Axiom cannot compute this integral +aa:=integrate(1/(x^m*(x^2-a^2)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | ---------------- d%L +--R ++ 2 2 n m +--I (- a + %L ) %L +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p65 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum7.input.pdf 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+$$\int{\frac{1}{a^2-x^2}}=\frac{1}{2a}\ln\left(\frac{a-x}{a+x}\right)$$ +$$\int{\frac{1}{a^2-x^2}}=-\frac{1}{a}\coth^{-1}\frac{x}{a}$$ +<<*>>= +)spool schaum8.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(1/(a^2-x^2),x) +--R +--R +--R log(x + a) - log(x - a) +--R (1) ----------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=1/(2*a)*log((a+x)/(a-x)) +--R +--R - x - a +--R log(-------) +--R x - a +--R (2) ------------ +--R 2a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R - x - a +--R log(x + a) - log(x - a) - log(-------) +--R x - a +--R (3) -------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 4 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 5 +dd:=divlog cc +--R +--R log(x + a) - log(- x - a) +--R (5) ------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 6 +logminus:=rule(log(x + a) - log(- x - a) == log(-1)) +--R +--I (6) log(x + a) - log(- x - a) + %I == log(- 1) + %I +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 7 14:163 Schaums and Axiom differ by a constant +ee:=logminus dd +--R +--R log(- 1) +--R (7) -------- +--R 2a +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.164~~~~~$\displaystyle\int{\frac{x~dx}{a^2-x^2}}$} +$$\int{\frac{x}{a^2-x^2}}=-\frac{1}{2}\ln(a^2-x^2)$$ +<<*>>= +)clear all + +--S 8 +aa:=integrate(x/(a^2-x^2),x) +--R +--R +--R 2 2 +--R log(x - a ) +--R (1) - ------------ +--R 2 +--R Type: Union(Expression Integer,...) +--E + +--S 9 +bb:=-1/2*log(a^2-x^2) +--R +--R 2 2 +--R log(- x + a ) +--R (2) - -------------- +--R 2 +--R Type: Expression Integer +--E + +--S 10 +cc:=aa-bb +--R +--R 2 2 2 2 +--R - log(x - a ) + log(- x + a ) +--R (3) ------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 11 +logminus1:=rule(-log(x^2-a^2)+log(-x^2+a^2) == log(-1)) +--R +--R 2 2 2 2 +--I (4) - log(x - a ) + log(- x + a ) + %H == log(- 1) + %H +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 12 14:164 Schaums and Axiom differ by a constant +dd:=logminus1 cc +--R +--R log(- 1) +--R (5) -------- +--R 2 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.165~~~~~$\displaystyle\int{\frac{x^2~dx}{a^2-x^2}}$} +$$\int{\frac{x^2}{a^2-x^2}}=-x+\frac{a}{2}\ln\left(\frac{a+x}{a-x}\right)$$ +<<*>>= +)clear all + +--S 13 +aa:=integrate(x^2/(a^2-x^2),x) +--R +--R +--R a log(x + a) - a log(x - a) - 2x +--R (1) -------------------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E + +--S 14 +bb:=-x+a/2*log((a+x)/(a-x)) +--R +--R - x - a +--R a log(-------) - 2x +--R x - a +--R (2) ------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 15 +cc:=aa-bb +--R +--R - x - a +--R a log(x + a) - a log(x - a) - a log(-------) +--R x - a +--R (3) -------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 16 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 17 +dd:=divlog cc +--R +--R a log(x + a) - a log(- x - a) +--R (5) ----------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 18 +logminusa:=rule(b*log(x + a) - b*log(- x - a) == b*log(-1)) +--R +--I (6) b log(x + a) - b log(- x - a) + %M == b log(- 1) + %M +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 19 14:165 Schaums and Axiom differ by a constant +ee:=logminusa dd +--R +--R a log(- 1) +--R (7) ---------- +--R 2 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.166~~~~~$\displaystyle\int{\frac{x^3~dx}{a^2-x^2}}$} +$$\int{\frac{x^3}{a^2-x^2}}=-\frac{x^2}{2}-\frac{a^2}{2}\ln(a^2-x^2)$$ + +<<*>>= +)clear all + +--S 20 +aa:=integrate(x^3/(a^2-x^2),x) +--R +--R +--R 2 2 2 2 +--R - a log(x - a ) - x +--R (1) --------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E + +--S 21 +bb:=-x^2/2-a^2/2*log(a^2-x^2) +--R +--R 2 2 2 2 +--R - a log(- x + a ) - x +--R (2) ----------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 22 +cc:=aa-bb +--R +--R 2 2 2 2 2 2 +--R - a log(x - a ) + a log(- x + a ) +--R (3) ----------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 23 +logminus1b:=rule(-b*log(x^2-a^2)+b*log(-x^2+a^2) == b*log(-1)) +--R +--R 2 2 2 2 +--I (4) - b log(x - a ) + b log(- x + a ) + %N == b log(- 1) + %N +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 24 14:166 Schaums and Axiom differ by a constant +dd:=logminus1b cc +--R +--R 2 +--R a log(- 1) +--R (5) ---------- +--R 2 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.167~~~~~$\displaystyle\int{\frac{dx}{x(a^2-x^2)}}$} +$$\int{\frac{1}{x(a^2-x^2)}}= +\frac{1}{2a^2}\ln\left(\frac{x^2}{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 25 +aa:=integrate(1/(x*(a^2-x^2)),x) +--R +--R +--R 2 2 +--R - log(x - a ) + 2log(x) +--R (1) ------------------------ +--R 2 +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 26 +bb:=1/(2*a^2)*log(x^2/(a^2-x^2)) +--R +--R 2 +--R x +--R log(- -------) +--R 2 2 +--R x - a +--R (2) -------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 27 +cc:=aa-bb +--R +--R 2 +--R 2 2 x +--R - log(x - a ) + 2log(x) - log(- -------) +--R 2 2 +--R x - a +--R (3) ----------------------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 28 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 29 +dd:=divlog cc +--R +--R 2 +--R 2log(x) - log(- x ) +--R (5) ------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 30 +logpowminus:=rule(log(-a^n) == n*log(a)+log(-1)) +--R +--R n +--R (6) log(- a ) == n log(a) + log(- 1) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 31 14:167 Schaums and Axiom differ by a constant +ee:=logpowminus dd +--R +--R log(- 1) +--R (7) - -------- +--R 2 +--R 2a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.168~~~~~$\displaystyle\int{\frac{dx}{x^2(a^2-x^2)}}$} +$$\int{\frac{1}{x^2(a^2-x^2)}}= +\frac{1}{a^2x}+\frac{1}{2a^3}\ln\left(\frac{a+x}{a-x}\right) +$$ +<<*>>= +)clear all + +--S 32 +aa:=integrate(1/(x^2*(a^2-x^2)),x) +--R +--R +--R x log(x + a) - x log(x - a) - 2a +--R (1) -------------------------------- +--R 3 +--R 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 33 +bb:=-1/(a^2*x)+1/(2*a^3)*log((a+x)/(a-x)) +--R +--R - x - a +--R x log(-------) - 2a +--R x - a +--R (2) ------------------- +--R 3 +--R 2a x +--R Type: Expression Integer +--E + +--S 34 +cc:=aa-bb +--R +--R - x - a +--R log(x + a) - log(x - a) - log(-------) +--R x - a +--R (3) -------------------------------------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 35 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 36 +dd:=divlog cc +--R +--R log(x + a) - log(- x - a) +--R (5) ------------------------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 37 +logminus:=rule(log(x + a) - log(- x - a) == log(-1)) +--R +--I (6) log(x + a) - log(- x - a) + %O == log(- 1) + %O +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 38 14:168 Schaums and Axiom differ by a constant +ee:=logminus dd +--R +--R log(- 1) +--R (7) -------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.169~~~~~$\displaystyle\int{\frac{dx}{x^3(a^2-x^2)}}$} +$$\int{\frac{1}{x^3(a^2-x^2)}}= +-\frac{1}{2a^2x^2}+\frac{1}{2a^4}\ln\left(\frac{x^2}{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 39 +aa:=integrate(1/(x^3*(a^2-x^2)),x) +--R +--R +--R 2 2 2 2 2 +--R - x log(x - a ) + 2x log(x) - a +--R (1) --------------------------------- +--R 4 2 +--R 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 40 +bb:=-1/(2*a^2*x^2)+1/(2*a^4)*log(x^2/(a^2-x^2)) +--R +--R 2 +--R 2 x 2 +--R x log(- -------) - a +--R 2 2 +--R x - a +--R (2) --------------------- +--R 4 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 41 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (3) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 42 +bb1:=divlog bb +--R +--R 2 2 2 2 2 2 +--R - x log(x - a ) + x log(- x ) - a +--R (4) ----------------------------------- +--R 4 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 43 +cc:=aa-bb1 +--R +--R 2 +--R 2log(x) - log(- x ) +--R (5) ------------------- +--R 4 +--R 2a +--R Type: Expression Integer +--E + +--S 44 +logminuspow:=rule(log(-x^n) == n*log(x)+log(-1)) +--R +--R n +--R (6) log(- x ) == n log(x) + log(- 1) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 45 14:169 Schaums and Axiom differ by a constant +dd:=logminuspow cc +--R +--R log(- 1) +--R (7) - -------- +--R 4 +--R 2a +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.170~~~~~$\displaystyle\int{\frac{dx}{(a^2-x^2)^2}}$} +$$\int{\frac{1}{(a^2-x^2)^2}}= +\frac{x}{2a^2(a^2-x^2)}+\frac{1}{4a^3}\ln\left(\frac{a+x}{a-x}\right) +$$ +<<*>>= +)clear all + +--S 46 +aa:=integrate(1/((a^2-x^2)^2),x) +--R +--R +--R 2 2 2 2 +--R (x - a )log(x + a) + (- x + a )log(x - a) - 2a x +--R (1) -------------------------------------------------- +--R 3 2 5 +--R 4a x - 4a +--R Type: Union(Expression Integer,...) +--E + +--S 47 +bb:=x/(2*a^2*(a^2-x^2))+1/(4*a^3)*log((a+x)/(a-x)) +--R +--R 2 2 - x - a +--R (x - a )log(-------) - 2a x +--R x - a +--R (2) ---------------------------- +--R 3 2 5 +--R 4a x - 4a +--R Type: Expression Integer +--E + +--S 48 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (3) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 49 +bb1:=divlog bb +--R +--R 2 2 2 2 +--R (- x + a )log(x - a) + (x - a )log(- x - a) - 2a x +--R (4) ---------------------------------------------------- +--R 3 2 5 +--R 4a x - 4a +--R Type: Expression Integer +--E + +--S 50 +cc:=aa-bb1 +--R +--R log(x + a) - log(- x - a) +--R (5) ------------------------- +--R 3 +--R 4a +--R Type: Expression Integer +--E + +--S 51 +logminus:=rule(log(x + a) - log(- x - a) == log(-1)) +--R +--I (6) log(x + a) - log(- x - a) + %P == log(- 1) + %P +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 52 14:170 Schaums and Axiom differ by a constant +dd:=logminus cc +--R +--R log(- 1) +--R (7) -------- +--R 3 +--R 4a +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.171~~~~~$\displaystyle\int{\frac{x~dx}{(a^2-x^2)^2}}$} +$$\int{\frac{x}{(a^2-x^2)^2}}= +\frac{1}{2(a^2-x^2)} +$$ +<<*>>= +)clear all + +--S 53 +aa:=integrate(x/((a^2-x^2)^2),x) +--R +--R +--R 1 +--R (1) - --------- +--R 2 2 +--R 2x - 2a +--R Type: Union(Expression Integer,...) +--E + +--S 54 +bb:=1/(2*(a^2-x^2)) +--R +--R 1 +--R (2) - --------- +--R 2 2 +--R 2x - 2a +--R Type: Fraction Polynomial Integer +--E + +--S 55 14:171 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.172~~~~~$\displaystyle\int{\frac{x^2dx}{(a^2-x^2)^2}}$} +$$\int{\frac{x^2}{(a^2-x^2)^2}}= +\frac{x}{2(a^2-x^2)}-\frac{1}{4a}\ln\left(\frac{a+x}{a-x}\right) +$$ +<<*>>= +)clear all + +--S 56 +aa:=integrate(x^2/((a^2-x^2)^2),x) +--R +--R +--R 2 2 2 2 +--R (- x + a )log(x + a) + (x - a )log(x - a) - 2a x +--R (1) -------------------------------------------------- +--R 2 3 +--R 4a x - 4a +--R Type: Union(Expression Integer,...) +--E + +--S 57 +bb:=x/(2*(a^2-x^2))-1/(4*a)*log((a+x)/(a-x)) +--R +--R 2 2 - x - a +--R (- x + a )log(-------) - 2a x +--R x - a +--R (2) ------------------------------ +--R 2 3 +--R 4a x - 4a +--R Type: Expression Integer +--E + +--S 58 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (3) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 59 +bb1:=divlog bb +--R +--R 2 2 2 2 +--R (x - a )log(x - a) + (- x + a )log(- x - a) - 2a x +--R (4) ---------------------------------------------------- +--R 2 3 +--R 4a x - 4a +--R Type: Expression Integer +--E + +--S 60 +cc:=aa-bb1 +--R +--R - log(x + a) + log(- x - a) +--R (5) --------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 61 +logminus2:=rule(-log(x + a) + log(- x - a) == log(-1)) +--R +--I (6) - log(x + a) + log(- x - a) + %S == log(- 1) + %S +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 62 14:172 Schaums and Axiom differ by a constant +dd:=logminus2 cc +--R +--R log(- 1) +--R (7) -------- +--R 4a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.173~~~~~$\displaystyle\int{\frac{x^3dx}{(a^2-x^2)^2}}$} +$$\int{\frac{x^3}{(a^2-x^2)^2}}= +\frac{a^2}{2(a^2-x^2)}+\frac{1}{2}\ln(a^2-x^2) +$$ +<<*>>= +)clear all + +--S 63 +aa:=integrate(x^3/((a^2-x^2)^2),x) +--R +--R +--R 2 2 2 2 2 +--R (x - a )log(x - a ) - a +--R (1) -------------------------- +--R 2 2 +--R 2x - 2a +--R Type: Union(Expression Integer,...) +--E + +--S 64 +bb:=a^2/(2*(a^2-x^2))+1/2*log(a^2-x^2) +--R +--R 2 2 2 2 2 +--R (x - a )log(- x + a ) - a +--R (2) ---------------------------- +--R 2 2 +--R 2x - 2a +--R Type: Expression Integer +--E + +--S 65 +cc:=aa-bb +--R +--R 2 2 2 2 +--R log(x - a ) - log(- x + a ) +--R (3) ----------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 66 +logminus3:=rule(log(x^2-a^2)-log(-x^2+a^2) == log(-1)) +--R +--R 2 2 2 2 +--I (4) log(x - a ) - log(- x + a ) + %T == log(- 1) + %T +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 67 14:173 Schaums and Axiom differ by a constant +dd:=logminus3 cc +--R +--R log(- 1) +--R (5) -------- +--R 2 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.174~~~~~$\displaystyle\int{\frac{dx}{x(a^2-x^2)^2}}$} +$$\int{\frac{1}{x(a^2-x^2)^2}}= +\frac{1}{2a^2(a^2-x^2)}+\frac{1}{2a^4}\ln\left(\frac{x^2}{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 68 +aa:=integrate(1/(x*(a^2-x^2)^2),x) +--R +--R +--R 2 2 2 2 2 2 2 +--R (- x + a )log(x - a ) + (2x - 2a )log(x) - a +--R (1) ------------------------------------------------ +--R 4 2 6 +--R 2a x - 2a +--R Type: Union(Expression Integer,...) +--E + +--S 69 +bb:=1/(2*a^2*(a^2-x^2))+1/(2*a^4)*log(x^2/(a^2-x^2)) +--R +--R 2 +--R 2 2 x 2 +--R (x - a )log(- -------) - a +--R 2 2 +--R x - a +--R (2) ---------------------------- +--R 4 2 6 +--R 2a x - 2a +--R Type: Expression Integer +--E + +--S 70 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (3) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 71 +bb1:=divlog bb +--R +--R 2 2 2 2 2 2 2 2 +--R (- x + a )log(x - a ) + (x - a )log(- x ) - a +--R (4) ------------------------------------------------- +--R 4 2 6 +--R 2a x - 2a +--R Type: Expression Integer +--E + +--S 72 +cc:=aa-bb1 +--R +--R 2 +--R 2log(x) - log(- x ) +--R (5) ------------------- +--R 4 +--R 2a +--R Type: Expression Integer +--E + +--S 73 +logpowminus:=rule(log(-a^n) == n*log(a)+log(-1)) +--R +--R n +--R (6) log(- a ) == n log(a) + log(- 1) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 74 14:174 Schaums and Axiom differ by a constant +dd:=logpowminus cc +--R +--R log(- 1) +--R (7) - -------- +--R 4 +--R 2a +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.175~~~~~$\displaystyle\int{\frac{dx}{x^2(a^2-x^2)^2}}$} +$$\int{\frac{1}{x^2(a^2-x^2)^2}}= +-\frac{1}{a^4x}+\frac{x}{2a^4(a^2-x^2)}+ +\frac{3}{4a^5}\ln\left(\frac{a+x}{a-x}\right) +$$ +<<*>>= +)clear all + +--S 75 +aa:=integrate(1/(x^2*(a^2-x^2)^2),x) +--R +--R 3 2 3 2 2 3 +--R (3x - 3a x)log(x + a) + (- 3x + 3a x)log(x - a) - 6a x + 4a +--R (1) --------------------------------------------------------------- +--R 5 3 7 +--R 4a x - 4a x +--R Type: Union(Expression Integer,...) +--E + +--S 76 +bb:=-1/(a^4*x)+x/(2*a^4*(a^2-x^2))+3/(4*a^5)*log((a+x)/(a-x)) +--R +--R 3 2 - x - a 2 3 +--R (3x - 3a x)log(-------) - 6a x + 4a +--R x - a +--R (2) -------------------------------------- +--R 5 3 7 +--R 4a x - 4a x +--R Type: Expression Integer +--E + +--S 77 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (3) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 78 +bb1:=divlog bb +--R +--R 3 2 3 2 2 3 +--R (- 3x + 3a x)log(x - a) + (3x - 3a x)log(- x - a) - 6a x + 4a +--R (4) ----------------------------------------------------------------- +--R 5 3 7 +--R 4a x - 4a x +--R Type: Expression Integer +--E + +--S 79 +cc:=aa-bb +--R +--R - x - a +--R 3log(x + a) - 3log(x - a) - 3log(-------) +--R x - a +--R (5) ----------------------------------------- +--R 5 +--R 4a +--R Type: Expression Integer +--E + +--S 80 +dd:=divlog cc +--R +--R 3log(x + a) - 3log(- x - a) +--R (6) --------------------------- +--R 5 +--R 4a +--R Type: Expression Integer +--E + +--S 81 +logminusb:=rule(b*log(x + a) - b*log(- x - a) == b*log(-1)) +--R +--I (7) b log(x + a) - b log(- x - a) + %U == b log(- 1) + %U +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 82 14:175 Schaums and Axiom differ by a constant +ee:=logminusb dd +--R +--R 3log(- 1) +--R (8) --------- +--R 5 +--R 4a +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.176~~~~~$\displaystyle\int{\frac{dx}{x^3(a^2-x^2)^2}}$} +$$\int{\frac{1}{x^3(a^2-x^2)^2}}= +\frac{1}{2a^4x^2}+\frac{1}{2a^4(a^2-x^2)}+ +\frac{1}{a^6}\ln\left(\frac{x^2}{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 83 +aa:=integrate(1/(x^3*(a^2-x^2)^2),x) +--R +--R +--R 4 2 2 2 2 4 2 2 2 2 4 +--R (- 2x + 2a x )log(x - a ) + (4x - 4a x )log(x) - 2a x + a +--R (1) -------------------------------------------------------------- +--R 6 4 8 2 +--R 2a x - 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 84 +bb:=-1/(2*a^4*x^2)+1/(2*a^4*(a^2-x^2))+1/a^6*log(x^2/(a^2-x^2)) +--R +--R 2 +--R 4 2 2 x 2 2 4 +--R (2x - 2a x )log(- -------) - 2a x + a +--R 2 2 +--R x - a +--R (2) ---------------------------------------- +--R 6 4 8 2 +--R 2a x - 2a x +--R Type: Expression Integer +--E + +--S 85 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (3) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 86 +bb1:=divlog bb +--R +--R 4 2 2 2 2 4 2 2 2 2 2 4 +--R (- 2x + 2a x )log(x - a ) + (2x - 2a x )log(- x ) - 2a x + a +--R (4) ----------------------------------------------------------------- +--R 6 4 8 2 +--R 2a x - 2a x +--R Type: Expression Integer +--E + +--S 87 +cc:=aa-bb1 +--R +--R 2 +--R 2log(x) - log(- x ) +--R (5) ------------------- +--R 6 +--R a +--R Type: Expression Integer +--E + +--S 88 +logpowminus:=rule(log(-a^n) == n*log(a)+log(-1)) +--R +--R n +--R (6) log(- a ) == n log(a) + log(- 1) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 89 14:176 Schaums and Axiom differ by a constant +dd:=logpowminus cc +--R +--R log(- 1) +--R (7) - -------- +--R 6 +--R a +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.177~~~~~$\displaystyle\int{\frac{dx}{(a^2-x^2)^n}}$} +$$\int{\frac{1}{(a^2-x^2)^n}}= +\frac{x}{2(n-1)a^2(a^2-x^2)^{n-1}}+ +\frac{2n-3}{(2n-2)a^2}\int{\frac{1}{(a^2-x^2)^{n-1}}} +$$ +<<*>>= +)clear all + +--S 90 14:177 Axiom cannot do this integration +aa:=integrate(1/((a^2-x^2)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | ----------- d%L +--R ++ 2 2 n +--I (a - %L ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.178~~~~~$\displaystyle\int{\frac{x~dx}{(a^2-x^2)^n}}$} +$$\int{\frac{x}{(a^2-x^2)^n}}= +\frac{1}{2(n-1)(a^2-x^2)^{n-1}} +$$ +<<*>>= +)clear all + +--S 91 +aa:=integrate(x/((a^2-x^2)^n),x) +--R +--R +--R 2 2 +--R - x + a +--R (1) -------------------------- +--R 2 2 +--R n log(- x + a ) +--R (2n - 2)%e +--R Type: Union(Expression Integer,...) +--E + +--S 92 +bb:=1/(2*(n-1)*(a^2-x^2)^(n-1)) +--R +--R 1 +--R (2) ------------------------ +--R 2 2 n - 1 +--R (2n - 2)(- x + a ) +--R Type: Expression Integer +--E + +--S 93 +cc:=aa-bb +--R +--R 2 2 +--R n log(- x + a ) 2 2 2 2 n - 1 +--R - %e + (- x + a )(- x + a ) +--R (3) -------------------------------------------------- +--R 2 2 +--R 2 2 n - 1 n log(- x + a ) +--R (2n - 2)(- x + a ) %e +--R Type: Expression Integer +--E + +--S 94 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 95 +dd:=explog cc +--R +--R 2 2 n 2 2 2 2 n - 1 +--R - (- x + a ) + (- x + a )(- x + a ) +--R (5) -------------------------------------------- +--R 2 2 n - 1 2 2 n +--R (2n - 2)(- x + a ) (- x + a ) +--R Type: Expression Integer +--E + +--S 96 14:178 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.179~~~~~$\displaystyle\int{\frac{dx}{x(a^2-x^2)^n}}$} +$$\int{\frac{1}{x(a^2-x^2)^n}}= +\frac{1}{2(n-1)a^2(a^2-x^2)^{n-1}}+ +\frac{1}{a^2}\int{\frac{1}{x(a^2-x^2)^{n-1}}} +$$ +<<*>>= +)clear all + +--S 97 14:179 Axiom cannot integrate this expression +aa:=integrate(1/(x*(a^2-x^2)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | -------------- d%L +--R ++ 2 2 n +--I %L (a - %L ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.180~~~~~$\displaystyle\int{\frac{x^mdx}{(a^2-x^2)^n}}$} +$$\int{\frac{x^m}{(a^2-x^2)^n}}= +a^2\int\frac{x^{m-2}}{(a^2-x^2)^n}- +\int{\frac{x^{m-2}}{(a^2-x^2)^{n-1}}} +$$ +<<*>>= +)clear all + +--S 98 14:180 Axiom cannot integrate this expression +aa:=integrate(x^m/((a^2-x^2)^n),x) +--R +--R +--R x m +--I ++ %L +--I (1) | ----------- d%L +--R ++ 2 2 n +--I (a - %L ) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.181~~~~~$\displaystyle\int{\frac{dx}{x^m(a^2-x^2)^n}}$} +$$\int{\frac{1}{x^m(a^2-x^2)^n}}= +\frac{1}{a^2}\int{\frac{1}{x^m(a^2-x^2)^{n-1}}}+ +\frac{1}{a^2}\int{\frac{1}{x^{m-2}(a^2-x^2)^n}} +$$ +<<*>>= +)clear all + +--S 99 14:181 Axiom cannot integrate this expression +aa:=integrate(1/(x^m*(a^2-x^2)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | -------------- d%L +--R ++ m 2 2 n +--I %L (a - %L ) +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p66 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum8.input.pdf b/src/axiom-website/CATS/schaum8.input.pdf new file mode 100644 index 0000000..9b904e9 --- /dev/null +++ b/src/axiom-website/CATS/schaum8.input.pdf @@ -0,0 +1,2710 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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--- /dev/null +++ b/src/axiom-website/CATS/schaum9.input.pamphlet @@ -0,0 +1,1700 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum9.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.182~~~~~$\displaystyle\int{\frac{dx}{\sqrt{x^2+a^2}}}$} +$$\int{\frac{1}{\sqrt{x^2+a^2}}}=\ln\left(x+\sqrt{x^2+a^2}\right)$$ +$$\int{\frac{1}{\sqrt{x^2+a^2}}}=\sinh^{-1}\frac{x}{a}$$ +<<*>>= +)spool schaum9.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(1/(sqrt(x^2+a^2)),x) +--R +--R +--R +-------+ +--R | 2 2 +--R (1) - log(\|x + a - x) +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=log(x+sqrt(x^2+a^2)) +--R +--R +-------+ +--R | 2 2 +--R (2) log(\|x + a + x) +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R (3) - log(\|x + a + x) - log(\|x + a - x) +--R Type: Expression Integer +--E + +--S 4 14:182 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 +--R (4) - log(a ) +--R Type: Expression Integer +--E + +@ +This is equal to $-\log(a^2)$ but Axiom cannot prove it. + +\section{\cite{1}:14.183~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{x^2+a^2}}}$} +$$\int{\frac{x}{\sqrt{x^2+a^2}}}=\sqrt{x^2+a^2}$$ +<<*>>= +)clear all + +--S 5 +aa:=integrate(x/(sqrt(x^2+a^2)),x) +--R +--R +--R +-------+ +--R | 2 2 2 2 +--R - x\|x + a + x + a +--R (1) ----------------------- +--R +-------+ +--R | 2 2 +--R \|x + a - x +--R Type: Union(Expression Integer,...) +--E + +--S 6 +bb:=sqrt(x^2+a^2) +--R +--R +-------+ +--R | 2 2 +--R (2) \|x + a +--R Type: Expression Integer +--E + +--S 7 14:183 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.184~~~~~$\displaystyle +\int{\frac{x^2~dx}{\sqrt{x^2+a^2}}}$} +$$\int{\frac{x^2}{\sqrt{x^2+a^2}}}= +\frac{x\sqrt{x^2+a^2}}{2}-\frac{a^2}{2}\ln\left(x+\sqrt{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 8 +aa:=integrate(x^2/sqrt(x^2+a^2),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 2 | 2 2 2 2 4 | 2 2 +--R (2a x\|x + a - 2a x - a )log(\|x + a - x) +--R + +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (- 2x - a x)\|x + a + 2x + 2a x +--R / +--R +-------+ +--R | 2 2 2 2 +--R 4x\|x + a - 4x - 2a +--R Type: Union(Expression Integer,...) +--E + +--S 9 +bb:=(x*sqrt(x^2+a^2))/2-a^2/2*log(x+sqrt(x^2+a^2)) +--R +--R +-------+ +-------+ +--R 2 | 2 2 | 2 2 +--R - a log(\|x + a + x) + x\|x + a +--R (2) ------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 10 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R a log(\|x + a + x) + a log(\|x + a - x) +--R (3) --------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 11 +logmul1:=rule(c*log(a)+c*log(b) == c*log(a*b)) +--R +--I (4) c log(b) + c log(a) + %K == c log(a b) + %K +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 12 14:184 Schaums and Axiom differ by a constant +dd:=logmul1 cc +--R +--R 2 2 +--R a log(a ) +--R (5) --------- +--R 2 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.185~~~~~$\displaystyle +\int{\frac{x^3~dx}{\sqrt{x^2+a^2}}}$} +$$\int{\frac{x^3}{\sqrt{x^2+a^2}}}= +\frac{(x^2+a^2)^{3/2}}{3}-a^2\sqrt{x^2+a^2} +$$ +<<*>>= +)clear all + +--S 13 +aa:=integrate(x^3/sqrt(x^2+a^2),x) +--R +--R +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x + 5a x + 6a x)\|x + a + 4x - 3a x - 9a x - 2a +--R (1) ------------------------------------------------------------ +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (12x + 3a )\|x + a - 12x - 9a x +--R Type: Union(Expression Integer,...) +--E + +--S 14 +bb:=(x^2+a^2)^(3/2)/3-a^2*sqrt(x^2+a^2) +--R +--R +-------+ +--R 2 2 | 2 2 +--R (x - 2a )\|x + a +--R (2) -------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 15 14:185 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.186~~~~~$\displaystyle\int{\frac{dx}{x\sqrt{x^2+a^2}}}$} +$$\int{\frac{1}{x\sqrt{x^2+a^2}}}= +-\frac{1}{a}\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 16 +aa:=integrate(1/(x*sqrt(x^2+a^2)),x) +--R +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - log(\|x + a - x + a) + log(\|x + a - x - a) +--R (1) --------------------------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 17 +bb:=-1/a*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R | 2 2 +--R \|x + a + a +--R log(--------------) +--R x +--R (2) - ------------------- +--R a +--R Type: Expression Integer +--E + +--S 18 +cc:=aa-bb +--R +--R (3) +--R +-------+ +--R +-------+ +-------+ | 2 2 +--R | 2 2 | 2 2 \|x + a + a +--R - log(\|x + a - x + a) + log(\|x + a - x - a) + log(--------------) +--R x +--R ------------------------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 19 +dd:=expandLog cc +--R +--R (4) +--R +-------+ +-------+ +-------+ +--R | 2 2 | 2 2 | 2 2 +--R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x - a) +--R + +--R - log(x) +--R / +--R a +--R Type: Expression Integer +--E + +--S 20 14:186 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R log(- 1) +--R (5) - -------- +--R a +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.187~~~~~$\displaystyle +\int{\frac{dx}{x^2\sqrt{x^2+a^2}}}$} +$$\int{\frac{1}{x^2\sqrt{x^2+a^2}}}= +-\frac{\sqrt{x^2+a^2}}{a^2x} +$$ +<<*>>= +)clear all + +--S 21 +aa:=integrate(1/(x^2*sqrt(x^2+a^2)),x) +--R +--R +--R 1 +--R (1) - ---------------- +--R +-------+ +--R | 2 2 2 +--R x\|x + a - x +--R Type: Union(Expression Integer,...) +--E + +--S 22 +bb:=-sqrt(x^2+a^2)/(a^2*x) +--R +--R +-------+ +--R | 2 2 +--R \|x + a +--R (2) - ---------- +--R 2 +--R a x +--R Type: Expression Integer +--E + +--S 23 14:187 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 1 +--R (3) - -- +--R 2 +--R a +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.188~~~~~$\displaystyle\int{\frac{dx}{x^3\sqrt{x^2+a^2}}}$} +$$\int{\frac{1}{x^3\sqrt{x^2+a^2}}}= +-\frac{\sqrt{x^2+a^2}}{2a^2x^2}+\frac{1}{2a^3} +\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 24 +aa:=integrate(1/(x^3*sqrt(x^2+a^2)),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 3 | 2 2 4 2 2 | 2 2 +--R (2x \|x + a - 2x - a x )log(\|x + a - x + a) +--R + +--R +-------+ +-------+ +--R 3 | 2 2 4 2 2 | 2 2 +--R (- 2x \|x + a + 2x + a x )log(\|x + a - x - a) +--R + +--R +-------+ +--R 2 3 | 2 2 3 3 +--R (2a x + a )\|x + a - 2a x - 2a x +--R / +--R +-------+ +--R 3 3 | 2 2 3 4 5 2 +--R 4a x \|x + a - 4a x - 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 25 +bb:=-sqrt(x^2+a^2)/(2*a^2*x^2)+1/(2*a^3)*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R | 2 2 +-------+ +--R 2 \|x + a + a | 2 2 +--R x log(--------------) - a\|x + a +--R x +--R (2) ----------------------------------- +--R 3 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 26 +cc:=aa-bb +--R +--R (3) +--R +-------+ +--R +-------+ +-------+ | 2 2 +--R | 2 2 | 2 2 \|x + a + a +--R log(\|x + a - x + a) - log(\|x + a - x - a) - log(--------------) +--R x +--R ----------------------------------------------------------------------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 27 +dd:=expandLog cc +--R +--R (4) +--R +-------+ +-------+ +-------+ +--R | 2 2 | 2 2 | 2 2 +--R - log(\|x + a + a) + log(\|x + a - x + a) - log(\|x + a - x - a) +--R + +--R log(x) +--R / +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 28 14:188 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R log(- 1) +--R (5) -------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.189~~~~~$\displaystyle\int{\sqrt{x^2+a^2}}~dx$} +$$\int{\sqrt{x^2+a^2}}= +\frac{x\sqrt{x^2+a^2}}{2}+\frac{a^2}{2}\ln\left(x+\sqrt{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 29 +aa:=integrate(sqrt(x^2+a^2),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 2 | 2 2 2 2 4 | 2 2 +--R (- 2a x\|x + a + 2a x + a )log(\|x + a - x) +--R + +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (- 2x - a x)\|x + a + 2x + 2a x +--R / +--R +-------+ +--R | 2 2 2 2 +--R 4x\|x + a - 4x - 2a +--R Type: Union(Expression Integer,...) +--E + +--S 30 +bb:=(x*sqrt(x^2+a^2))/2+a^2/2*log(x+sqrt(x^2+a^2)) +--R +--R +-------+ +-------+ +--R 2 | 2 2 | 2 2 +--R a log(\|x + a + x) + x\|x + a +--R (2) ----------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 31 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R - a log(\|x + a + x) - a log(\|x + a - x) +--R (3) ----------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 32 14:189 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 2 +--R a log(a ) +--R (4) - --------- +--R 2 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.190~~~~~$\displaystyle\int{x\sqrt{x^2+a^2}}~dx$} +$$\int{x\sqrt{x^2+a^2}}= +\frac{(x^2+a^2)^{3/2}}{3} +$$ +<<*>>= +)clear all + +--S 33 +aa:=integrate(x*sqrt(x^2+a^2),x) +--R +--R +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x - 7a x - 3a x)\|x + a + 4x + 9a x + 6a x + a +--R (1) ----------------------------------------------------------- +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (12x + 3a )\|x + a - 12x - 9a x +--R Type: Union(Expression Integer,...) +--E + +--S 34 +bb:=(x^2+a^2)^(3/2)/3 +--R +--R +-------+ +--R 2 2 | 2 2 +--R (x + a )\|x + a +--R (2) ------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 35 14:190 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.191~~~~~$\displaystyle +\int{x^2\sqrt{x^2+a^2}}~dx$} +$$\int{x^2\sqrt{x^2+a^2}}= +\frac{x(x^2+a^2)^{3/2}}{4}-\frac{a^2x\sqrt{x^2+a^2}}{8}- +\frac{a^4}{8}\ln\left(x+\sqrt{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 36 +aa:=integrate(x^2*sqrt(x^2+a^2),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 4 3 6 | 2 2 4 4 6 2 8 | 2 2 +--R ((8a x + 4a x)\|x + a - 8a x - 8a x - a )log(\|x + a - x) +--R + +--R +-------+ +--R 7 2 5 4 3 6 | 2 2 8 2 6 4 4 6 2 +--R (- 16x - 24a x - 10a x - a x)\|x + a + 16x + 32a x + 20a x + 4a x +--R / +--R +-------+ +--R 3 2 | 2 2 4 2 2 4 +--R (64x + 32a x)\|x + a - 64x - 64a x - 8a +--R Type: Union(Expression Integer,...) +--E + +--S 37 +bb:=(x*(x^2+a^2)^(3/2))/4-(a^2*x*sqrt(x^2+a^2))/8-a^4/8*log(x+sqrt(x^2+a^2)) +--R +--R +-------+ +-------+ +--R 4 | 2 2 3 2 | 2 2 +--R - a log(\|x + a + x) + (2x + a x)\|x + a +--R (2) ----------------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 38 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 4 | 2 2 4 | 2 2 +--R a log(\|x + a + x) + a log(\|x + a - x) +--R (3) --------------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 39 14:191 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 4 2 +--R a log(a ) +--R (4) --------- +--R 8 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.192~~~~~$\displaystyle +\int{x^3\sqrt{x^2+a^2}}~dx$} +$$\int{x^3\sqrt{x^2+a^2}}= +\frac{(x^2+a^2)^{5/2}}{5}-\frac{a^2(x^2+a^2)^{3/2}}{3} +$$ +<<*>>= +)clear all + +--S 40 +aa:=integrate(x^3*sqrt(x^2+a^2),x) +--R +--R +--R (1) +--R +-------+ +--R 9 2 7 4 5 6 3 8 | 2 2 10 2 8 +--R (- 48x - 76a x - 3a x + 35a x + 10a x)\|x + a + 48x + 100a x +--R + +--R 4 6 6 4 8 2 10 +--R 35a x - 40a x - 25a x - 2a +--R / +--R +-------+ +--R 4 2 2 4 | 2 2 5 2 3 4 +--R (240x + 180a x + 15a )\|x + a - 240x - 300a x - 75a x +--R Type: Union(Expression Integer,...) +--E + +--S 41 +bb:=(x^2+a^2)^(5/2)/5-(a^2*(x^2+a^2)^(3/2))/3 +--R +--R +-------+ +--R 4 2 2 4 | 2 2 +--R (3x + a x - 2a )\|x + a +--R (2) ---------------------------- +--R 15 +--R Type: Expression Integer +--E + +--S 42 14:192 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.193~~~~~$\displaystyle +\int{\frac{\sqrt{x^2+a^2}}{x}}~dx$} +$$\int{\frac{\sqrt{x^2+a^2}}{x}}= +\sqrt{x^2+a^2}-a\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 43 +aa:=integrate(sqrt(x^2+a^2)/x,x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R (- a\|x + a + a x)log(\|x + a - x + a) +--R + +--R +-------+ +-------+ +-------+ +--R | 2 2 | 2 2 | 2 2 2 2 +--R (a\|x + a - a x)log(\|x + a - x - a) - x\|x + a + x + a +--R / +--R +-------+ +--R | 2 2 +--R \|x + a - x +--R Type: Union(Expression Integer,...) +--E + +--S 44 +bb:=sqrt(x^2+a^2)-a*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R | 2 2 +-------+ +--R \|x + a + a | 2 2 +--R (2) - a log(--------------) + \|x + a +--R x +--R Type: Expression Integer +--E + +--S 45 +cc:=aa-bb +--R +--R (3) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - a log(\|x + a - x + a) + a log(\|x + a - x - a) +--R + +--R +-------+ +--R | 2 2 +--R \|x + a + a +--R a log(--------------) +--R x +--R Type: Expression Integer +--E + +--S 46 +dd:=expandLog cc +--R +--R (4) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R a log(\|x + a + a) - a log(\|x + a - x + a) +--R + +--R +-------+ +--R | 2 2 +--R a log(\|x + a - x - a) - a log(x) +--R Type: Expression Integer +--E + +--S 47 14:193 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R (5) - a log(- 1) +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.194~~~~~$\displaystyle +\int{\frac{\sqrt{x^2+a^2}}{x^2}}~dx$} +$$\int{\frac{\sqrt{x^2+a^2}}{x^2}}= +-\frac{\sqrt{x^2+a^2}}{x}+\ln\left(x+\sqrt{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 48 +aa:=integrate(sqrt(x^2+a^2)/x^2,x) +--R +--R +--R +-------+ +-------+ +--R | 2 2 2 | 2 2 2 +--R (- x\|x + a + x )log(\|x + a - x) - a +--R (1) -------------------------------------------- +--R +-------+ +--R | 2 2 2 +--R x\|x + a - x +--R Type: Union(Expression Integer,...) +--E + +--S 49 +bb:=-sqrt(x^2+a^2)/x+log(x+sqrt(x^2+a^2)) +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R x log(\|x + a + x) - \|x + a +--R (2) ---------------------------------- +--R x +--R Type: Expression Integer +--E + +--S 50 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R (3) - log(\|x + a + x) - log(\|x + a - x) - 1 +--R Type: Expression Integer +--E + +--S 51 14:194 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 +--R (4) - log(a ) - 1 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.195~~~~~$\displaystyle +\int{\frac{\sqrt{x^2+a^2}}{x^3}}~dx$} +$$\int{\frac{\sqrt{x^2+a^2}}{x^3}}= +-\frac{\sqrt{x^2+a^2}}{2x^2}-\frac{1}{2a} +\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 52 +aa:=integrate(sqrt(x^2+a^2)/x^3,x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 3 | 2 2 4 2 2 | 2 2 +--R (- 2x \|x + a + 2x + a x )log(\|x + a - x + a) +--R + +--R +-------+ +-------+ +--R 3 | 2 2 4 2 2 | 2 2 +--R (2x \|x + a - 2x - a x )log(\|x + a - x - a) +--R + +--R +-------+ +--R 2 3 | 2 2 3 3 +--R (2a x + a )\|x + a - 2a x - 2a x +--R / +--R +-------+ +--R 3 | 2 2 4 3 2 +--R 4a x \|x + a - 4a x - 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 53 +bb:=-sqrt(x^2+a^2)/(2*x^2)-1/(2*a)*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R | 2 2 +-------+ +--R 2 \|x + a + a | 2 2 +--R - x log(--------------) - a\|x + a +--R x +--R (2) ------------------------------------- +--R 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 54 +cc:=aa-bb +--R +--R (3) +--R +-------+ +--R +-------+ +-------+ | 2 2 +--R | 2 2 | 2 2 \|x + a + a +--R - log(\|x + a - x + a) + log(\|x + a - x - a) + log(--------------) +--R x +--R ------------------------------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 55 +dd:=expandLog cc +--R +--R (4) +--R +-------+ +-------+ +-------+ +--R | 2 2 | 2 2 | 2 2 +--R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x - a) +--R + +--R - log(x) +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 56 14:195 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R log(- 1) +--R (5) - -------- +--R 2a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.196~~~~~$\displaystyle\int{\frac{dx}{(x^2+a^2)^{3/2}}}$} +$$\int{\frac{1}{(x^2+a^2)^{3/2}}}= +\frac{x}{a^2\sqrt{x^2+a^2}} +$$ +<<*>>= +)clear all + +--S 57 +aa:=integrate(1/(x^2+a^2)^(3/2),x) +--R +--R +--R 1 +--R (1) - --------------------- +--R +-------+ +--R | 2 2 2 2 +--R x\|x + a - x - a +--R Type: Union(Expression Integer,...) +--E + +--S 58 +bb:=x/(a^2*sqrt(x^2+a^2)) +--R +--R x +--R (2) ------------ +--R +-------+ +--R 2 | 2 2 +--R a \|x + a +--R Type: Expression Integer +--E + +--S 59 14:196 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 1 +--R (3) -- +--R 2 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.197~~~~~$\displaystyle +\int{\frac{x~dx}{(x^2+a^2)^{3/2}}}$} +$$\int{\frac{x}{(x^2+a^2)^{3/2}}}= +\frac{-1}{\sqrt{x^2+a^2}} +$$ +<<*>>= +)clear all + +--S 60 +aa:=integrate(x/(x^2+a^2)^(3/2),x) +--R +--R +--R +-------+ +--R | 2 2 +--R \|x + a - x +--R (1) --------------------- +--R +-------+ +--R | 2 2 2 2 +--R x\|x + a - x - a +--R Type: Union(Expression Integer,...) +--E + +--S 61 +bb:=-1/sqrt(x^2+a^2) +--R +--R 1 +--R (2) - ---------- +--R +-------+ +--R | 2 2 +--R \|x + a +--R Type: Expression Integer +--E + +--S 62 14:197 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.198~~~~~$\displaystyle +\int{\frac{x^2dx}{(x^2+a^2)^{3/2}}}$} +$$\int{\frac{x^2}{(x^2+a^2)^{3/2}}}= +\frac{-x}{\sqrt{x^2+a^2}}+\ln\left(x+\sqrt{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 63 +aa:=integrate(x^2/(x^2+a^2)^(3/2),x) +--R +--R +--R +-------+ +-------+ +--R | 2 2 2 2 | 2 2 2 +--R (- x\|x + a + x + a )log(\|x + a - x) + a +--R (1) ------------------------------------------------- +--R +-------+ +--R | 2 2 2 2 +--R x\|x + a - x - a +--R Type: Union(Expression Integer,...) +--E + +--S 64 +bb:=-x/sqrt(x^2+a^2)+log(x+sqrt(x^2+a^2)) +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R \|x + a log(\|x + a + x) - x +--R (2) --------------------------------- +--R +-------+ +--R | 2 2 +--R \|x + a +--R Type: Expression Integer +--E + +--S 65 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R (3) - log(\|x + a + x) - log(\|x + a - x) - 1 +--R Type: Expression Integer +--E + +--S 66 14:198 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 +--R (4) - log(a ) - 1 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.199~~~~~$\displaystyle +\int{\frac{x^3dx}{(x^2+a^2)^{3/2}}}$} +$$\int{\frac{x^3}{(x^2+a^2)^{3/2}}}= +\sqrt{x^2+a^2}+\frac{a^2}{\sqrt{x^2+a^2}} +$$ +<<*>>= +)clear all + +--S 67 +aa:=integrate(x^3/(x^2+a^2)^(3/2),x) +--R +--R +--R +-------+ +--R 3 2 | 2 2 4 2 2 4 +--R (- 2x - 4a x)\|x + a + 2x + 5a x + 2a +--R (1) -------------------------------------------- +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (2x + a )\|x + a - 2x - 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 68 +bb:=sqrt(x^2+a^2)+a^2/sqrt(x^2+a^2) +--R +--R 2 2 +--R x + 2a +--R (2) ---------- +--R +-------+ +--R | 2 2 +--R \|x + a +--R Type: Expression Integer +--E + +--S 69 14:199 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.200~~~~~$\displaystyle +\int{\frac{dx}{x(x^2+a^2)^{3/2}}}$} +$$\int{\frac{1}{x(x^2+a^2)^{3/2}}}= +\frac{1}{a^2\sqrt{x^2+a^2}}- +\frac{1}{a^3}\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 70 +aa:=integrate(1/(x*(x^2+a^2)^(3/2)),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R | 2 2 2 2 | 2 2 +--R (- x\|x + a + x + a )log(\|x + a - x + a) +--R + +--R +-------+ +-------+ +-------+ +--R | 2 2 2 2 | 2 2 | 2 2 +--R (x\|x + a - x - a )log(\|x + a - x - a) - a\|x + a + a x +--R / +--R +-------+ +--R 3 | 2 2 3 2 5 +--R a x\|x + a - a x - a +--R Type: Union(Expression Integer,...) +--E + +--S 71 +bb:=1/(a^2*sqrt(x^2+a^2))-1/a^3*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 \|x + a + a +--R - \|x + a log(--------------) + a +--R x +--R (2) ----------------------------------- +--R +-------+ +--R 3 | 2 2 +--R a \|x + a +--R Type: Expression Integer +--E + +--S 72 +cc:=aa-bb +--R +--R (3) +--R +-------+ +--R +-------+ +-------+ | 2 2 +--R | 2 2 | 2 2 \|x + a + a +--R - log(\|x + a - x + a) + log(\|x + a - x - a) + log(--------------) +--R x +--R ------------------------------------------------------------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 73 +dd:=expandLog cc +--R +--R (4) +--R +-------+ +-------+ +-------+ +--R | 2 2 | 2 2 | 2 2 +--R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x - a) +--R + +--R - log(x) +--R / +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 74 14:200 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R log(- 1) +--R (5) - -------- +--R 3 +--R a +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.201~~~~~$\displaystyle +\int{\frac{dx}{x^2(x^2+a^2)^{3/2}}}$} +$$\int{\frac{1}{x^2(x^2+a^2)^{3/2}}}= +-\frac{\sqrt{x^2+a^2}}{a^4x}-\frac{x}{a^4\sqrt{x^2+a^2}} +$$ +<<*>>= +)clear all + +--S 75 +aa:=integrate(1/(x^2*(x^2+a^2)^(3/2)),x) +--R +--R +--R 1 +--R (1) - ----------------------------------- +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (2x + a x)\|x + a - 2x - 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 76 +bb:=-sqrt(x^2+a^2)/(a^4*x)-x/(a^4*sqrt(x^2+a^2)) +--R +--R 2 2 +--R - 2x - a +--R (2) ------------- +--R +-------+ +--R 4 | 2 2 +--R a x\|x + a +--R Type: Expression Integer +--E + +--S 77 14:201 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 2 +--R (3) - -- +--R 4 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.202~~~~~$\displaystyle +\int{\frac{dx}{x^3(x^2+a^2)^{3/2}}}$} +$$\int{\frac{1}{x^3(x^2+a^2)^{3/2}}}= +\frac{-1}{2a^2x^2\sqrt{x^2+a^2}}- +\frac{3}{2a^4\sqrt{x^2+a^2}}+ +\frac{3}{2a^5}\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 78 +aa:=integrate(1/(x^3*(x^2+a^2)^(3/2)),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 5 2 3 | 2 2 6 2 4 4 2 | 2 2 +--R ((12x + 9a x )\|x + a - 12x - 15a x - 3a x )log(\|x + a - x + a) +--R + +--R +-------+ +--R 5 2 3 | 2 2 6 2 4 4 2 +--R ((- 12x - 9a x )\|x + a + 12x + 15a x + 3a x ) +--R * +--R +-------+ +--R | 2 2 +--R log(\|x + a - x - a) +--R + +--R +-------+ +--R 4 3 2 5 | 2 2 5 3 3 5 +--R (12a x + 7a x + a )\|x + a - 12a x - 13a x - 3a x +--R / +--R +-------+ +--R 5 5 7 3 | 2 2 5 6 7 4 9 2 +--R (8a x + 6a x )\|x + a - 8a x - 10a x - 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 79 +bb:=-1/(2*a^2*x^2*sqrt(x^2+a^2))-3/(2*a^4*sqrt(x^2+a^2))+3/(2*a^5)*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R +-------+ | 2 2 +--R 2 | 2 2 \|x + a + a 2 3 +--R 3x \|x + a log(--------------) - 3a x - a +--R x +--R (2) --------------------------------------------- +--R +-------+ +--R 5 2 | 2 2 +--R 2a x \|x + a +--R Type: Expression Integer +--E + +--S 80 +cc:=aa-bb +--R +--R (3) +--R +-------+ +--R +-------+ +-------+ | 2 2 +--R | 2 2 | 2 2 \|x + a + a +--R 3log(\|x + a - x + a) - 3log(\|x + a - x - a) - 3log(--------------) +--R x +--R -------------------------------------------------------------------------- +--R 5 +--R 2a +--R Type: Expression Integer +--E + +--S 81 +dd:=expandLog cc +--R +--R (4) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 3log(\|x + a + a) + 3log(\|x + a - x + a) +--R + +--R +-------+ +--R | 2 2 +--R - 3log(\|x + a - x - a) + 3log(x) +--R / +--R 5 +--R 2a +--R Type: Expression Integer +--E + +--S 82 14:202 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R 3log(- 1) +--R (5) --------- +--R 5 +--R 2a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.203~~~~~$\displaystyle\int{(x^2+a^2)^{3/2}}~dx$} +$$\int{(x^2+a^2)^{3/2}}= +\frac{x(x^2+a^2)^{3/2}}{4}+\frac{3a^2x\sqrt{x^2+a^2}}{8}+ +\frac{3}{8}a^4\ln\left(x+\sqrt{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 83 +aa:=integrate((x^2+a^2)^(3/2),x) +--R +--R (1) +--R +-------+ +-------+ +--R 4 3 6 | 2 2 4 4 6 2 8 | 2 2 +--R ((- 24a x - 12a x)\|x + a + 24a x + 24a x + 3a )log(\|x + a - x) +--R + +--R +-------+ +--R 7 2 5 4 3 6 | 2 2 8 2 6 4 4 +--R (- 16x - 56a x - 42a x - 5a x)\|x + a + 16x + 64a x + 68a x +--R + +--R 6 2 +--R 20a x +--R / +--R +-------+ +--R 3 2 | 2 2 4 2 2 4 +--R (64x + 32a x)\|x + a - 64x - 64a x - 8a +--R Type: Union(Expression Integer,...) +--E + +--S 84 +bb:=(x*(x^2+a^2)^(3/2))/4+(3*a^2*x*sqrt(x^2+a^2))/8+3/8*a^4*log(x+sqrt(x^2+a^2)) +--R +--R +-------+ +-------+ +--R 4 | 2 2 3 2 | 2 2 +--R 3a log(\|x + a + x) + (2x + 5a x)\|x + a +--R (2) ----------------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 85 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 4 | 2 2 4 | 2 2 +--R - 3a log(\|x + a + x) - 3a log(\|x + a - x) +--R (3) ------------------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 86 14:203 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 4 2 +--R 3a log(a ) +--R (4) - ---------- +--R 8 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.204~~~~~$\displaystyle\int{x(x^2+a^2)^{3/2}}~dx$} +$$\int{x(x^2+a^2)^{3/2}}=\frac{(x^2+a^2)^{5/2}}{5}$$ +<<*>>= +)clear all + +--S 87 +aa:=integrate(x*(x^2+a^2)^(3/2),x) +--R +--R +--R (1) +--R +-------+ +--R 9 2 7 4 5 6 3 8 | 2 2 10 2 8 +--R (- 16x - 52a x - 61a x - 30a x - 5a x)\|x + a + 16x + 60a x +--R + +--R 4 6 6 4 8 2 10 +--R 85a x + 55a x + 15a x + a +--R / +--R +-------+ +--R 4 2 2 4 | 2 2 5 2 3 4 +--R (80x + 60a x + 5a )\|x + a - 80x - 100a x - 25a x +--R Type: Union(Expression Integer,...) +--E + +--S 88 +bb:=(x^2+a^2)^(5/2)/5 +--R +--R +-------+ +--R 4 2 2 4 | 2 2 +--R (x + 2a x + a )\|x + a +--R (2) --------------------------- +--R 5 +--R Type: Expression Integer +--E + +--S 89 14:204 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.205~~~~~$\displaystyle\int{x^2(x^2+a^2)^{3/2}}~dx$} +$$\int{x^2(x^2+a^2)^{3/2}}= +\frac{x(x^2+a^2)^{5/2}}{6}-\frac{a^2x(x^2+a^2)^{3/2}}{24}- +\frac{a^4x\sqrt{x^2+a^2}}{16}- +\frac{a^6}{16}\ln\left(x+\sqrt{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 90 +aa:=integrate(x^2*(x^2+a^2)^(3/2),x) +--R +--R +--R (1) +--R +-------+ +--R 6 5 8 3 10 | 2 2 6 6 8 4 10 2 +--R (96a x + 96a x + 18a x)\|x + a - 96a x - 144a x - 54a x +--R + +--R 12 +--R - 3a +--R * +--R +-------+ +--R | 2 2 +--R log(\|x + a - x) +--R + +--R +-------+ +--R 11 2 9 4 7 6 5 8 3 10 | 2 2 +--R (- 256x - 832a x - 912a x - 404a x - 68a x - 3a x)\|x + a +--R + +--R 12 2 10 4 8 6 6 8 4 10 2 +--R 256x + 960a x + 1296a x + 772a x + 198a x + 18a x +--R / +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (1536x + 1536a x + 288a x)\|x + a - 1536x - 2304a x - 864a x - 48a +--R Type: Union(Expression Integer,...) +--E + +--S 91 +bb:=(x*(x^2+a^2)^(5/2))/6-(a^2*x*(x^2+a^2)^(3/2))/24-(a^4*x*sqrt(x^2+a^2))/16-a^6/16*log(x+sqrt(x^2+a^2)) +--R +--R +-------+ +-------+ +--R 6 | 2 2 5 2 3 4 | 2 2 +--R - 3a log(\|x + a + x) + (8x + 14a x + 3a x)\|x + a +--R (2) ---------------------------------------------------------- +--R 48 +--R Type: Expression Integer +--E + +--S 92 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 6 | 2 2 6 | 2 2 +--R a log(\|x + a + x) + a log(\|x + a - x) +--R (3) --------------------------------------------- +--R 16 +--R Type: Expression Integer +--E + +--S 93 14:205 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 6 2 +--R a log(a ) +--R (4) --------- +--R 16 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.206~~~~~$\displaystyle\int{x^3(x^2+a^2)^{3/2}}~dx$} +$$\int{x^3(x^2+a^2)^{3/2}}= +\frac{(x^2+a^2)^{7/2}}{7}-\frac{a^2(x^2+a^2)^{5/2}}{5} +$$ +<<*>>= +)clear all + +--S 94 +aa:=integrate(x^3*(x^2+a^2)^(3/2),x) +--R +--R +--R (1) +--R 13 2 11 4 9 6 7 8 5 10 3 +--R - 320x - 1072a x - 1240a x - 467a x + 112a x + 105a x +--R + +--R 12 +--R 14a x +--R * +--R +-------+ +--R | 2 2 +--R \|x + a +--R + +--R 14 2 12 4 10 6 8 8 6 10 4 12 2 +--R 320x + 1232a x + 1736a x + 973a x + 21a x - 175a x - 49a x +--R + +--R 14 +--R - 2a +--R / +--R +-------+ +--R 6 2 4 4 2 6 | 2 2 7 2 5 +--R (2240x + 2800a x + 840a x + 35a )\|x + a - 2240x - 3920a x +--R + +--R 4 3 6 +--R - 1960a x - 245a x +--R Type: Union(Expression Integer,...) +--E + +--S 95 +bb:=(x^2+a^2)^(7/2)/7-(a^2*(x^2+a^2)^(5/2))/5 +--R +--R +-------+ +--R 6 2 4 4 2 6 | 2 2 +--R (5x + 8a x + a x - 2a )\|x + a +--R (2) ------------------------------------ +--R 35 +--R Type: Expression Integer +--E + +--S 96 14:206 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.207~~~~~$\displaystyle +\int{\frac{(x^2+a^2)^{3/2}}{x}}~dx$} +$$\int{\frac{(x^2+a^2)^{3/2}}{x}}= +\frac{(x^2+a^2)^{3/2}}{3}+a^2\sqrt{x^2+a^2}- +a^3\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 97 +aa:=integrate((x^2+a^2)^(3/2)/x,x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 3 2 5 | 2 2 3 3 5 | 2 2 +--R ((- 12a x - 3a )\|x + a + 12a x + 9a x)log(\|x + a - x + a) +--R + +--R +-------+ +-------+ +--R 3 2 5 | 2 2 3 3 5 | 2 2 +--R ((12a x + 3a )\|x + a - 12a x - 9a x)log(\|x + a - x - a) +--R + +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x - 19a x - 12a x)\|x + a + 4x + 21a x + 21a x + 4a +--R / +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (12x + 3a )\|x + a - 12x - 9a x +--R Type: Union(Expression Integer,...) +--E + +--S 98 +bb:=(x^2+a^2)^(3/2)/3+a^2*sqrt(x^2+a^2)-a^3*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R | 2 2 +-------+ +--R 3 \|x + a + a 2 2 | 2 2 +--R - 3a log(--------------) + (x + 4a )\|x + a +--R x +--R (2) ----------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 99 +cc:=aa-bb +--R +--R (3) +--R +-------+ +-------+ +--R 3 | 2 2 3 | 2 2 +--R - a log(\|x + a - x + a) + a log(\|x + a - x - a) +--R + +--R +-------+ +--R | 2 2 +--R 3 \|x + a + a +--R a log(--------------) +--R x +--R Type: Expression Integer +--E + +--S 100 +dd:=expandLog cc +--R +--R (4) +--R +-------+ +-------+ +--R 3 | 2 2 3 | 2 2 +--R a log(\|x + a + a) - a log(\|x + a - x + a) +--R + +--R +-------+ +--R 3 | 2 2 3 +--R a log(\|x + a - x - a) - a log(x) +--R Type: Expression Integer +--E + +--S 101 14:207 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R 3 +--R (5) - a log(- 1) +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.208~~~~~$\displaystyle +\int{\frac{(x^2+a^2)^{3/2}}{x^2}}~dx$} +$$\int{\frac{(x^2+a^2)^{3/2}}{x^2}}= +-\frac{(x^2+a^2)^{3/2}}{x}+\frac{3x\sqrt{x^2+a^2}}{2}+ +\frac{3}{2}a^2\ln\left(x+\sqrt{x^2+a^2}\right) +$$ +<<*>>= +)clear all + +--S 102 +aa:=integrate((x^2+a^2)^{3/2}/x^2,x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 2 3 4 | 2 2 2 4 4 2 | 2 2 +--R ((- 12a x - 3a x)\|x + a + 12a x + 9a x )log(\|x + a - x) +--R + +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x - 3a x + 4a x)\|x + a + 4x + 5a x - 3a x - 2a +--R / +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (8x + 2a x)\|x + a - 8x - 6a x +--R Type: Union(Expression Integer,...) +--E + +--S 103 +bb:=-(x^2+a^2)^(3/2)/x+(3*x*sqrt(x^2+a^2))/2+3/2*a^2*log(x+sqrt(x^2+a^2)) +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 2 | 2 2 +--R 3a x log(\|x + a + x) + (x - 2a )\|x + a +--R (2) ----------------------------------------------- +--R 2x +--R Type: Expression Integer +--E + +--S 104 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 2 +--R - 3a log(\|x + a + x) - 3a log(\|x + a - x) - 2a +--R (3) ------------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 105 14:208 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 2 2 +--R - 3a log(a ) - 2a +--R (4) ------------------ +--R 2 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.209~~~~~$\displaystyle +\int{\frac{(x^2+a^2)^{3/2}}{x^3}}~dx$} +$$\int{\frac{(x^2+a^2)^{3/2}}{x^3}}= +-\frac{(x^2+a^2)^{3/2}}{2x^2}+\frac{3}{2}\sqrt{x^2+a^2}- +\frac{3}{2}a\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 106 +aa:=integrate((x^2+a^2)^(3/2)/x^3,x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 4 3 2 | 2 2 5 3 3 | 2 2 +--R ((- 12a x - 3a x )\|x + a + 12a x + 9a x )log(\|x + a - x + a) +--R + +--R +-------+ +-------+ +--R 4 3 2 | 2 2 5 3 3 | 2 2 +--R ((12a x + 3a x )\|x + a - 12a x - 9a x )log(\|x + a - x - a) +--R + +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 8x - 2a x + 3a x)\|x + a + 8x + 6a x - 3a x - a +--R / +--R +-------+ +--R 4 2 2 | 2 2 5 2 3 +--R (8x + 2a x )\|x + a - 8x - 6a x +--R Type: Union(Expression Integer,...) +--E + +--S 107 +bb:=-(x^2+a^2)^(3/2)/(2*x^2)+3/2*sqrt(x^2+a^2)-3/2*a*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R | 2 2 +-------+ +--R 2 \|x + a + a 2 2 | 2 2 +--R - 3a x log(--------------) + (2x - a )\|x + a +--R x +--R (2) ------------------------------------------------- +--R 2 +--R 2x +--R Type: Expression Integer +--E + +--S 108 +cc:=aa-bb +--R +--R (3) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 3a log(\|x + a - x + a) + 3a log(\|x + a - x - a) +--R + +--R +-------+ +--R | 2 2 +--R \|x + a + a +--R 3a log(--------------) +--R x +--R / +--R 2 +--R Type: Expression Integer +--E + +--S 109 +dd:=expandLog cc +--R +--R (4) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 3a log(\|x + a + a) - 3a log(\|x + a - x + a) +--R + +--R +-------+ +--R | 2 2 +--R 3a log(\|x + a - x - a) - 3a log(x) +--R / +--R 2 +--R Type: Expression Integer +--E + +--S 110 14:209 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R 3a log(- 1) +--R (5) - ----------- +--R 2 +--R Type: Expression Integer +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p67-68 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum9.input.pdf b/src/axiom-website/CATS/schaum9.input.pdf new file mode 100644 index 0000000..93e4430 --- /dev/null +++ b/src/axiom-website/CATS/schaum9.input.pdf @@ -0,0 +1,3328 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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100644 index 0000000..f25430b --- /dev/null +++ b/src/axiom-website/bookvol10.2abb.html @@ -0,0 +1,121 @@ + + + + Axiom Computer Algebra System + + + + + +

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Axiom Abbreviated Category and Domain graph

+
+ choose here: + 0.1 or + 0.25 or + 0.5 or + 1.0 or + 1.5 or ... +
+
+ + +
+ + + diff --git a/src/axiom-website/bookvol10.2abb.svg b/src/axiom-website/bookvol10.2abb.svg new file mode 100644 index 0000000..91ca8a2 --- /dev/null +++ b/src/axiom-website/bookvol10.2abb.svg @@ -0,0 +1,926 @@ + + +]> + + + + + +dotabb + + +CATEGORY + + +CATEGORY + + + +ABELGRP + + +ABELGRP + + + +CABMON + + +CABMON + + + +ABELGRP->CABMON + + + + +ABELMON + + +ABELMON + + + +CABMON->ABELMON + + + + +ABELSG + + +ABELSG + + + +ABELMON->ABELSG + + + + +SETCAT + + +SETCAT + + + +ABELSG->SETCAT + + + + +REPDB + +REPDB + + +ABELSG->REPDB + + + + +BASTYPE + + +BASTYPE + + + +SETCAT->BASTYPE + + + + +KOERCE + + +KOERCE + + + +SETCAT->KOERCE + + + + +AGG + + +AGG + + + +TYPE + + +TYPE + + + +AGG->TYPE + + + + +TYPE->CATEGORY + + + + +ALAGG + + +ALAGG + + + +TBAGG + + +TBAGG + + + +ALAGG->TBAGG + + + + +LSAGG + + +LSAGG + + + +ALAGG->LSAGG + + + + +IXAGG + + +IXAGG + + + +TBAGG->IXAGG + + + + +KDAGG + + +KDAGG + + + +TBAGG->KDAGG + + + + +FLAGG + + +FLAGG + + + +LSAGG->FLAGG + + + + +ELAGG + + +ELAGG + + + +LSAGG->ELAGG + + + + +A1AGG + + +A1AGG + + + +A1AGG->FLAGG + + + + +LNAGG + + +LNAGG + + + +FLAGG->LNAGG + + + + +BASTYPE->CATEGORY + + + + +BGAGG + + +BGAGG + + + +HOAGG + + +HOAGG + + + +BGAGG->HOAGG + + + + +HOAGG->AGG + + + + +BMODULE + + +BMODULE + + + +LMODULE + + +LMODULE + + + +BMODULE->LMODULE + + + + +RMODULE + + +RMODULE + + + +BMODULE->RMODULE + + + + +LMODULE->ABELGRP + + + + +RMODULE->ABELGRP + + + + +BRAGG + + +BRAGG + + + +RCAGG + + +RCAGG + + + +BRAGG->RCAGG + + + + +RCAGG->HOAGG + + + + +BTAGG + + +BTAGG + + + +BTAGG->A1AGG + + + + +ORDSET + + +ORDSET + + + +BTAGG->ORDSET + + + + +LOGIC + +LOGIC + + +BTAGG->LOGIC + + + + +ORDSET->SETCAT + + + + +CLAGG + + +CLAGG + + + +CLAGG->HOAGG + + + + +DIAGG + + +DIAGG + + + +DIOPS + + +DIOPS + + + +DIAGG->DIOPS + + + + +DIOPS->BGAGG + + + + +DIOPS->CLAGG + + + + +DLAGG + + +DLAGG + + + +DLAGG->RCAGG + + + + +DQAGG + + +DQAGG + + + +SKAGG + + +SKAGG + + + +DQAGG->SKAGG + + + + +QUAGG + + +QUAGG + + + +DQAGG->QUAGG + + + + +SKAGG->BGAGG + + + + +QUAGG->BGAGG + + + + +ELAGG->LNAGG + + + + +LNAGG->CLAGG + + + + +LNAGG->IXAGG + + + + +ELTAB + + +ELTAB + + + +ELTAB->CATEGORY + + + + +ELTAGG + + +ELTAGG + + + +ELTAGG->ELTAB + + + + +FINITE + + +FINITE + + + +FINITE->SETCAT + + + + +FSAGG + + +FSAGG + + + +FSAGG->DIAGG + + + + +SETAGG + + +SETAGG + + + +FSAGG->SETAGG + + + + +SETAGG->SETCAT + + + + +SETAGG->CLAGG + + + + +IXAGG->HOAGG + + + + +IXAGG->ELTAGG + + + + +KDAGG->DIAGG + + + + +KOERCE->CATEGORY + + + + +KONVERT + + +KONVERT + + + +KONVERT->CATEGORY + + + + +MDAGG + + +MDAGG + + + +MDAGG->DIOPS + + + + +MONOID + + +MONOID + + + +SGROUP + + +SGROUP + + + +MONOID->SGROUP + + + + +SGROUP->SETCAT + + + + +MSETAGG + + +MSETAGG + + + +MSETAGG->SETAGG + + + + +MSETAGG->MDAGG + + + + +OAGROUP + + +OAGROUP + + + +OAGROUP->ABELGRP + + + + +OCAMON + + +OCAMON + + + +OAGROUP->OCAMON + + + + +OCAMON->CABMON + + + + +OAMON + + +OAMON + + + +OCAMON->OAMON + + + + +OAMON->ABELMON + + + + +OASGP + + +OASGP + + + +OAMON->OASGP + + + + +OASGP->ABELMON + + + + +OASGP->ORDSET + + + + +OAMONS + + +OAMONS + + + +OAMONS->OCAMON + + + + +OMSAGG + + +OMSAGG + + + +OMSAGG->MSETAGG + + + + +PRQAGG + + +PRQAGG + + + +OMSAGG->PRQAGG + + + + +PRQAGG->BGAGG + + + + +ORDFIN + + +ORDFIN + + + +ORDFIN->ORDSET + + + + +ORDFIN->FINITE + + + + +ORDMON + + +ORDMON + + + +ORDMON->ORDSET + + + + +ORDMON->MONOID + + + + +ORDRING + + +ORDRING + + + +ORDRING->MONOID + + + + +ORDRING->OAGROUP + + + + +RING + + +RING + + + +ORDRING->RING + + + + +RING->LMODULE + + + + +RING->MONOID + + + + +RNG + + +RNG + + + +RING->RNG + + + + +RETRACT + + +RETRACT + + + +RETRACT->CATEGORY + + + + +RNG->ABELGRP + + + + +RNG->SGROUP + + + + +SRAGG + + +SRAGG + + + +SRAGG->A1AGG + + + + +STAGG + + +STAGG + + + +STAGG->RCAGG + + + + +STAGG->LNAGG + + + + +STEP + + +STEP + + + +STEP->SETCAT + + + + +URAGG + + +URAGG + + + +URAGG->RCAGG + + + + + diff --git a/src/axiom-website/bookvol10.2full.html b/src/axiom-website/bookvol10.2full.html new file mode 100644 index 0000000..d45ab3a --- /dev/null +++ b/src/axiom-website/bookvol10.2full.html @@ -0,0 +1,121 @@ + + + + Axiom Computer Algebra System + + + + + + +
+ [ + + Home + + ] +   + [ + + Screenshots + + ] +   + [ + + FAQ + + ] +   + [ + + Download + + ] +   + [ + + Documentation + + ] +   + [ + + Current State + + ] +   + [ + + Community + + ] +   + [ + + Developers + + ] +   + [ + + Patches + + ] +
+ +
+
+ + +

Axiom Abbreviated Category and Domain graph

+
+ choose here: + 0.1 or + 0.25 or + 0.5 or + 1.0 or + 1.5 or ... +
+
+ + +
+ + + diff --git a/src/axiom-website/bookvol10.2full.svg b/src/axiom-website/bookvol10.2full.svg new file mode 100644 index 0000000..8b5c3f1 --- /dev/null +++ b/src/axiom-website/bookvol10.2full.svg @@ -0,0 +1,1438 @@ + + +]> + + + + + +dotfull + + +Category + + +Category + + + +AbelianGroup() + + +AbelianGroup() + + + +CancellationAbelianMonoid() + + +CancellationAbelianMonoid() + + + +AbelianGroup()->CancellationAbelianMonoid() + + + + +AbelianMonoid() + + +AbelianMonoid() + + + +CancellationAbelianMonoid()->AbelianMonoid() + + + + +AbelianSemiGroup() + + +AbelianSemiGroup() + + + +AbelianMonoid()->AbelianSemiGroup() + + + + +SetCategory() + + +SetCategory() + + + +AbelianSemiGroup()->SetCategory() + + + + +RepeatedDoubling(SetCategory) + +RepeatedDoubling(SetCategory) + + +AbelianSemiGroup()->RepeatedDoubling(SetCategory) + + + + +BasicType() + + +BasicType() + + + +SetCategory()->BasicType() + + + + +CoercibleTo(OutputForm) + + +CoercibleTo(OutputForm) + + + +SetCategory()->CoercibleTo(OutputForm) + + + + +Aggregate() + + +Aggregate() + + + +Type() + + +Type() + + + +Aggregate()->Type() + + + + +Type()->Category + + + + +AssociationListAggregate(a:SetCategory,b:SetCategory) + + +AssociationListAggregate(a:SetCategory,b:SetCategory) + + + +TableAggregate(a:SetCategory,b:SetCategory) + + +TableAggregate(a:SetCategory,b:SetCategory) + + + +AssociationListAggregate(a:SetCategory,b:SetCategory)->TableAggregate(a:SetCategory,b:SetCategory) + + + + +ListAggregate(Record(a:SetCategory,b:SetCategory)) + + +ListAggregate(Record(a:SetCategory,b:SetCategory)) + + + +AssociationListAggregate(a:SetCategory,b:SetCategory)->ListAggregate(Record(a:SetCategory,b:SetCategory)) + + + + +IndexedAggregate(a:SetCategory,b:SetCategory) + + +IndexedAggregate(a:SetCategory,b:SetCategory) + + + +TableAggregate(a:SetCategory,b:SetCategory)->IndexedAggregate(a:SetCategory,b:SetCategory) + + + + +KeyedDictionary(a:SetCategory,b:SetCategory) + + +KeyedDictionary(a:SetCategory,b:SetCategory) + + + +TableAggregate(a:SetCategory,b:SetCategory)->KeyedDictionary(a:SetCategory,b:SetCategory) + + + + +ListAggregate(a:Type) + + +ListAggregate(a:Type) + + + +ListAggregate(Record(a:SetCategory,b:SetCategory))->ListAggregate(a:Type) + + + + +OneDimensionalArrayAggregate(a:Type) + + +OneDimensionalArrayAggregate(a:Type) + + + +FiniteLinearAggregate(a:Type) + + +FiniteLinearAggregate(a:Type) + + + +OneDimensionalArrayAggregate(a:Type)->FiniteLinearAggregate(a:Type) + + + + +LinearAggregate(a:Type) + + +LinearAggregate(a:Type) + + + +FiniteLinearAggregate(a:Type)->LinearAggregate(a:Type) + + + + +OneDimensionalArrayAggregate(Character) + + +OneDimensionalArrayAggregate(Character) + + + +OneDimensionalArrayAggregate(Character)->OneDimensionalArrayAggregate(a:Type) + + + + +OneDimensionalArrayAggregate(Boolean) + + +OneDimensionalArrayAggregate(Boolean) + + + +OneDimensionalArrayAggregate(Boolean)->OneDimensionalArrayAggregate(a:Type) + + + + +BasicType()->Category + + + + +BagAggregate(a:Type) + + +BagAggregate(a:Type) + + + +HomogeneousAggregate(a:Type) + + +HomogeneousAggregate(a:Type) + + + +BagAggregate(a:Type)->HomogeneousAggregate(a:Type) + + + + +HomogeneousAggregate(a:Type)->Aggregate() + + + + +BagAggregate(a:SetCategory) + + +BagAggregate(a:SetCategory) + + + +BagAggregate(a:SetCategory)->BagAggregate(a:Type) + + + + +BiModule(a:Ring,b:Ring) + + +BiModule(a:Ring,b:Ring) + + + +LeftModule(a:Ring) + + +LeftModule(a:Ring) + + + +BiModule(a:Ring,b:Ring)->LeftModule(a:Ring) + + + + +RightModule(a:Ring) + + +RightModule(a:Ring) + + + +BiModule(a:Ring,b:Ring)->RightModule(a:Ring) + + + + +LeftModule(a:Rng) + + +LeftModule(a:Rng) + + + +LeftModule(a:Ring)->LeftModule(a:Rng) + + + + +RightModule(a:Rng) + + +RightModule(a:Rng) + + + +RightModule(a:Ring)->RightModule(a:Rng) + + + + +BiModule(a:CommutativeRing,b:CommutativeRing) + + +BiModule(a:CommutativeRing,b:CommutativeRing) + + + +BiModule(a:CommutativeRing,b:CommutativeRing)->BiModule(a:Ring,b:Ring) + + + + +BiModule(a:Ring,b:OrderedAbelianMonoid) + + +BiModule(a:Ring,b:OrderedAbelianMonoid) + + + +BiModule(a:Ring,b:OrderedAbelianMonoid)->BiModule(a:Ring,b:Ring) + + + + +BinaryRecursiveAggregate(a:Type) + + +BinaryRecursiveAggregate(a:Type) + + + +RecursiveAggregate(a:Type) + + +RecursiveAggregate(a:Type) + + + +BinaryRecursiveAggregate(a:Type)->RecursiveAggregate(a:Type) + + + + +RecursiveAggregate(a:Type)->HomogeneousAggregate(a:Type) + + + + +BitAggregate() + + +BitAggregate() + + + +BitAggregate()->OneDimensionalArrayAggregate(Boolean) + + + + +OrderedSet() + + +OrderedSet() + + + +BitAggregate()->OrderedSet() + + + + +Logic() + +Logic() + + +BitAggregate()->Logic() + + + + +OrderedSet()->SetCategory() + + + + +Collection(a:Type) + + +Collection(a:Type) + + + +Collection(a:Type)->HomogeneousAggregate(a:Type) + + + + +Collection(a:SetCategory) + + +Collection(a:SetCategory) + + + +Collection(a:SetCategory)->Collection(a:Type) + + + + +Dictionary(a:SetCategory) + + +Dictionary(a:SetCategory) + + + +DictionaryOperations(a:SetCategory) + + +DictionaryOperations(a:SetCategory) + + + +Dictionary(a:SetCategory)->DictionaryOperations(a:SetCategory) + + + + +DictionaryOperations(a:SetCategory)->BagAggregate(a:SetCategory) + + + + +DictionaryOperations(a:SetCategory)->Collection(a:SetCategory) + + + + +Dictionary(Record(a:SetCategory,b:SetCategory)) + + +Dictionary(Record(a:SetCategory,b:SetCategory)) + + + +Dictionary(Record(a:SetCategory,b:SetCategory))->Dictionary(a:SetCategory) + + + + +DoublyLinkedAggregate(a:Type) + + +DoublyLinkedAggregate(a:Type) + + + +DoublyLinkedAggregate(a:Type)->RecursiveAggregate(a:Type) + + + + +DequeueAggregate(a:Type) + + +DequeueAggregate(a:Type) + + + +StackAggregate(a:Type) + + +StackAggregate(a:Type) + + + +DequeueAggregate(a:Type)->StackAggregate(a:Type) + + + + +QueueAggregate(a:Type) + + +QueueAggregate(a:Type) + + + +DequeueAggregate(a:Type)->QueueAggregate(a:Type) + + + + +StackAggregate(a:Type)->BagAggregate(a:Type) + + + + +QueueAggregate(a:Type)->BagAggregate(a:Type) + + + + +DequeueAggregate(a:SetCategory) + + +DequeueAggregate(a:SetCategory) + + + +DequeueAggregate(a:SetCategory)->DequeueAggregate(a:Type) + + + + +ExtensibleLinearAggregate(a:Type) + + +ExtensibleLinearAggregate(a:Type) + + + +ExtensibleLinearAggregate(a:Type)->LinearAggregate(a:Type) + + + + +LinearAggregate(a:Type)->Collection(a:Type) + + + + +IndexedAggregate(b:Integer,a:Type) + + +IndexedAggregate(b:Integer,a:Type) + + + +LinearAggregate(a:Type)->IndexedAggregate(b:Integer,a:Type) + + + + +Eltable(a:SetCategory,b:Type) + + +Eltable(a:SetCategory,b:Type) + + + +Eltable(a:SetCategory,b:Type)->Category + + + + +EltableAggregate(a:SetCategory,b:Type) + + +EltableAggregate(a:SetCategory,b:Type) + + + +EltableAggregate(a:SetCategory,b:Type)->Eltable(a:SetCategory,b:Type) + + + + +Finite() + + +Finite() + + + +Finite()->SetCategory() + + + + +FiniteSetAggregate(a:SetCategory) + + +FiniteSetAggregate(a:SetCategory) + + + +FiniteSetAggregate(a:SetCategory)->Dictionary(a:SetCategory) + + + + +SetAggregate(a:SetCategory) + + +SetAggregate(a:SetCategory) + + + +FiniteSetAggregate(a:SetCategory)->SetAggregate(a:SetCategory) + + + + +SetAggregate(a:SetCategory)->SetCategory() + + + + +SetAggregate(a:SetCategory)->Collection(a:SetCategory) + + + + +IndexedAggregate(a:SetCategory,b:Type) + + +IndexedAggregate(a:SetCategory,b:Type) + + + +IndexedAggregate(a:SetCategory,b:Type)->HomogeneousAggregate(a:Type) + + + + +IndexedAggregate(a:SetCategory,b:Type)->EltableAggregate(a:SetCategory,b:Type) + + + + +IndexedAggregate(a:SetCategory,b:SetCategory)->IndexedAggregate(a:SetCategory,b:Type) + + + + +IndexedAggregate(b:Integer,a:Type)->IndexedAggregate(a:SetCategory,b:Type) + + + + +KeyedDictionary(a:SetCategory,b:SetCategory)->Dictionary(Record(a:SetCategory,b:SetCategory)) + + + + +CoercibleTo(a:Type) + + +CoercibleTo(a:Type) + + + +CoercibleTo(a:Type)->Category + + + + +CoercibleTo(OutputForm)->CoercibleTo(a:Type) + + + + +ConvertibleTo(a:Type) + + +ConvertibleTo(a:Type) + + + +ConvertibleTo(a:Type)->Category + + + + +ConvertibleTo(DoubleFloat) + + +ConvertibleTo(DoubleFloat) + + + +ConvertibleTo(DoubleFloat)->ConvertibleTo(a:Type) + + + + +ConvertibleTo(Float) + + +ConvertibleTo(Float) + + + +ConvertibleTo(Float)->ConvertibleTo(a:Type) + + + + +ConvertibleTo(InputForm) + + +ConvertibleTo(InputForm) + + + +ConvertibleTo(InputForm)->ConvertibleTo(a:Type) + + + + +ConvertibleTo(Integer) + + +ConvertibleTo(Integer) + + + +ConvertibleTo(Integer)->ConvertibleTo(a:Type) + + + + +ConvertibleTo(Pattern(Integer)) + + +ConvertibleTo(Pattern(Integer)) + + + +ConvertibleTo(Pattern(Integer))->ConvertibleTo(a:Type) + + + + +ConvertibleTo(Pattern(Float)) + + +ConvertibleTo(Pattern(Float)) + + + +ConvertibleTo(Pattern(Float))->ConvertibleTo(a:Type) + + + + +ConvertibleTo(Complex(Float)) + + +ConvertibleTo(Complex(Float)) + + + +ConvertibleTo(Complex(Float))->ConvertibleTo(a:Type) + + + + +ConvertibleTo(Complex(DoubleFloat)) + + +ConvertibleTo(Complex(DoubleFloat)) + + + +ConvertibleTo(Complex(DoubleFloat))->ConvertibleTo(a:Type) + + + + +ConvertibleTo(String) + + +ConvertibleTo(String) + + + +ConvertibleTo(String)->ConvertibleTo(a:Type) + + + + +ConvertibleTo(Symbol) + + +ConvertibleTo(Symbol) + + + +ConvertibleTo(Symbol)->ConvertibleTo(a:Type) + + + + +ConvertibleTo(SExpression) + + +ConvertibleTo(SExpression) + + + +ConvertibleTo(SExpression)->ConvertibleTo(a:Type) + + + + +ConvertibleTo(Pattern(Base)) + + +ConvertibleTo(Pattern(Base)) + + + +ConvertibleTo(Pattern(Base))->ConvertibleTo(a:Type) + + + + +ConvertibleTo(List(Integer)) + + +ConvertibleTo(List(Integer)) + + + +ConvertibleTo(List(Integer))->ConvertibleTo(a:Type) + + + + +ConvertibleTo(List(Character)) + + +ConvertibleTo(List(Character)) + + + +ConvertibleTo(List(Character))->ConvertibleTo(a:Type) + + + + +ConvertibleTo(UnivariatePolynomialCategory(CommutativeRing)) + + +ConvertibleTo(UnivariatePolynomialCategory(CommutativeRing)) + + + +ConvertibleTo(UnivariatePolynomialCategory(CommutativeRing))->ConvertibleTo(a:Type) + + + + +LeftModule(a:Rng)->AbelianGroup() + + + + +ListAggregate(a:Type)->FiniteLinearAggregate(a:Type) + + + + +ListAggregate(a:Type)->ExtensibleLinearAggregate(a:Type) + + + + +StreamAggregate(a:Type) + + +StreamAggregate(a:Type) + + + +ListAggregate(a:Type)->StreamAggregate(a:Type) + + + + +StreamAggregate(a:Type)->LinearAggregate(a:Type) + + + + +UnaryRecursiveAggregate(a:Type) + + +UnaryRecursiveAggregate(a:Type) + + + +StreamAggregate(a:Type)->UnaryRecursiveAggregate(a:Type) + + + + +MultiDictionary(a:SetCategory) + + +MultiDictionary(a:SetCategory) + + + +MultiDictionary(a:SetCategory)->DictionaryOperations(a:SetCategory) + + + + +Monoid() + + +Monoid() + + + +SemiGroup() + + +SemiGroup() + + + +Monoid()->SemiGroup() + + + + +SemiGroup()->SetCategory() + + + + +RepeatedSquaring(SemiGroup) + +RepeatedSquaring(SemiGroup) + + +SemiGroup()->RepeatedSquaring(SemiGroup) + + + + +MultisetAggregate(a:SetCategory) + + +MultisetAggregate(a:SetCategory) + + + +MultisetAggregate(a:SetCategory)->SetAggregate(a:SetCategory) + + + + +MultisetAggregate(a:SetCategory)->MultiDictionary(a:SetCategory) + + + + +OrderedAbelianGroup() + + +OrderedAbelianGroup() + + + +OrderedAbelianGroup()->AbelianGroup() + + + + +OrderedCancellationAbelianMonoid() + + +OrderedCancellationAbelianMonoid() + + + +OrderedAbelianGroup()->OrderedCancellationAbelianMonoid() + + + + +OrderedCancellationAbelianMonoid()->CancellationAbelianMonoid() + + + + +OrderedAbelianMonoid() + + +OrderedAbelianMonoid() + + + +OrderedCancellationAbelianMonoid()->OrderedAbelianMonoid() + + + + +OrderedAbelianMonoid()->AbelianMonoid() + + + + +OrderedAbelianSemiGroup() + + +OrderedAbelianSemiGroup() + + + +OrderedAbelianMonoid()->OrderedAbelianSemiGroup() + + + + +OrderedAbelianSemiGroup()->AbelianMonoid() + + + + +OrderedAbelianSemiGroup()->OrderedSet() + + + + +OAMONS + + +OAMONS + + + +OCAMON + +OCAMON + + +OAMONS->OCAMON + + + + +OrderedMultisetAggregate(a:SetCategory) + + +OrderedMultisetAggregate(a:SetCategory) + + + +OrderedMultisetAggregate(a:SetCategory)->MultisetAggregate(a:SetCategory) + + + + +PriorityQueueAggregate(a:SetCategory) + + +PriorityQueueAggregate(a:SetCategory) + + + +OrderedMultisetAggregate(a:SetCategory)->PriorityQueueAggregate(a:SetCategory) + + + + +PriorityQueueAggregate(a:Type) + + +PriorityQueueAggregate(a:Type) + + + +PriorityQueueAggregate(a:SetCategory)->PriorityQueueAggregate(a:Type) + + + + +OrderedFinite() + + +OrderedFinite() + + + +OrderedFinite()->OrderedSet() + + + + +OrderedFinite()->Finite() + + + + +OrderedMonoid() + + +OrderedMonoid() + + + +OrderedMonoid()->OrderedSet() + + + + +OrderedMonoid()->Monoid() + + + + +OrderedRing() + + +OrderedRing() + + + +OrderedRing()->Monoid() + + + + +OrderedRing()->OrderedAbelianGroup() + + + + +Ring() + + +Ring() + + + +OrderedRing()->Ring() + + + + +Ring()->LeftModule(a:Ring) + + + + +Ring()->Monoid() + + + + +Rng() + + +Rng() + + + +Ring()->Rng() + + + + +PriorityQueueAggregate(a:Type)->BagAggregate(a:Type) + + + + +PriorityQueueAggregate(a:OrderedSet) + + +PriorityQueueAggregate(a:OrderedSet) + + + +PriorityQueueAggregate(a:OrderedSet)->PriorityQueueAggregate(a:SetCategory) + + + + +QueueAggregate(a:SetCategory) + + +QueueAggregate(a:SetCategory) + + + +QueueAggregate(a:SetCategory)->QueueAggregate(a:Type) + + + + +RetractableTo(a:Type) + + +RetractableTo(a:Type) + + + +RetractableTo(a:Type)->Category + + + + +RetractableTo(SetCategory) + + +RetractableTo(SetCategory) + + + +RetractableTo(SetCategory)->RetractableTo(a:Type) + + + + +RetractableTo(Symbol) + + +RetractableTo(Symbol) + + + +RetractableTo(Symbol)->RetractableTo(a:Type) + + + + +RetractableTo(Integer) + + +RetractableTo(Integer) + + + +RetractableTo(Integer)->RetractableTo(a:Type) + + + + +RetractableTo(NonNegativeInteger) + + +RetractableTo(NonNegativeInteger) + + + +RetractableTo(NonNegativeInteger)->RetractableTo(a:Type) + + + + +RetractableTo(Fraction(Integer)) + + +RetractableTo(Fraction(Integer)) + + + +RetractableTo(Fraction(Integer))->RetractableTo(a:Type) + + + + +RetractableTo(Float) + + +RetractableTo(Float) + + + +RetractableTo(Float)->RetractableTo(a:Type) + + + + +RetractableTo(Kernel(ExpressionSpace)) + + +RetractableTo(Kernel(ExpressionSpace)) + + + +RetractableTo(Kernel(ExpressionSpace))->RetractableTo(a:Type) + + + + +RetractableTo(CommutativeRing) + + +RetractableTo(CommutativeRing) + + + +RetractableTo(CommutativeRing)->RetractableTo(a:Type) + + + + +Rng()->AbelianGroup() + + + + +Rng()->SemiGroup() + + + + +RightModule(a:Rng)->AbelianGroup() + + + + +StackAggregate(a:SetCategory) + + +StackAggregate(a:SetCategory) + + + +StackAggregate(a:SetCategory)->StackAggregate(a:Type) + + + + +StringAggregate() + + +StringAggregate() + + + +StringAggregate()->OneDimensionalArrayAggregate(Character) + + + + +UnaryRecursiveAggregate(a:Type)->RecursiveAggregate(a:Type) + + + + +StepThrough() + + +StepThrough() + + + +StepThrough()->SetCategory() + + + + + diff --git a/src/axiom-website/community.html b/src/axiom-website/community.html new file mode 100644 index 0000000..20314fd --- /dev/null +++ b/src/axiom-website/community.html @@ -0,0 +1,248 @@ + + + + Axiom Computer Algebra System + + + + +
+ [ + + Home + + ] +   + [ + + Screenshots + + ] +   + [ + + FAQ + + ] +   + [ + + Download + + ] +   + [ + + Documentation + + ] +   + [ + + Current State + + ] +   + [ + + Community + + ] +   + [ + + Developers + + ] +   + [ + + Patches + + ] +
+ +
+
+
+

Axiom community, mailing lists

+ + Several + mailing lists are available: + +
+
axiom-mail + (Archive) +
General discussion on Axiom + +
axiom-developer + (Archive) +
When you have issues to compile Axiom, when you have issues with + Axiom internals + +
axiom-legal + (Archive) +
All legal issues, like license issues + +
axiom-math + (Archive) +
Discussion of math theory and philosophy related to Axiom +
+ + +

Axiom Releases

+ +

+ Axiom is released about every two months on a best-effort basis. + The "golden" released sources are maintained in GIT, CVS and ARCH. +

+ +

+ Axiom release (Git) is hosted on GitHub. +

+ +

+ Axiom release (CVS) is hosted on Savannah. +

+ +

+ Axiom release (CVS) is hosted on Sourceforge. +

+ +

+ Axiom release (ARCH) is hosted on arch.axiom-developer.org +

+ + +

Axiom Development

+ +

+ Axiom is continuously developed and a "silver" version is available. + The development sources are maintained in git and SVN. +

+ +

+ Axiom development (SVN) is hosted on Sourceforge. +

+ +

+ Axiom development (git) is hosted on git.axiom-developer.org +

+ + +

+ Here is a list of people that have contributed to Axiom (to November, 2008) + 

+Cyril Alberga          Roy Adler              Christian Aistleitner
+Richard Anderson       George Andrews         S.J. Atkins
+Henry Baker            Stephen Balzac         Yurij Baransky
+David R. Barton        Gerald Baumgartner     Gilbert Baumslag
+Jay Belanger           David Bindel           Fred Blair
+Vladimir Bondarenko    Mark Botch
+Alexandre Bouyer       Peter A. Broadbery     Martin Brock
+Manuel Bronstein       Florian Bundschuh      Luanne Burns
+William Burge
+Quentin Carpent        Robert Caviness        Bruce Char
+Ondrej Certik          Cheekai Chin           David V. Chudnovsky
+Gregory V. Chudnovsky  Josh Cohen             Christophe Conil
+Don Coppersmith        George Corliss         Robert Corless
+Gary Cornell           Meino Cramer           Claire Di Crescenzo
+David Cyganski
+Timothy Daly Sr.       Timothy Daly Jr.       James H. Davenport
+Didier Deshommes       Michael Dewar
+Jean Della Dora        Gabriel Dos Reis       Claire DiCrescendo
+Sam Dooley             Lionel Ducos           Martin Dunstan
+Brian Dupee            Dominique Duval
+Robert Edwards         Heow Eide-Goodman      Lars Erickson
+Richard Fateman        Bertfried Fauser       Stuart Feldman
+Brian Ford             Albrecht Fortenbacher  George Frances
+Constantine Frangos    Timothy Freeman        Korrinn Fu
+Marc Gaetano           Rudiger Gebauer        Kathy Gerber
+Patricia Gianni        Samantha Goldrich      Holger Gollan
+Teresa Gomez-Diaz      Laureano Gonzalez-Vega Stephen Gortler
+Johannes Grabmeier     Matt Grayson           Klaus Ebbe Grue
+James Griesmer         Vladimir Grinberg      Oswald Gschnitzer
+Jocelyn Guidry
+Steve Hague            Satoshi Hamaguchi      Mike Hansen
+Richard Harke          Vilya Harvey           Martin Hassner
+Arthur S. Hathaway     Dan Hatton             Waldek Hebisch
+Ralf Hemmecke          Henderson              Antoine Hersen
+Gernot Hueber
+Pietro Iglio
+Alejandro Jakubi       Richard Jenks
+Kai Kaminski           Grant Keady            Tony Kennedy
+Paul Kosinski          Klaus Kusche           Bernhard Kutzler
+Larry Lambe            Franz Lehner           Frederic Lehobey
+Michel Levaud          Howard Levy            Liu Xiaojun
+Rudiger Loos           Michael Lucks          Richard Luczak
+Camm Maguire           Francois Maltey        Alasdair McAndrew
+Bob McElrath           Michael McGettrick     Ian Meikle
+David Mentre           Victor S. Miller       Gerard Milmeister
+Mohammed Mobarak       H. Michael Moeller     Michael Monagan
+Marc Moreno-Maza       Scott Morrison         Joel Moses
+Mark Murray
+William Naylor         C. Andrew Neff         John Nelder
+Godfrey Nolan          Arthur Norman          Jinzhong Niu
+Michael O'Connor       Summat Oemrawsingh     Kostas Oikonomou
+Humberto Ortiz-Zuazaga  
+Julian A. Padget       Bill Page              Susan Pelzel
+Michel Petitot         Didier Pinchon         Ayal Pinkus
+Jose Alfredo Portes
+Claude Quitte
+Arthur C. Ralfs        Norman Ramsey          Anatoly Raportirenko
+Michael Richardson     Renaud Rioboo          Jean Rivlin
+Nicolas Robidoux       Simon Robinson         Raymond Rogers
+Michael Rothstein      Martin Rubey
+Philip Santas          Alfred Scheerhorn      William Schelter
+Gerhard Schneider      Martin Schoenert       Marshall Schor
+Frithjof Schulze       Fritz Schwarz          Nick Simicich
+William Sit            Elena Smirnova         Jonathan Steinbach
+Fabio Stumbo           Christine Sundaresan   Robert Sutor
+Moss E. Sweedler       Eugene Surowitz
+Max Tegmark            James Thatcher         Balbir Thomas
+Mike Thomas            Dylan Thurston         Barry Trager
+Themos T. Tsikas
+Gregory Vanuxem
+Bernhard Wall          Stephen Watt           Jaap Weel
+Juergen Weiss          M. Weller              Mark Wegman
+James Wen              Thorsten Werther       Michael Wester
+John M. Wiley          Berhard Will           Clifton J. Williamson
+Stephen Wilson         Shmuel Winograd        Robert Wisbauer
+Sandra Wityak          Waldemar Wiwianka      Knut Wolf
+Clifford Yapp          David Yun
+Vadim Zhytnikov        Richard Zippel         Evelyn Zoernack
+Bruno Zuercher         Dan Zwillinger
+
+

+ + +
+ + + diff --git a/src/axiom-website/currentstate.html b/src/axiom-website/currentstate.html new file mode 100644 index 0000000..cca960f --- /dev/null +++ b/src/axiom-website/currentstate.html @@ -0,0 +1,811 @@ + + + + Axiom Website + + + + +
+ [ + + Home + + ] +   + [ + + Screenshots + + ] +   + [ + + FAQ + + ] +   + [ + + Download + + ] +   + [ + + Documentation + + ] +   + [ + + Current State + + ] +   + [ + + Community + + ] +   + [ + + Developers + + ] +   + [ + + Patches + + ] +
+ +
+
+
+ This information was last updated on July 23, 2008 +
+ 
+
+Axiom was released on March 25, 2008. You can view the 
+release notes for each release.
+
+This TODO list represents abbreviated notes to categorize the work directions
+and note progress (completed items have a - prefix). Items are expanded as
+the work thread moves into that subject. There is no priority and no schedule.
+The ratio is (# of completed task per section)/(total completed tasks) and
+is a measure of the ratio of effort between areas. Completed tasks are marked
+with a - in column 1 (+ indicates changes since last update)
+
+There are approximately 285 tasks listed. These tasks are broken into the
+following groups:
+
+ Pristine sources is the main thread of developing Axiom
+ Splits are pieces of work to give away
+ Joins are pieces of work to incorporate or cooperate with
+ New Development is a list of ideas for new directions or needed work
+ Research is a list of ideas that need investigation
+ Community is a list of supporting tasks for the algebra community
+
+**************************************************************************
+PRISTINE SOURCES 
+ Motivation: create a new source tree and slowly migrate original NAG/IBM
+             sources into the tree. This will ensure that only things that
+             are properly required, used and tested get shipped. Further
+             ensure that everything exists under make/cvs control.
+
+ All of the tasks have been completed except
+
+  recover numerics
+  port to Windows
+
+**************************************************************************
+SPLITS 
+ Motivation: several pieces of the system have nothing to do with algebra
+             and either have old technology that needs to be rewritten or
+             common technology that ought to be contributed to other efforts.
+
+
+ FIREFOX BROWSER
+  Motivation: We need a portable, useful front end to Axiom that combines
+  the separate pieces of the Axiom command line, hyperdoc, and graphics
+  into a unified way of working with pamphlet files.
+
+ * Rewrite Hyperdoc Pages
+    This task is to extract the information from the current
+    hyperdoc system and make it available in the browser.
+      rewrite all pages using AJAX
+      add additional pages
+      rearrange the order for clarity
+      dynamically create category/domain/packages pages
+      dynamically create operations pages 
+      allow ASQ style lookups
+
+ * Integrate Graphics
+    This task is to extract the functionality from the graphics
+    and make it work within the browser.
++     collect the graphics into a book volume
+      document the graphics
+      refactor graphics to extract X11 API
+      make API compatible version of graphics for CANVAS element
+      enable dynamic drawing into CANVAS
+      enable popup page with CANVAS and control menu elements
+ 
+ * Write Notebook Front End
+    This task is to make a new, browser-based front end to Axiom.
+      create new API domain to expose history
+      make new context sensitive menus
+      make card/div/Record objects 
+      enable create/destroy/rearrange/import/export of card decks
+      unify card decks and pamphlets
+
+
+ AXIOM TEST SUITE->COMMON TEST SUITE
+  Motivation: Algebra needs a hand-verified test suite which ought to be
+              shared with other computer algebra systems. Needs a common
+              syntax, could be a driver for previous item.
+   rewrite regression test files with results from MMA, Maple, Maxima
+   reformat files to use a common theory/practice format
+   organize files into a useful taxonomy
+
+
+**************************************************************************
+JOINS 
+ Motivation: there are several CA systems with very interesting features
+             and/or algebra. Axiom needs to import these ideas or find
+             ways to cooperate with these projects.
+
+
+
+
+ GAP->AXIOM
+  Motivation: GAP has much stronger facilities for manipulating groups.
+              Axiom can do group manipulation but GAP needs to be studied.
+
+
+
+
+ OCTAVE->AXIOM
+  Motivation: Octave has the capability to be a replacement for the NAG
+              numeric libraries. Either an interface has to be retargetted
+              or the code has to be ported to Aldor or Lisp.
+-  Test rewriting Fortran code into Aldor
++  BLAS converted to pamphlet form
+   BLAS integrated into Axiom
+
+
+
+
+
+ MAGNUS->AXIOM
+  Motivation: Magnus is strong in infinite group theory. Like GAP it needs
+              to be studied and imported.
+-  Discussion and proposal for work in this area with CCNY
++  Magnus is being rebuilt as a literate program and will eventually be
++  merged with Axiom
+   Rebuild Magnus as a literate program
+   Extract and categorize the magnus algorithms
+   Provide Axiom implementations of the algorithms
+
+
+
+
+
+ ASTRONOMER->AXIOM
+  Motivation: Internal representation of polynomials that can be scaled to
+              ~250k terms needs to be imported.
+
+This task is dead.
+
+
+
+
+ COMBINAT->AXIOM
+  Motivation: External basis representation of large polynomials needs to
+              be imported.
+
+This task is dead.
+
+
+
+
+
+ MATHEMATICA->AXIOM
+  Motivation: Workbooks need to be imported.
+
+This task has changed. Rather than focus on the mathematica workbooks
+we plan to create and use the new Axiom firefox browser as a notebook
+front end similar to Sage.
+
+
+
+
+**************************************************************************
+NEW DEVELOPMENT 
+
+
+
+
+ EXTERNALS DOCUMENTATION
+  Motivation: in order to allow users to use the system we need to collect
+              and organize the documentation and help facilities.
+- The Axiom book was rewritten and is now available as part of the normal
+- distribution. 
+- The Axiom tutorial book was published at Lulu.com
+- The firefox browser was created
++ Additional pages from hyperdoc need firefox rewrites
+  The .dvi files need to be accessible in the firefox hyperdoc
++ )help files are needed for more domains
++ )display operation needs examples for operations 
+
+
+
+ INTERNALS DOCUMENTATION
+  Motivation: in order to allow multiple developers to work on the system
+              the structure of the system needs to be explained.
+-  Directory structure
+-   document shell script
+-   Makefile structure
+-   document how the makefile tree works
+-  Lisp structure
+-   src structure
+-    algebra 
+-   boot
+-    clef
+-    doc
+-    etc
+-   graph
+-    htex
+-    hyper
+-    include
+-    input
+-    interp
+-   install
+-   lib
+-    lsp
+-   noweb
+-    paste
+-    sman
+
+  Motivation: restructure Axiom into literate book volumes
+- Volume 0: Jenks book
+- Volume 1: Tutorial
+  Volume 2: User's Guide
+  Volume 3: Programmer's Guide
++ Volume 4: Developer's Guide
++ Volume 5: Interpreter
+  Volume 6: Axiom Command
++ Volume 7: Hyperdoc
++ Volume 8: Graphics
+  Volume 9: Compiler
+  Volume 10: Algebra
+  Volume 11: Firefox Hyperdoc
+
+
+
+ ALGEGRA DOCUMENTATION
+  LITERATE PROGRAMMING 
+   Motivation: algebra code should be able to include research papers
+               and documentation of design choices, space, time and
+               complexity. Researchers should be encouraged to write
+               literate programs. Literate programs should be easily
+               turned into book sections or chapters. 
+-   seek appropriate tools
+-     tangle, weave
+-     cweb
+-     noweb
+-   seek feedback
+-   create preliminary setup
+-   create example pamphlet (DHMATRIX)
+-     define doc structure
+-     prototype
+-   modify/rewrite noweb
+-     make undefined chunks transparent
+-     example generation
+      test case generation
+-   create booklet example (MATRIX)
++     define booklet structure
++     prototype
+-   integrate into make
+-   integrate into obj/mnt
++   integrate example (DHMATRIX2) into make
++   integrate book example into make
+    write pamphlets for type hierarchy
+      write SetCategory pamphlet
+      integrate into make
+      integrate into test
+    write pamphlets for data structure hierarchy
+      write aggregate pamphlet
+      integrate into make
+      integrate into test
+    create example pamphlet from scratch (PRIMES is in P algorithm)
+      recreate paper in tex
+      write algorithm
+      merge into build
++   create example pamphlet from planarity paper
+      recreate paper in tex
+      write algorithm
+      merge into build
+    create complex pamphlet (Barry Trager's Integration thesis)
+-     recreate thesis in tex
+      factor thesis into booklets
+      factor booklets into pamphlets
+
+
+
+ SINGLE-STEPPING
+  Motivation: Users of algebra should be able to step thru algorithms at
+              some "appropriate" level of detail. Hard to do as most 
+              algebra algorithms differ from hand methods.
+
+
+
+ LIBRARY CLEANUP
+  Motivation: FullyExplicitRingOver needs a better theory basis, etc.
+
+
+
+
+ GCL BUILD
+  Motivation: run on a popular common lisp
+-  Download GCL
+-  Build image in /spad/lsp/gcl
+-  Build Axiom on image
+
+
+
+ CMUCL BUILD
+  Motivation: run on a popular common lisp
+   Download CMUCL sources
+   Build image in /spad/lsp/cmucl
+   Build Axiom on image
+
+
+
+ CCL BUILD
+  Motivation: run on a popular common lisp
+-  Get final CCL sources from Arthur
+-  Build image in /spad/lsp/ccl
+-  Build Axiom on image
+
+This effort is dead.
+              
+**************************************************************************
+RESEARCH 
+
+ AXIOM JOURNAL
+  Motivation: Provide a motivation and place to publish Axiom algorithms
+              in a literate programming style. Journal development will
+              require refereed papers.
+-  Discuss with Community
+   Contact TOMS at ACM
+   Create License for pamphlet submission
+-    This is likely to be the creative commons license
+-  Create Prototype online site (www.nongnu.org/axiom/Journal)
+     Create Journal directory
+     Structure pages similar to Journal
+     Integrate into CVS
+     Allow dynamic referee lists
+   Research online/offline references to Axiom in literature
+   Create citations bibliography
+   Get permission to mirror/host published papers
+-  Give presentation to Univ of Pisa re: literate journal
+-  Worked with ACM to provide electronic ISSAC journal
+
+
+
+ 
++CRYSTAL
++ Motivation: Soon it will be possible to have all prior published math
++             online and locally available. We need to design and build
++             a user interface model that supports researchers in math
++             by providing semantic searches, multi-faceted problem views,
++             real-time problem classification, and an "intensional stance"
++             assistant user interface.
++ Define model of interaction
+  Prototype semantic markup
+  Prototype semantic classification with KROPS
+  Prototype facet viewer
+  Prototype classifier
+  Prototype intensional stance assisant
+
+
+
+
+ COMPLEXITY
+  Motivation: Axiom is fully layered. Research results in the lowest 
+              layers can be leveraged to generate results in higher layers.
+              Use Axiom against itself to create towers of complexity.
+              Insist on complexity results in documentation.
+
+
+
+
+ TEXTBOOK OBJECT
+  Motivation: Axiom interpreter objects don't have a standard name for
+              their parts and do not have a manipulation language for
+              rewriting parts. 
+              idea: have a "tree" primitive for each type
+              idea: define a graph of tree to tree maps that label each 
+                    edge with the transformation applied to the subtree
+
+
+
+
+ TEXTBOOK CONTEXT
+  Motivation: Equations are ambiguous when standing alone. Textbooks provide
+              context in the surrounding text. This is not adequately 
+              captured by the type system. A new approach to this issue
+              is required.
+
+
+
+
+ OPEN PROBLEMS LIST
+  Motivation: Magnus maintains a community agreed-upon list of "open problems"
+              This is an excellent source of PhD work and a good focus
+              mechanism. Create and maintain this.
+
+
+
+
+ PROVISOS
+  Motivation: Assume is a weak, global mechanism for capturing provisos.
+              A new proviso object needs to exist that wraps existing 
+              objects, provides scope for provisos and provides an
+              algebra of provisos based on type (arithmetic, boolean,
+              typed, etc).
+-  Review the literature to extract proviso forms.
+-  Classify proviso forms.
+-  Draft research paper
+   Develop an arithmetic of proviso forms.
+
+
+
+
+ INTERVAL ARITHMETIC
+  Motivation: This is a fundamentally stronger way to represent uncertainty
+              in results. It needs to be lifted from the numeric domain and
+              defined symbolically. It needs to be integrated into the
+              proviso algebra.
+
+   Document interval.as algebra domain
+
+
+
+
++INDEFINITES
++ Motivation: Research needs to be done on how to represent and compute
++             with indefinite objects from domains. An indefinite object
++             has a fixed type (such as Integer) but a non-specified value.
++             Thus it is possible to say that x+1 is of type Indefinite Integer
++             since 1 is an Integer and x is an Indefinite Integer.
++ Submit NSF grant proposal (awarded)
++ Conduct research on indefinites
+   Construct domain examples
+
+
+
+
++WEB ACCESS TO AXIOM
++ Motivation: There are several research efforts to make computer algebra
++             available on the web. Axiom needs to participate and lead.
++ Work with ORCCA
+   rebuild OPENMATH support
+   update to latest OPENMATH standards
++ Work with MathAction
++  integrate automatically running Axiom into web pages
+
+
+
+
+ DOMAIN INTEGRATION
+  Motivation: Axiom would be strongly useful in domains such as robotics.
+              Areas such as backsolving of joint position can be fruitful
+              areas of research. Deep integration in domains (DSP, Vision)
+              needs to be pushed.
++ The Doyen effort is the current direction for this. Axiom will integrate
++ with the rest of the sciences thru this effort.
+
+
+
+
+ SEMI-ANALYTIC SOLUTIONS
+  Motivation: Axiom could provide parallel tasks that would continuously
+              try to extract analytic solutions from incoming data. For
+              example, in vision finding analytic edges to feed to a
+              parallel recognizer algorithm.
+
+
+
+
+ GAMES INTEGRATION
+  Motivation: Axiom can be used to model games in novel ways. For example,
+              flight games with pursuit curves that live in a non-uniform
+              space (planet gravity, changing density across air/water).
+
+
+
+
+ CONSTRAINED MODELS
+  Motivation: Force motion that always remains analytic. Thus the final
+              solution is known to be analytic. For example, constrain
+              intersecting pipes to known boundary conditions. Computational
+              geometry seems the most fruitful area.
+
+
+
+
+ VALIDATION and VERIFICATION
+  Motivation: Algebra systems need a high level of confidence in the results.
+              Hand checking of the results is difficult and expensive. 
+              Automatically substituting results into the original equations
+              could raise confidence. Auditing of boundaries of algebra
+              algorithms. Precondition/Postcondition results for domains.
+              Audited test suites used for regression testing.
+
+
+
+
+ ALGEBRA MANIPULATION LANGUAGE
+  Motivation: Algebra systems have interpreters that embody a lot of 
+              knowledge about the actions available to manipulate equations.
+              This knowledge needs a language. The language needs to be
+              crafted so that it can be added to domains and applied
+              in symbolic form by a generalized simplification routine.
+   Review the literature to extract manipulation sentences.
+   Classify manipulation forms.
+   Draft manipulation paper
+
+
+
+
+ BOYER-MOORE THEOREM PROVER INTEGRATION
+  Motivation: Computational logic is a branch of computer mathematics
+              that is not currently available in Axiom. The Boyer-Moore
+              theorem prover, written in common lisp, provides a good
+              general purpose platform to study the interaction of the
+              theorem proving systems with Axiom
+-  Contact Boyer & Moore
+-  Contact Chandy & Misra
+-  Download ACL2
+   Build ACL2
+   Invoke ACL2 from Axiom
+   Integrate ACL2 into Axiom
+     Use Dijkstra's methods against SetCategory
+   Draft research paper
+
+
+
+
+ MetaPRL THEOREM PROVER Application
+  Motivation: MetaPRL is a Higher-order Logic proof system that is well
+              suited to proving Axiom-style languages. The proofs from
+              this system would give higher confidence levels in the
+              correctness of Axiom's algorithms.
+-  Contact Artimov 
+-   Review nuprl
+-   Review MetaPRL
+-   Construct trial example
+-     Collect all sorting routines in Axiom
+      Model the required subset of Axiom's language 
+      Construct a proof of the sort algorithms
+
+**************************************************************************
+COMMUNITY 
+ 
+ COLLECT AXIOM ALGEBRA DEVELOPERS
+  Motivation: Work in Axiom has continued over the years but has not
+              been integrated. Find this work and integrate it.
+ 
+ COLLECT AXIOM KNOWN BUGS
++ Motivation: Axiom has known bugs. Create a known bug list.
++   Integrate fixes for catdef
++   Integrate fixes for pleqn
+ 
+ TRAIN AXIOM INTERNAL DEVELOPERS
+  Motivation: Many of the subtasks listed above can be split off, done in
+              parallel and lead independently. Find people who are interested,
+              especially external projects (eg work with Gnuplot to enhance
+              it with Axiom's abilities).
+-  Set up cvs on Tenkan
+-  Set up accounts for developers
+-  Import current Axiom tree into CVS
+-  Try system builds on developers machines
+-  Exchange code via CVS
+-  Expand platforms
+-    Cygwin
+-    Mingw
+-    FreeBSD
+
+ SUPPORT FRENCH WORK
+  Motivation: An organized community will help focus new Axiom development.
+-  Lyon conference presentation
+-  Set up OSCAS group
+-  Announce OSCAS group
+-  Moderate OSCAS group
+  This is a dead branch of effort.
+
+ SET UP GOLD REPOSITORIES
+  Motivation: Community needs all the usual tools enabled like bug list,
+              mailing list, cvs, etc.
+- savannah.nongnu.org (CVS)
+- sourceforge.net (CVS)
+- arch.axiom-developer.org (ARCH)
++ github (GIT)
+
+ SET UP SILVER REPOSITORIES
+  Motivation: Expose leading edge work
+- sourceforge.net (SVN)
+- git.axiom-developer.org (GIT)
+  savannah.nongnu.org (GIT)
+
+
+ ROSETTA
+  Motivation: the community needs to see and use the many open source
+              computer algebra systems so they can compare and contrast
+              the various systems. This will help focus the issues of
+              what is currently possible and how hard it is to develop
+              a large, general purpose system. Also useful for teaching.
+-  Build Rosetta CD
+-   collect open source systems
+-   build them on a common subtree
+-   write Rosetta document
+-   write them to CD such that they can be run with installation
+-   distribute CD to conferences, teachers, etc
++  Update Rosetta CD
++   add Axiom
++   fix mistakes
++   add paragraph per table
++   update Rosetta document for Giac
++   update Rosetta document for DoCon
++   update Rosetta document for Macsyma
++   write Meta-Rosetta document to expound on differences
++   integrate systems with browser front-end
++   build broken systems
++    feed back diffs to original projects
++   add new systems
++   find new customers
++ The Rosetta effort is joining with the Quantian effort and the result
++ will be known as Doyen. This work is in process.
+  Build Doyen
+   Create live CD
+   Create standalone MathAction
+   Integrate Axiom into standalone MathAction
+   Build coordination tools between daughter Doyen and mother Doyen site
+
+ AXIOM WEBSITE
+  Motivation: maintain a website so that progress on making
+              Axiom open-source is made visible and the community has
+              a place to discuss issues and ideas (not necessarily
+              related to Axiom but to the community in general).
+-  create website
+-  build interest list
+-  create and maintain update schedule
+
+
+ EXPLORE FUNDING MODELS
+  Motivation: Find creative ways to pay people who work on Axiom. Everybody
+              wants free developers but paid work will motivate development.
+   Academic degree (BA in open source? BA in computer algebra?)
+- Developed an open source lab at City College of NY. Lab is up and running
+- Lab is part of the capstone projects for CS undergrads
+   Grant model by line of code
+-   contact IBM
+-   contact SUN
++   contact NSF
+    contact INRIA
+    contact DARPA 
++   explore other contacts
+   Award Model 
+-   Schelter Award
+-    contact UTexas
+-    contact FSF
+     develop plan
+      define award type and frequency
+      find sponsor organization
+-       Efraim Armendariz 
+-       Brad Kuhn 
+      find volunteers to judge award
+    Oscars/Emmy/Tony award model for CA at ISSAC?
+-   Jenks Award
+   Pay for support model
++  Pay for travel model
++   Contact Google
++   Contact IBM
++   Contact SUN
++   Contact Sage
++   Contact NSF  
+
+
+
+Completed, dead, or abandoned tasks + 
+ HYPERTEX->BROWSER/TEX
+  Motivation: hypertex was pre-browser technology. It should be rewritten
+              to fit into today's technology. At least HTML, possibly XML.
+- This work has forked into two paths. The first path is to recreate the
+- browser from the hypertex sources. The second path is to rebuild the
+- whole effort into a standard browser-based tool. Ultimately this will
+- probably get merged with the Doyen science platform effort 
+- Recreate Hypertex
+- Create standard browser
+-   create standalone browser version
+-   merge into Doyen
+
+
+ ASQ->JAVADOC/DOC++
+  Motivation: asq is pre-javadoc. It should be rewritten to fit into the
+              browser/tex new world of docs
+- The ASQ tool needs to get redone so that standard html pages get built
+- during system build and made available. They should have a javadoc-like
+- look and feel.
+- Create ASQ-HTML page generator
+
+This task is complete.
+
+
+
+
+ PLOT->GNUPLOT
+  Motivation: plot has nothing to do with algebra but has some nice features
+              such as data feedback from graph manipulation. These facilities
+              should be moved to Gnuplot and the plot function retargetted.
+  The graphics have been rebuilt and currently work. The graphics can only
+  be run from a file at the moment because the connecting process, called
+  sman, which manages all of the axiom processes is not yet working again.
+  Rebuild sman
+  Connect sman to graphics
+  Connect graphics to gnuplot
+
+This task was abandoned.
+
+
+
+
+The 2D I/O direction has changed. The current plan is to use the
+firefox browser as a universal front-end to Axiom.
+
+ READ-EVAL-PRINT->TeXmacs
+  Motivation: the time has come for 2d input. TeXmacs provides this. There
+              needs to be a much tighter integration of TeXmacs and Axiom.
+   Connect to TeXmacs community
+-  Build TeXmacs on Linux
+-  Build TexMacs on Cygwin
+   Build TexMacs on Windows
+   pick up the notebook interface ideas from Sage
+
+
+ AXIOM I/O->COMMON SYNTAX
+  Motivation: Rosetta should not exist. Algebra syntax should standardize.
+              Existing committee work needs feedback from Axiom.
+
+The Rosetta effort is complete.
+
+
+ + + + diff --git a/src/axiom-website/developers.html b/src/axiom-website/developers.html new file mode 100644 index 0000000..baaae86 --- /dev/null +++ b/src/axiom-website/developers.html @@ -0,0 +1,263 @@ + + + + Axiom Computer Algebra System + + + + +
+ [ + + Home + + ] +   + [ + + Screenshots + + ] +   + [ + + FAQ + + ] +   + [ + + Download + + ] +   + [ + + Documentation + + ] +   + [ + + Current State + + ] +   + [ + + Community + + ] +   + [ + + Developers + + ] +   + [ + + Patches + + ] +
+ +
+
+
+

Axiom Development

+ +Axiom development is now hosted on an Arch server +running on axiom.developer.org. +
+
+There is an irc channel where developers can find other developers. It is: +
+server: irc.freenode.net +
+channel:#axiom-developer +
+
+You only need to do these setup steps once. +In order to access the arch sources you need to let arch know who +you are with the command: +
+
+  tla my-id "First Last <addy@host.com>"
+
+
+Next you need to register an archive: +
+
+  tla register-archive arch@axiom-developer.org--axiom http://axiom-developer.org/archive/axiom
+
+
+Then set up a default archive +
+
+  tla my-default-archive arch@axiom-developer.org--axiom
+
+
+
+In order to get the latest source for the Axiom main line type: +
+
+  tla get axiom--main--1
+
+
+
+There are several branches available. Each branch is used to develop +a particular idea and get a stable, working version before merging +it back to the main line. The currently available branches are: +
+
+axiom--main--1       the main development branch
+     this branch will be mirrored to the CVS on savannah when it
+     is stable and tested. 
+     Currently in step 4 of development.
+
+axiom--hyperdoc--1   development of hyperdoc
+     NOTE: This branch has been merged and is now dead.
+     this branch contains code for building hyperdoc. 
+
+axiom--BSD--1        port to BSD
+     work with Mark Murray <mark at grondar.org>
+     Currently in step 1 of development.
+
+axiom--MACOSX--1     port to MACOSX
+     work with Chuck Miller <cfm at ms.unimelb.edu.au>
+     Currently in step 1 of development.
+
+book--main--1        axiom book
+     work with community to clean up the book for printing
+     Currently in step 1 of development.
+
+axiom--solaris--1    port to solaris
+     work with Kostas Oikonomou <ko at research.att.com>
+     Currently in step 1 of development.
+
+axiom--graphics--1   finish graphics integration
+     NOTE: This branch has been merged and is now dead.
+     work on integration, testing of graphics.
+
+axiom--windows--1    port to windows
+     work with < Mike Thomas <mike.thomas at brisbane.paradigmgeo.com>
+     Currently in step 1 of development.
+
+axiom--language--1   explore axiom language modifications
+     work with Stephen Wilson <wilsons at multiboard.com>
+     Currently in step 1 of development.
+
+axiom--sbcl--1       port axiom to steel bank common lisp
+     work with Tim Daly Jr. <tim at tenkan.org> and
+               Nate Daly <nate at tenkan.org>
+     Currently in step 1 of development.
+
+zlc--main--1        add a zero learning curve interface to axiom
+     work with Jinzhong Niu <jniu at gc.cuny.edu>
+               Xaiowei Xu <xuxw at yahoo.com>
+     Currently in step 1 of development.
+
+axiom--algebra--1    prototype algebra code
+     Currently in step 1 of development.
+
+axiom--GUI--1    portable GUI interface
+     work with Kai Kaminski (kai.kaminski@gmail.com)
+     Currently in step 1 of development.
+
+
+
+
+If you need write access to the archive you need to follow these steps: +
+Create a key by typing: +
+
+ssh-keygen -t dsa
+
+This will create a file called .ssh/id_dsa.pub. You need to send the +contents of this file to Tim Daly <daly@idsi.net> +so your interactions can be enabled. +
+In order to access the arch sources you need to let arch know who +you are with the command: +
+
+  tla my-id "First Last <addy@host.com>"
+
+
+Next you need to register an archive: +
+
+  tla register-archive arch@axiom-developer.org--axiom sftp://arch@axiom-developer.org/home/arch/archive/axiom
+
+Notice that you're using sftp rather than http. The sftp function +uses the secure keys to enable ftp access to the sources. The http +function is read-only. +
+Then set up a default archive +
+
+  tla my-default-archive arch@axiom-developer.org--axiom
+
+
+
+In order to get the latest source for the Axiom main line type: +
+
+  tla get axiom--main--1
+
+
+Now you can change the sources. Once you've made changes they need +to be sent back (commit) to the host. In order to do the commit +function in tla you need a log file that summarizes the changes. +You can create and edit the log file with: +
+
+  emacs -nw `tla make-log`
+
+
+or for the vi fans +
+  vi `tla make-log`
+
+Finally you commit the changes with: +
+
+  tla commit
+
+
+
+The official website for arch is +here +
+More information on arch is available +here +
+ + SourceForge.net Logo + + + + diff --git a/src/axiom-website/dhmatrix.spad.pdf b/src/axiom-website/dhmatrix.spad.pdf new file mode 100644 index 0000000..5fecda9 Binary files /dev/null and b/src/axiom-website/dhmatrix.spad.pdf differ diff --git a/src/axiom-website/diff.html b/src/axiom-website/diff.html new file mode 100644 index 0000000..5228d8a --- /dev/null +++ b/src/axiom-website/diff.html @@ -0,0 +1,78 @@ + + + + Submitting patch files + + + +
+
+In general a patch file is of the form
+
+cd the.dir
+diff -Naur file.pamphlet file.pamphlet.new >the.dir.file.pamphlet.patch
+
+this has several features:
+  * it tells me where the patch is being applied
+  * the Naur options give me context
+  * i can see the impact of a change to multiple files
+
+I read the patches before applying them so be sure to document the
+reasons for the change in the pamphlet file. It may seem trivial
+but remember that I didn't do the initial analysis so I, and others,
+will have to understand after the fact.
+
+In most cases in a pamphlet file if you're changing a few lines
+of code or a whole function it should be documented thus:
+
+
+...
+
+(defun foo ()
+  (list
+    "this is ok code"
+    "this is broken code"
+    "this is also broken"
+    "this is ok"
+  )
+)
+ 
+turns into
+
+\subsection{foo list fix}
+This code used to read:
+\begin{verbatim}
+    "this is broken code"
+    "this is also broken"
+\end{verbatim}
+but clearly the elements of the list are wrong. We are going to
+print this list for the user so we don't want them to know anything
+is broken. Thus we have wonderful new code that will inspire confidence.
+This list is printed with the [[printlist]] function.
+<>=
+    "this is great code"
+    "this inspires confidence"
+@
+
+
+...
+
+(defun foo ()
+  (list
+    "this is ok code"
+<>
+    "this is ok"
+  )
+)
+
+and, yes, I do know that this is tedious.
+
+In general, it is also useful to update the pamphlet files with 
+documentation-only changes as you understand what a block of code
+is intended to do. Most of this information has been lost to history
+and the world can leverage your efforts at understanding if you take
+the time to document it. In many ways this is as important as fixing
+the code.
+
+ + diff --git a/src/axiom-website/documentation.html b/src/axiom-website/documentation.html new file mode 100644 index 0000000..a31b83c --- /dev/null +++ b/src/axiom-website/documentation.html @@ -0,0 +1,257 @@ + + + + Axiom Computer Algebra System + + + + +
+ [ + + Home + + ] +   + [ + + Screenshots + + ] +   + [ + + FAQ + + ] +   + [ + + Download + + ] +   + [ + + Documentation + + ] +   + [ + + Current State + + ] +   + [ + + Community + + ] +   + [ + + Developers + + ] +   + [ + + Patches + + ] +
+ +
+
+
+

Why Literate Programming?

+ +Once a program has been developed and the developers have moved +on to other tasks it needs to be maintained. The fundamental problem +is that if you modify code you didn't write, you don't see the big +picture and you don't understand the reasons why the code is written +the way it is. Thus changing code without a larger context is almost +always going to introduce new bugs. +

+

+The only way to correctly change code is to deeply understand the +implications of the change. This requires a deep understanding of +the code and an awareness of the big picture. Yet the "why" of code +is rarely ever written down in standard programming practice. +The goal is only to elaborate the "how" so the machine can perform +the task. The programmer communicates with the machine. +

+

+Literate programming, as used in Axiom, is an attempt to communicate +with other users, developers, and researchers in addition to the +machine. The goal is to have the program read like a story so that +others can understand the rational, the theory, the choices, the +implications, and the implementation context as well as the "how". +

+

+This code is intended to live forever but it is highly probable +that you will not. Write to communicate with the next person to +pick up the torch. When you explore code, write down what you learn. +When you change code, explain why you made your choices. When you +write new code explain what others need to know to maintain it. +

+ + +

User documentation

+ +The Axiom system is gradually being documented in a set of volumes. +These change with every update to the system since they contain the +actual system source code. The volumes listed here are updated every +other month when the system is distributed. The current volume set is: + + +

+ Axiom is being reworked to use the Firefox browser as the new + front end. Static pages + from the new hyperdoc show some of the details. These pages + will be "live" in the new Axiom hyperdoc. + +

+ The algebra graph in its current form is + available. You'll need a reasonably new browser to see the graph. + This is a work in progress. + +

+ A first tutorial is available at: + http://www.dcs.st-and.ac.uk/~mnd/documentation/axiom_tutorial/ +

+ +

+ The Rosetta + (src) + (pdf) + document is a comparison of nearly equivalent commands between + many different computer algebra systems. +

+
+ + + diff --git a/src/axiom-website/dotabb b/src/axiom-website/dotabb new file mode 100644 index 0000000..4482d7f --- /dev/null +++ b/src/axiom-website/dotabb @@ -0,0 +1,646 @@ +digraph AxiomSept2008 { + ranksep=1.25; + node [shape=box, color=yellow, style=filled]; + +"ABELGRP" [color=lightblue,style=filled]; +"ABELGRP" -> "CABMON" + +"ABELMON" [color=lightblue,style=filled]; +"ABELMON" -> "ABELSG" + +"ABELSG" [color=lightblue,style=filled]; +"ABELSG" -> "SETCAT" + +"ACF" [color=lightblue,style=filled]; +"ACF" -> "FIELD" +"ACF" -> "RADCAT" + +"ACPLOT" -> "PPCURVE" + + +"AGG" [color=lightblue,style=filled]; +"AGG" -> "TYPE" + +"AHYP" [color=lightblue,style=filled]; +"AHYP" -> "CATEGORY" + +"ALAGG" [color=lightblue,style=filled]; +"ALAGG" -> "TBAGG" +"ALAGG" -> "LSAGG" + +"ALGEBRA" [color=lightblue,style=filled]; +"ALGEBRA" -> "RING" +"ALGEBRA" -> "MODULE" + + +"AMR" [color=lightblue,style=filled]; +"AMR" -> "RING" +"AMR" -> "BMODULE" + +"AN" -> "ES" +"AN" -> "ACF" +"AN" -> "RETRACT" +"AN" -> "LINEXP" +"AN" -> "REAL" +"AN" -> "CHARZ" +"AN" -> "KONVERT" +"AN" -> "DIFRING" + +"ANY" -> "SETCAT" + + +"ARRAY1" -> "A1AGG" + + +"ARRAY2" -> "ARR2CAT" +"ARRAY2" -> "IIARRAY2" + +"ARR2CAT" [color=lightblue,style=filled]; +"ARR2CAT" -> "HOAGG" + +"ASTACK" -> "STACK" + +"ATRIG" [color=lightblue,style=filled]; +"ATRIG" -> "CATEGORY" + +"ATTREG" [color=lightblue,style=filled]; +"ATTREG" -> "CATEGORY" + +"A1AGG" [color=lightblue,style=filled]; +"A1AGG" -> "FLAGG" + + +"BASTYPE" [color=lightblue,style=filled]; +"BASTYPE" -> "CATEGORY" + + +"BGAGG" [color=lightblue,style=filled]; +"BGAGG" -> "HOAGG" + +"BITS" -> "BTAGG" + +"BMODULE" [color=lightblue,style=filled]; +"BMODULE" -> "LMODULE" +"BMODULE" -> "RMODULE" + +"BOOLEAN" -> "ORDSET" +"BOOLEAN" -> "FINITE" +"BOOLEAN" -> "LOGIC" +"BOOLEAN" -> "KONVERT" + +"BRAGG" [color=lightblue,style=filled]; +"BRAGG" -> "RCAGG" + + +"BTAGG" [color=lightblue,style=filled]; +"BTAGG" -> "ORDSET" +"BTAGG" -> "LOGIC" +"BTAGG" -> "A1AGG" + +"CABMON" [color=lightblue,style=filled]; +"CABMON" -> "ABELMON" + +"CARD" -> "ORDSET" +"CARD" -> "ABELMON" +"CARD" -> "MONOID" +"CARD" -> "RETRACT" + +"CARTEN" -> "GRALG" +"CARTEN" -> "GRMOD" + + +"CATEGORY" [color=lightblue,style=filled]; + +"CHARNZ" [color=lightblue,style=filled]; +"CHARNZ" -> "RING" + +"CHARZ" [color=lightblue,style=filled]; +"CHARZ" -> "RING" + + +"CLAGG" [color=lightblue,style=filled]; +"CLAGG" -> "HOAGG" + +"CLIF" -> "RING" +"CLIF" -> "ALGEBRA" +"CLIF" -> "VSPACE" + + +"CFCAT" [color=lightblue,style=filled]; +"CFCAT" -> "CATEGORY" + + +"COLOR" -> "ABELSG" + + + +"COMBOPC" [color=lightblue,style=filled]; +"COMBOPC" -> "CFCAT" + +"COMRING" [color=lightblue,style=filled]; +"COMRING" -> "RING" +"COMRING" -> "BMODULE" + +"CONTFRAC" -> "ALGEBRA" +"CONTFRAC" -> "FIELD" + + + + + +"DBASE" -> "SETCAT" + +"DEQUEUE" -> "DQAGG" + +"DFLOAT" -> "FPS" +"DFLOAT" -> "DIFRING" +"DFLOAT" -> "OM" +"DFLOAT" -> "TRANFUN" +"DFLOAT" -> "SPFCAT" +"DFLOAT" -> "KONVERT" + +"DIAGG" [color=lightblue,style=filled]; +"DIAGG" -> "DIOPS" + +"DIFEXT" [color=lightblue,style=filled]; +"DIFEXT" -> "RING" + +"DIFRING" [color=lightblue,style=filled]; +"DIFRING" -> "RING" + +"DIOPS" [color=lightblue,style=filled]; +"DIOPS" -> "BGAGG" +"DIOPS" -> "CLAGG" + +"DIVRING" [color=lightblue,style=filled]; +"DIVRING" -> "ENTIRER" +"DIVRING" -> "ALGEBRA" + +"DLAGG" [color=lightblue,style=filled]; +"DLAGG" -> "RCAGG" + +"DLIST" -> "LSAGG" + +"DQAGG" [color=lightblue,style=filled]; +"DQAGG" -> "SKAGG" +"DQAGG" -> "QUAGG" + +"ELAGG" [color=lightblue,style=filled]; +"ELAGG" -> "LNAGG" + +"ELEMFUN" [color=lightblue,style=filled]; +"ELEMFUN" -> "CATEGORY" + +"ELTAB" [color=lightblue,style=filled]; +"ELTAB" -> "CATEGORY" + +"ELTAGG" [color=lightblue,style=filled]; +"ELTAGG" -> "ELTAB" + +"ENTIRER" [color=lightblue,style=filled]; +"ENTIRER" -> "RING" +"ENTIRER" -> "BMODULE" + +"ES" [color=lightblue,style=filled]; +"ES" -> "ORDSET" +"ES" -> "RETRACT" +"ES" -> "IEVALAB" +"ES" -> "EVALAB" + + + +"EUCDOM" [color=lightblue,style=filled]; +"EUCDOM" -> "PID" + +"EVALAB" [color=lightblue,style=filled]; +"EVALAB" -> "IEVALAB" + +"FAMR" [color=lightblue,style=filled]; +"FAMR" -> "AMR" +"FAMR" -> "FRETRCT" + +"FARRAY" -> "IFARRAY" + +"FFCAT" [color=lightblue,style=filled]; +"FFCAT" -> "MONOGEN" + + +"FIELD" [color=lightblue,style=filled]; +"FIELD" -> "EUCDOM" +"FIELD" -> "UFD" +"FIELD" -> "DIVRING" + +"FINITE" [color=lightblue,style=filled]; +"FINITE" -> "SETCAT" + +"FINRALG" [color=lightblue,style=filled]; +"FINRALG" -> "ALGEBRA" + +"FLAGG" [color=lightblue,style=filled]; +"FLAGG" -> "LNAGG" + + +"FLINEXP" [color=lightblue,style=filled]; +"FLINEXP" -> "LINEXP" + +"FPATMAB" [color=lightblue,style=filled]; +"FPATMAB" -> "TYPE" + +"FPS" [color=lightblue,style=filled]; +"FPS" -> "RNS" + +"FRAMALG" [color=lightblue,style=filled]; +"FRAMALG" -> "FINRALG" + +"FRETRCT" [color=lightblue,style=filled]; +"FRETRCT" -> "RETRACT" + +"FSAGG" [color=lightblue,style=filled]; +"FSAGG" -> "DIAGG" +"FSAGG" -> "SETAGG" + + + +"GCDDOM" [color=lightblue,style=filled]; +"GCDDOM" -> "INTDOM" + +"GRALG" [color=lightblue,style=filled]; +"GRALG" -> "GRMOD" +"GRALG" -> "RETRACT" + +"GRMOD" [color=lightblue,style=filled]; +"GRMOD" -> "SETCAT" + +"GROUP" [color=lightblue,style=filled]; +"GROUP" -> "MONOID" + +"HEAP" -> "PRQAGG" + +"HOAGG" [color=lightblue,style=filled]; +"HOAGG" -> "AGG" + +"HYPCAT" [color=lightblue,style=filled]; +"HYPCAT" -> "CATEGORY" + + +"IAN" -> "ES" +"IAN" -> "ACF" +"IAN" -> "RETRACT" +"IAN" -> "LINEXP" +"IAN" -> "REAL" +"IAN" -> "CHARZ" +"IAN" -> "KONVERT" +"IAN" -> "DIFRING" + +"IARRAY1" -> "A1AGG" + +"IARRAY2" -> "ARR2CAT" +"IARRAY2" -> "IIARRAY2" + +"IBITS" -> "BTAGG" + +"ICARD" -> "ORDSET" + +"IEVALAB" [color=lightblue,style=filled]; +"IEVALAB" -> "CATEGORY" + +"IFARRAY" -> "A1AGG" +"IFARRAY" -> "ELAGG" + +"IIARRAY2" -> "ARR2CAT" + + +"INS" [color=lightblue,style=filled]; +"INS" -> "UFD" +"INS" -> "EUCDOM" +"INS" -> "OINTDOM" +"INS" -> "DIFRING" +"INS" -> "KONVERT" +"INS" -> "RETRACT" +"INS" -> "LINEXP" +"INS" -> "PATMAB" +"INS" -> "CFCAT" +"INS" -> "REAL" +"INS" -> "CHARZ" +"INS" -> "STEP" + +"INT" -> "INS" +"INT" -> "KONVERT" +"INT" -> "OM" + +"INTDOM" [color=lightblue,style=filled]; +"INTDOM" -> "COMRING" +"INTDOM" -> "ALGEBRA" +"INTDOM" -> "ENTIRER" + + + +"IXAGG" [color=lightblue,style=filled]; +"IXAGG" -> "HOAGG" +"IXAGG" -> "ELTAGG" + +"KDAGG" [color=lightblue,style=filled]; +"KDAGG" -> "DIAGG" + +"KOERCE" [color=lightblue,style=filled]; +"KOERCE" -> "CATEGORY" + +"KONVERT" [color=lightblue,style=filled]; +"KONVERT" -> "CATEGORY" + +"LFCAT" [color=lightblue,style=filled]; +"LFCAT" -> "PRIMCAT" +"LFCAT" -> "TRANFUN" + +"LINEXP" [color=lightblue,style=filled]; +"LINEXP" -> "RING" + +"LOGIC" [color=lightblue,style=filled]; +"LOGIC" -> "BASTYPE" + +"LMODULE" [color=lightblue,style=filled]; +"LMODULE" -> "ABELGRP" + +"LNAGG" [color=lightblue,style=filled]; +"LNAGG" -> "IXAGG" +"LNAGG" -> "CLAGG" + +"LSAGG" [color=lightblue,style=filled]; +"LSAGG" -> "FLAGG" +"LSAGG" -> "ELAGG" + +"MDAGG" [color=lightblue,style=filled]; +"MDAGG" -> "DIOPS" + + +"MODULE" [color=lightblue,style=filled]; +"MODULE" -> "BMODULE" + +"MONOID" [color=lightblue,style=filled]; +"MONOID" -> "SGROUP" + +"MONOGEN" [color=lightblue,style=filled]; +"MONOGEN" -> "FRAMALG" +"MONOGEN" -> "COMRING" +"MONOGEN" -> "KONVERT" +"MONOGEN" -> "FRETRCT" +"MONOGEN" -> "FLINEXP" + + + +"MSETAGG" [color=lightblue,style=filled]; +"MSETAGG" -> "MDAGG" +"MSETAGG" -> "SETAGG" + + +"MTSCAT" [color=lightblue,style=filled]; +"MTSCAT" -> "PDRING" +"MTSCAT" -> "PSCAT" +"MTSCAT" -> "IEVALAB" +"MTSCAT" -> "EVALAB" + + +"NONE" -> "SETCAT" + + + +"NNI" -> "OAMONS" +"NNI" -> "MONOID" + +"OAGROUP" [color=lightblue,style=filled]; +"OAGROUP" -> "OCAMON" +"OAGROUP" -> "ABELGRP" + +"OAMON" [color=lightblue,style=filled]; +"OAMON" -> "OASGP" +"OAMON" -> "ABELMON" + +"OAMONS" [color=lightblue,style=filled]; +"OAMONS" -> "OCAMON" + +"OASGP" [color=lightblue,style=filled]; +"OASGP" -> "ORDSET" +"OASGP" -> "ABELMON" + +"OCAMON" [color=lightblue,style=filled]; +"OCAMON" -> "OAMON" +"OCAMON" -> "CABMON" + +"OINTDOM" [color=lightblue,style=filled]; +"OINTDOM" -> "INTDOM" +"OINTDOM" -> "ORDRING" + +"OM" [color=lightblue,style=filled]; +"OM" -> "CATEGORY" + +"OMSAGG" [color=lightblue,style=filled]; +"OMSAGG" -> "MSETAGG" +"OMSAGG" -> "PRQAGG" + +"ONECOMP" -> "SETCAT" +"ONECOMP" -> "FRETRCT" + + + +"ORDCOMP" -> "SETCAT" +"ORDCOMP" -> "FRETRCT" + + +"ORDFIN" [color=lightblue,style=filled]; +"ORDFIN" -> "ORDSET" +"ORDFIN" -> "FINITE" + +"ORDMON" [color=lightblue,style=filled]; +"ORDMON" -> "ORDSET" +"ORDMON" -> "MONOID" + +"ORDRING" [color=lightblue,style=filled]; +"ORDRING" -> "OAGROUP" +"ORDRING" -> "RING" +"ORDRING" -> "MONOID" + +"ORDSET" [color=lightblue,style=filled]; +"ORDSET" -> "SETCAT" + +"PALETTE" -> "SETCAT" + +"PATAB" [color=lightblue,style=filled]; +"PATAB" -> "CATEGORY" + +"PATLRES" -> "SETCAT" + +"PATMAB" [color=lightblue,style=filled]; +"PATMAB" -> "SETCAT" + +"PATRES" -> "SETCAT" + + +"PATTERN" -> "SETCAT" +"PATTERN" -> "RETRACT" + + + +"PDRING" [color=lightblue,style=filled]; +"PDRING" -> "RING" + +"PFECAT" [color=lightblue,style=filled]; +"PFECAT" -> "UFD" + +"PI" -> "OASGP" +"PI" -> "MONOID" + +"PID" [color=lightblue,style=filled]; +"PID" -> "GCDDOM" + + + + + + + +"PRIMARR" -> "A1AGG" + + +"PRIMCAT" [color=lightblue,style=filled]; +"PRIMCAT" -> "CATEGORY" + +"PRQAGG" [color=lightblue,style=filled]; +"PRQAGG" -> "BGAGG" + +"PSCAT" [color=lightblue,style=filled]; +"PSCAT" -> "AMR" + +"RADFF" -> "FFCAT" +"RADFF" -> "SAE" + + +"REF" -> "TYPE" + +"QFORM" -> "ABELGRP" + +"QUAGG" [color=lightblue,style=filled]; +"QUAGG" -> "BGAGG" + +"QUEUE" -> "QUAGG" + +"QEQUAT" -> "KOERCE" + +"RCAGG" [color=lightblue,style=filled]; +"RCAGG" -> "HOAGG" + +"RADCAT" [color=lightblue,style=filled]; +"RADCAT" -> "CATEGORY" + +"REAL" [color=lightblue,style=filled]; +"REAL" -> "KONVERT" + + +"RETRACT" [color=lightblue,style=filled]; +"RETRACT" -> "CATEGORY" + + +"RING" [color=lightblue,style=filled]; +"RING" -> "RNG" +"RING" -> "MONOID" +"RING" -> "LMODULE" + +"RMODULE" [color=lightblue,style=filled]; +"RMODULE" -> "ABELGRP" + +"RNG" [color=lightblue,style=filled]; +"RNG" -> "ABELGRP" +"RNG" -> "SGROUP" + +"RNS" [color=lightblue,style=filled]; +"RNS" -> "FIELD" +"RNS" -> "ORDRING" +"RNS" -> "REAL" +"RNS" -> "RETRACT" +"RNS" -> "RADCAT" +"RNS" -> "KONVERT" +"RNS" -> "PATMAB" +"RNS" -> "CHARZ" + +"ROMAN" -> "INS" + +"SAE" -> "MONOGEN" + + + +"SAOS" -> "ORDSET" + +"SETAGG" [color=lightblue,style=filled]; +"SETAGG" -> "SETCAT" +"SETAGG" -> "CLAGG" + +"SETCAT" [color=lightblue,style=filled]; +"SETCAT" -> "BASTYPE" +"SETCAT" -> "KOERCE" + +"SGROUP" [color=lightblue,style=filled]; +"SGROUP" -> "SETCAT" + +"SINT" -> "INS" +"SINT" -> "LOGIC" +"SINT" -> "OM" + +"SKAGG" [color=lightblue,style=filled]; +"SKAGG" -> "BGAGG" + +"SRAGG" [color=lightblue,style=filled]; +"SRAGG" -> "A1AGG" + +"STACK" -> "SKAGG" + +"STAGG" [color=lightblue,style=filled]; +"STAGG" -> "RCAGG" +"STAGG" -> "LNAGG" + +"STEP" [color=lightblue,style=filled]; +"STEP" -> "SETCAT" + +"SPFCAT" [color=lightblue,style=filled]; +"SPFCAT" -> "CATEGORY" + + +"TBAGG" [color=lightblue,style=filled]; +"TBAGG" -> "KDAGG" +"TBAGG" -> "IXAGG" + +"TRANFUN" [color=lightblue,style=filled]; +"TRANFUN" -> "TRIGCAT" +"TRANFUN" -> "ATRIG" +"TRANFUN" -> "HYPCAT" +"TRANFUN" -> "AHYP" +"TRANFUN" -> "ELEMFUN" + +"TRIGCAT" [color=lightblue,style=filled]; +"TRIGCAT" -> "CATEGORY" + +"TUPLE" -> "PRIMARR" + +"TYPE" [color=lightblue,style=filled]; +"TYPE" -> "CATEGORY" + +"UFD" [color=lightblue,style=filled]; +"UFD" -> "GCDDOM" + +"ULSCAT" [color=lightblue,style=filled]; +"ULSCAT" -> "UPSCAT" + +"URAGG" [color=lightblue,style=filled]; +"URAGG" -> "RCAGG" + +"UPSCAT" [color=lightblue,style=filled]; +"UPSCAT" -> "PSCAT" + +"UPXSCAT" [color=lightblue,style=filled]; +"UPXSCAT" -> "UPSCAT" + +"UTSCAT" [color=lightblue,style=filled]; +"UTSCAT" -> "UPSCAT" + +"VSPACE" [color=lightblue,style=filled]; +"VSPACE" -> "MODULE" + +} diff --git a/src/axiom-website/dotabb.html b/src/axiom-website/dotabb.html new file mode 100644 index 0000000..3ce1a05 --- /dev/null +++ b/src/axiom-website/dotabb.html @@ -0,0 +1,46 @@ + + +Axiom Abbreviated Category and Domain graph + + + + +

Axiom Abbreviated Category and Domain graph

+ + +
+ choose here: + 0.1 or + 0.25 or + 0.5 or + 1.0 or + 1.5 or ... +
+ +
+ + +
+ + + diff --git a/src/axiom-website/dotabb.svg b/src/axiom-website/dotabb.svg new file mode 100644 index 0000000..daeb754 --- /dev/null +++ b/src/axiom-website/dotabb.svg @@ -0,0 +1,2352 @@ + + +]> + + + + + +AxiomSept2008 + + +ABELGRP + +ABELGRP + + +CABMON + +CABMON + + +ABELGRP->CABMON + + + + +ABELMON + +ABELMON + + +CABMON->ABELMON + + + + +ABELSG + +ABELSG + + +ABELMON->ABELSG + + + + +SETCAT + +SETCAT + + +ABELSG->SETCAT + + + + +BASTYPE + +BASTYPE + + +SETCAT->BASTYPE + + + + +KOERCE + +KOERCE + + +SETCAT->KOERCE + + + + +ACF + +ACF + + +FIELD + +FIELD + + +ACF->FIELD + + + + +RADCAT + +RADCAT + + +ACF->RADCAT + + + + +DIVRING + +DIVRING + + +FIELD->DIVRING + + + + +EUCDOM + +EUCDOM + + +FIELD->EUCDOM + + + + +UFD + +UFD + + +FIELD->UFD + + + + +CATEGORY + +CATEGORY + + +RADCAT->CATEGORY + + + + +ACPLOT + +ACPLOT + + +PPCURVE + +PPCURVE + + +ACPLOT->PPCURVE + + + + +AGG + +AGG + + +TYPE + +TYPE + + +AGG->TYPE + + + + +TYPE->CATEGORY + + + + +AHYP + +AHYP + + +AHYP->CATEGORY + 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+ + +SAOS + +SAOS + + +SAOS->ORDSET + + + + +SINT + +SINT + + +SINT->LOGIC + + + + +SINT->OM + + + + +SINT->INS + + + + +SRAGG + +SRAGG + + +SRAGG->A1AGG + + + + +STAGG + +STAGG + + +STAGG->RCAGG + + + + +STAGG->LNAGG + + + + +TRIGCAT->CATEGORY + + + + +TUPLE + +TUPLE + + +TUPLE->PRIMARR + + + + +ULSCAT + +ULSCAT + + +UPSCAT + +UPSCAT + + +ULSCAT->UPSCAT + + + + +UPSCAT->PSCAT + + + + +URAGG + +URAGG + + +URAGG->RCAGG + + + + +UPXSCAT + +UPXSCAT + + +UPXSCAT->UPSCAT + + + + +UTSCAT + +UTSCAT + + +UTSCAT->UPSCAT + + + + + diff --git a/src/axiom-website/download.html b/src/axiom-website/download.html new file mode 100644 index 0000000..767259b --- /dev/null +++ b/src/axiom-website/download.html @@ -0,0 +1,1050 @@ + + + + Axiom Computer Algebra System + + + + +
+ [ + + Home + + ] +   + [ + + Screenshots + + ] +   + [ + + FAQ + + ] +   + [ + + Download + + ] +   + [ + + Documentation + + ] +   + [ + + Current State + + ] +   + [ + + Community + + ] +   + [ + + Developers + + ] +   + [ + + Patches + + ] +
+ +
+
+

Pre-compiled binaries

+ + Axiom has been compiled to run on various platforms. +
+
+
+

+ This table contains links to various tar-gzipped version of files. + In general you need to know the name of the file you download, + usually something ending in .tgz (tar-gzip). You also need to know + where the file gets untarred, this is referred to as (where) below. + When you cd to the (where) location you should see the top level + Makefile for Axiom, the changelog, etc. +

+

+ Axiom builds on various platforms and uses the convention that the last + name in the AXIOM shell variable denotes the type of system. This is + referred to as the SYSNAME. You need to know which SYSNAME you downloaded. +

+

+ To use one of these binaries just do: +

+  download the binary and untar it.
+  cd axiom
+  export AXIOM=`pwd`/mnt/SYSNAME <= replace SYSNAME with actual name
+  export PATH=$AXIOM/bin:$PATH
+  axiom
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + SYSNAME      + + + + Nov 2007 + + + + Jan 2008 + + + + Mar 2008 + + + + May 2008 + + + + July 2008 + + + + Sept 2008 + + + + Nov 2008 + +
+ + debian + + + + + + + src + bin + + + + src + bin + + + + src + bin + + + + + + + src + bin + + + + src + bin + +
+ + Doyen + + + + + + + src + iso + + + + src + iso + + + + + + + src + iso + + + + + + + src + +
+ + fedora5 + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + +
+ + fedora6 + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + +
+ + fedora7 + + + + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + +
+ + fedora8 + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + +
+ + fedora8-64 + + + + + + + + + + src + bin + + + + src + bin + + + + + + + + + + src + +
+ + fedora9 + + + + + + + + + + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + +
+ + macosxppc + + + + + + + src + bin + + + + src + bin + + + + + + + + + + + + + +
+ + opensuse + + + + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + +
+ + redhat72 + + + + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + +
+ + redhat9 + + + + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + +
+ + ubuntu + + + + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + + + + src + bin + +
+ + ubuntu64 + + + + + + + + + + + + + + + + + + + + + + src + bin + +
+ + +

Older versions

+ + + Fedora Core 1 (binary) + + +
+ + + Fedora Core 2 (binary) + + +
+ + + Fedora Core 2, AMD 64 bit processor (binary) + + +
+ +
+
+

Source code

+

+Axiom source code is maintained in a Gold and Silver +version. The Gold version is the "released" version. Gold +versions are released every two months. +
+The Silver version is the current "bleeding edge" that +contains changes which will be tested and released into Gold +every two months. Unless you need a recent feature or bug fix, or are +working as a developer, there is no reason to use Silver + + +

GOLD SOURCES

+

Tarball

+The Gold (November 2008) release of Axiom is available. +
+The source code tarball from November, 2008 is +here + +
+wget http://axiom.axiom-developer.org/axiom-website/downloads/axiom-july2008-src.tgz
+tar -zxf axiom-july2008-src.tgz
+cd axiom
+export AXIOM=`pwd`/mnt/ (see table below)
+export PATH=$AXIOM/bin:$PATH
+make
+
+ + +

GIT

+You can clone the git repository from GitHub: + +
+git-clone git://github.com/daly/axiom.git
+cd axiom
+export AXIOM=`pwd`/mnt/ (see table below)
+export PATH=$AXIOM/bin:$PATH
+make
+
+ + +

CVS

+ You can also download the source tree from CVS and compile it. To + download the code from sourceforge, do: + +
+export CVS_RSH="ssh"
+cvs -z3 -d:pserver:anonymous@axiom.cvs.sourceforge.net:/cvsroot/axiom co -P axiom 
+cd axiom
+export AXIOM=`pwd`/mnt/ (see table below)
+export PATH=$AXIOM/bin:$PATH
+make
+
+ +Or you can download the sourcecode from savannah: +
+export CVS_RSH="ssh"
+cvs -z3 -d:pserver:anonymous@cvs.savannah.nongnu.org:/sources/axiom co -P axiom 
+cd axiom
+export AXIOM=`pwd`/mnt/ (see table below)
+export PATH=$AXIOM/bin:$PATH
+make
+
+ + +
+

Compile notes

+

In general, various systems insist on moving critical files + around or, worse yet, don't install needed files. These notes + show particular details for known systems + +

Ubuntu

+ 
+echo 0 >/proc/sys/kernel/randomize_va_space
+apt-get install cvs m4 libxpm-dev libxt-dev x-dev libx11-dev libxext-dev gettext git-core texlive gawk
+cvs -z3 -d:pserver:anonymous@axiom.cvs.sourceforge.net:/cvsroot/axiom co -P axiom
+cd axiom
+export AXIOM=`pwd`/mnt/ubuntu
+export PATH=$AXIOM/bin:$PATH
+make
+
+ +

OpenSuSE

+ 
+echo 0 >/proc/sys/kernel/exec_shield
+echo 0 >/proc/sys/kernel/randomize_va_space
+rpm -i texlive-latex-2007-69.noarch.rpm
+cvs -z3 -d:pserver:anonymous@axiom.cvs.sourceforge.net:/cvsroot/axiom co -P axiom
+cd axiom
+export AXIOM=`pwd`/mnt/opensuse
+export PATH=$AXIOM/bin:$PATH
+make
+
+ +

debian

+ 
+You might have to add a line to /etc/apt/sources.list like:
+  deb http://ftp.debian.org/debian etch main contrib
+and then do:
+apt-get update
+
+Next you need to install these packages:
+
+apt-get install gcc libc6-dev build-essential debhelper g++ g++-4.1 gcl 
+apt-get install gettext gs-gpl html2text intltool-debian libgmp3-dev
+apt-get install libgmp3c2 libgmpxx4 libice-dev libxau-dev libxaw-headers
+apt-get install libxaw7-dev libxdmcp-dev libxext-dev libxmu-dev libxmu-headers
+apt-get install libxpm-dev libxt-dev po-debconf x-dev x11proto-core-dev
+apt-get install x11proto-input-dev x11proto-kb-dev x11proto-xext-dev
+apt-get install xtrans-dev libncurses5-dev libreadline5-dev libsm-dev
+apt-get install libstdc++6-4.1-dev libx11-dev gawk
+cvs -z3 -d:pserver:anonymous@axiom.cvs.sourceforge.net:/cvsroot/axiom co -P axiom
+cd axiom
+export AXIOM=`pwd`/mnt/debian
+export PATH=$AXIOM/bin:$PATH
+make
+
+ +

Mac OSX PPC

+The MAC port has a few issues. The known problems are: +
+The pseudo-terminals (/dev/pty) don't work so the hyperdoc/graphics fails.
+This is still under study.
+
+The )browse command works but you have to use the latest firefox because
+safari does not seem to know about the http request object.
+
+There is a nasty interaction between CVS and OSX. Apparently CVS won't
+let you delete directories. OSX considers two names that differ only
+by case to be the same thing. Axiom did a global downcase of all
+filenames but CVS doesn't want to delete directories so the uppercase
+ones overwrite the lowercase ones. The fix is to use the sources
+from git rather than from CVS.
+
+ 
+install xcode from http://developer.apple.com/tools/download
+download and untar the sources from the apple website.
+
+cd axiom
+export AXIOM=`pwd`/mnt/macosxppc
+export PATH=/sw/bin:$AXIOM/bin:$PATH
+make
+
+ + + +

Doyen

+

The Doyen image was created by Jose Alfredo Portes

+ +
+

+ The book is included in the binary distribution. To view it type: +
+ + xdvi (where)/mnt/(SYSNAME)/doc/book.dvi + +

+

+ The tutorial is also included in the binary distribution. To view it type: +
+ + xdvi (where)/mnt/(SYSNAME)/doc/bookvol1.dvi + +

+
+ Axiom is also available as a pre-compiled package in: + + + +
+In general, you download the binary to a location and then type: +
+  tar -zxf binaryname.tgz
+     this creates a directory called "mnt"
+     in "mnt" there is a directory which is the sysname.
+     for instance, you'll see mnt/fedora5 so the sysname is fedora5
+  export AXIOM=`pwd`/mnt/(sysname)
+  export PATH=$AXIOM/bin:$PATH
+  axiom
+
+You do not need the source code to run Axiom. +Everything is in the binary. +If you do choose to build from the sources you download the source +and then type: +
+  tar -zxf sourcename.tgz
+    this creates a directory called "axiom"
+    the sysname is given in the column head of the following table
+  cd axiom
+  export AXIOM=`pwd`/mnt/(sysname)
+  export PATH=$AXIOM/bin:$PATH
+  make
+
+Note that if the make fails on some systems it may be due to the +security setup. As root you can try: +
+ echo 0 >/proc/sys/kernel/exec-shield
+ echo 0 >/proc/sys/kernel/randomize_va_space
+
+Both of these settings have caused build problems. + + + + diff --git a/src/axiom-website/endpaper.pdf b/src/axiom-website/endpaper.pdf new file mode 100644 index 0000000..45471f9 Binary files /dev/null and b/src/axiom-website/endpaper.pdf differ diff --git a/src/axiom-website/faq.html b/src/axiom-website/faq.html new file mode 100644 index 0000000..d94ea84 --- /dev/null +++ b/src/axiom-website/faq.html @@ -0,0 +1,255 @@ + + + + Axiom Computer Algebra System + + + + +
+ [ + + Home + + ] +   + [ + + Screenshots + + ] +   + [ + + FAQ + + ] +   + [ + + Download + + ] +   + [ + + Documentation + + ] +   + [ + + Current State + + ] +   + [ + + Community + + ] +   + [ + + Developers + + ] +   + [ + + Patches + + ] +
+ +
+
+ +
+

FAQ

+ +
+
+ + What is Axiom license? + +
+
+ Axiom is free software, available under a BSD like license. For + more details, please have a look in the licences + available in the CVS repository. +
+ +
+ + What is the future of Axiom? + +
+
+ + Tim Daly is Lead Axiom Developer. He has a fairly detailed agenda for Axiom. + +
+ +
+ + What is Axiom size? + +
+
+ + Axiom is: +
    +
  • 92 MB of source code
    • +
  • about 56 MB once compiled
    • +
  • 403591 lines for the interpreted in 220 files
    • +
  • 255790 lines of algebra in 371 files
    • +
+ +
+ +
+ + On which system is known to run Axiom? + +
+
+ + People have compiled and run Axiom on: +
    + +
  • Debian GNU/Linux 3.0, on i386, Sparc64. On PowerPC + architecture, Axiom will not build because it + requires gcc-3.3 (which supports -mlong-calls option, + supporting relocs of more tha 24 bits). +
    • + +
  • Debian GNU/Linux sid, on mipsel, ia64, i386, ppc, alpha, + and sparc (and more + to come!)
    • + +
  • RedHat Linux 7.3 and 9.0 on i386
    • + +
  • Slackware Linux 8.0.01 on Intel ProLiant ML530 + 2@800MHz
    • + +
  • Red Hat Linux 8.0 on Intel ProLiant ML530 2@1.0GHz + -- Red Hat Enterprise Linux ES release 2.1 (Panama)
    • + +
  • SuSE Linux Ent Svr 8.0 on Intel ProLiant DL360 G2 + 2@1.4GHz
    • + +
  • Red Flag Linux 4.0
    • + +
  • Microsoft Windows
    • +
+ +
+ +
+ + How do I submit a patch? + +
+
+
+In general a patch file is of the form
+
+cd the.dir
+diff -Naur file.pamphlet file.pamphlet.new >the.dir.file.pamphlet.patch
+send the patch to the Axiom mailing list 
+<axiom-developer@nongngu.org> or
+Tim Daly <daly@axiom-developer.org>
+
+this has several features:
+  * it tells me where the patch is being applied
+  * the Naur options give me context
+  * I can see the impact of a change to multiple files
+
+I read the patches before applying them so be sure to document the
+reasons for the change in the pamphlet file. It may seem trivial
+but remember that I didn't do the initial analysis so I, and others,
+will have to understand after the fact.
+
+In most cases in a pamphlet file if you're changing a few lines
+of code or a whole function it should be documented thus:
+
+
+...
+
+(defun foo ()
+  (list
+    "this is ok code"
+    "this is broken code"
+    "this is also broken"
+    "this is ok"
+  )
+)
+ 
+turns into
+
+\subsection{foo list fix}
+This code used to read:
+\begin{verbatim}
+    "this is broken code"
+    "this is also broken"
+\end{verbatim}
+but clearly the elements of the list are wrong. We are going to
+print this list for the user so we don't want them to know anything
+is broken. Thus we have wonderful new code that will inspire confidence.
+This list is printed with the [[printlist]] function.
+<<foo list fix>>=
+    "this is great code"
+    "this inspires confidence"
+@
+
+
+...
+
+(defun foo ()
+  (list
+    "this is ok code"
+<<foo list fix>>
+    "this is ok"
+  )
+)
+
+and, yes, I do know that this is tedious.
+
+In general, it is also useful to update the pamphlet files with 
+documentation-only changes as you understand what a block of code
+is intended to do. Most of this information has been lost to history
+and the world can leverage your efforts at understanding if you take
+the time to document it. In many ways this is as important as fixing
+the code.
+
+
+ +
+
+ + + diff --git a/src/axiom-website/index.html b/src/axiom-website/index.html new file mode 100644 index 0000000..741592d --- /dev/null +++ b/src/axiom-website/index.html @@ -0,0 +1,214 @@ + + + + Axiom Computer Algebra System + + + + +
+ [ + + Home + + ] +   + [ + + Screenshots + + ] +   + [ + + FAQ + + ] +   + [ + + Download + + ] +   + [ + + Documentation + + ] +   + [ + + Current State + + ] +   + [ + + Community + + ] +   + [ + + Developers + + ] +   + [ + + Patches + + ] +
+ +
+
+DOCUMENTATION: + + + +
+

+ What is Axiom? +

+

+ + Axiom has been in development since 1971. At that time, it was called + Scratchpad. Scratchpad was a large, general purpose computer algebra + system that was originally developed by IBM under the direction of + Richard Jenks. The project started in 1971 and evolved slowly. Barry + Trager was key to the technical direction of the project. Scratchpad + developed over a 20 year stretch and was basically considered as a + research platform for developing new ideas in computational + mathematics. In the 1990s, as IBM's fortunes slid, the Scratchpad + project was renamed to Axiom, sold to the Numerical Algorithms Group + (NAG) in England and became a commercial system. As part of the + Scratchpad project at IBM in Yorktown + + Tim Daly + + worked on all aspects + of the system and eventually helped transfer the product to NAG. For a + variety of reasons it never became a financial success and NAG + withdrew it from the market in October, 2001. + +

+

+ + NAG agreed to release Axiom as free software. The basic motivation was + that Axiom represents something different from other programs in a lot + of ways. Primarily because of its foundation in mathematics the Axiom + system will potentially be useful 30 years from now. In its current + state it represents about 30 years and 300 man-years of research + work. To strive to keep such a large collection of knowledge alive + seems a worthwhile goal. + +

+ + Efforts are underway to extend this software to +

    +
  • (a) develop a better user interface +
  • (b) make it useful as a teaching tool +
  • (c) develop an algebra server protocol +
  • (d) integrate additional mathematics +
  • (e) rebuild the algebra in a literate programming style +
  • (f) integrate logic programming +
  • (g) develop an Axiom Journal with refereed submissions. +
+

+
+

+ + Axiom is a general purpose Computer Algebra system. It is useful for + research and development of mathematical algorithms. It defines a + strongly typed, mathematically correct type hierarchy. It has a + programming language and a built-in compiler. + +

+

+ Axiom development was partially supported by + CAISS, + the Center for Algorithms and Interactive Scientific Software. + CAISS is a joint effort of the Computer Science and Mathematics + Departments of The City College of New York, part of the City + University system. Support by CAISS and CCNY is gratefully acknowledged. + In particular, the support by +
+ 

+   Matthew Goldstein  CUNY Chancellor
+   Zeev Dagan         CCNY Provost
+   Maria Tamargo      CCNY Dean of Science
+   Joseph Barba       CCNY Dean of Engineering
+   Gilbert Baumslag   CCNY Distinguish Professor, Director of CAISS
+   Douglas Troeger    CCNY Computer Science Chair
+   Ed Grossman        CCNY Mathematics Chair
+
+ +

+ + SourceForge.net Logo + +
Last Update: November 2008
+ + + + diff --git a/src/axiom-website/nov2008abb.svg b/src/axiom-website/nov2008abb.svg new file mode 100644 index 0000000..e22a237 --- /dev/null +++ b/src/axiom-website/nov2008abb.svg @@ -0,0 +1,3537 @@ + + +]> + + + + + +dotabb + + +CATEGORY + + +CATEGORY + + + +AHYP + + +AHYP + + + +AHYP->CATEGORY + + + + +ATRIG + + +ATRIG + + + +ATRIG->CATEGORY + + + + +ATTREG + + +ATTREG + + + +ATTREG->CATEGORY + + + + +BASTYPE + + +BASTYPE + + + +BASTYPE->CATEGORY + + + + +KOERCE + + +KOERCE + + + +KOERCE->CATEGORY + + + + +CFCAT + + +CFCAT + + + +CFCAT->CATEGORY + + + + +KONVERT + + +KONVERT + + + +KONVERT->CATEGORY + + + + +ELEMFUN + + +ELEMFUN + + + +ELEMFUN->CATEGORY + + + + +ELTAB + + +ELTAB + + + +ELTAB->CATEGORY + + + + +HYPCAT + + +HYPCAT + + + +HYPCAT->CATEGORY + + + + +IEVALAB + + +IEVALAB + + + +IEVALAB->CATEGORY + + + + +OM + + +OM + + + +OM->CATEGORY + + + + +PTRANFN + + +PTRANFN + + + +PTRANFN->CATEGORY + + + + +PATAB + + +PATAB + + + +PATAB->CATEGORY + + + + +PRIMCAT + + +PRIMCAT 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+ +COMPCAT->ORDSET + + + + +COMPCAT->COMRING + + + + +COMPCAT->DIFEXT + + + + +COMPCAT->FLINEXP + + + + +COMPCAT->MONOGEN + + + + +FFCAT + + +FFCAT + + + +FFCAT->MONOGEN + + + + + diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html new file mode 100644 index 0000000..b717fce --- /dev/null +++ b/src/axiom-website/patches.html @@ -0,0 +1,740 @@ + + + + Axiom Computer Algebra System + + + + +
+ [ + + Home + + ] +   + [ + + Screenshots + + ] +   + [ + + FAQ + + ] +   + [ + + Download + + ] +   + [ + + Documentation + + ] +   + [ + + Current State + + ] +   + [ + + Community + + ] +   + [ + + Developers + + ] +   + [ + + Patches + + ] +
+ +
+
+ +RELEASES + + +
+

November 2007 release

+ +November 2007 Release Notes +

+
+20070721.01.tpd.patch +update contributor name list
+20070721.02.tpd.patch +make evalSharpOne declare arg specials (Waldek Hebisch)
+20070722.01.tpd.patch +cleanup latex warnings
+20070810.01.tpd.patch +remove metameta
+20070811.01.tpd.patch +add mathml (Arthur Ralfs)
+20070811.02.tpd.patch +remove spurious trace debug calls (Greg Vanuxem)
+20070811.03.tpd.patch +fix digits-by-radix (Steve Wilson)
+20070812.01.tpd.patch +remerge input branch
+20070812.02.tpd.patch +mathml license change
+20070812.03.tpd.patch +newton.spad added
+20070819.01.tpd.patch +move commands into bookvol6
+20070821.01.tpd.patch +remove bookvol6.idx file
+20070821.02.tpd.patch +move axiom command to bookvol6
+20070822.01.tpd.patch +document the axiom command
+20070822.02.tpd.patch +add )help files
+20070823.01.tpd.patch +add )help files
+20070824.01.tpd.patch +add )help documentation to algebra sources
+20070826.01.tpd.patch +add )help files
+20070826.02.tpd.patch +add )help files
+20070826.03.tpd.patch +update VERSION variable
+20070901.01.tpd.patch +add )help files
+20070903.01.tpd.patch +add )help files
+20070905.01.tpd.patch +add )help files
+20070906.01.tpd.patch +add )help files
+20070907.01.tpd.patch +copy axiom command to int
+20070907.02.tpd.patch +make regression respect NOISE variable
+20070909.01.tpd.patch +remove duplicate newton.spad
+20070913.01.tpd.patch +schaum1.input added
+20070914.01.tpd.patch +fix )hd restart (Jose Portes)
+20070914.02.tpd.patch +remove double )spool from regression tests
+20070914.03.tpd.patch +fix bad bracing of )hd command
+20070915.01.tpd.patch +cleanup regression tests
+20070915.02.tpd.patch +100 integrate((z^a+1)^b,z) infinite loop
+20070916.01.tpd.patch +103 solve(z=z,z)
+20070921.01.tpd.patch +101 laplace(log(z),z,w)
+20070927.01.tpd.patch +add pfaffian regression test
+20070929.01.tpd.patch +remove pfaffian regression test
+20071003.01.tpd.patch +add AxiomServer domain
+20071004.01.tpd.patch +kamke0 ODE regression tests
+20071004.02.tpd.patch +kamke1 ODE regression tests
+20071005.01.tpd.patch +kamke2 ODE regression tests
+20071005.02.tpd.patch +kamke3 ODE regression tests
+20071005.03.tpd.patch +kamke4 ODE regression tests
+20071005.04.tpd.patch +kamke5 ODE regression tests
+20071006.01.tpd.patch +kamke6 ODE regression tests
+20071006.02.tpd.patch +kamke7 ODE regression tests
+20071013.01.acr.patch +add Arthur Ralfs license
+20071014.01.acr.patch +use getContentType(pathvar) in axserver (Arthur Ralfs)
+20071020.01.acr.patch +fix axserver to use new return values (Arthur Ralfs)
+20071118.01.tpd.patch +7010 (209), 7011 fix i-output bugs
+20071119.01.tpd.patch +add fedora 6,7,8 stanzas
+ +
+

January 2008 Release

+ +January 2008 Release Notes +

+
+20071129.01.tpd.patch +add zips/axiomfonts.tgz for mathml
+20071129.02.tpd.patch +remove some mathml fonts, change instructions
+20071205.01.tpd.patch +7014, 7016 fix continuedFraction mathml
+20071206.01.tpd.patch +7020 find right sourcefile for )show
+20071208.01.tpd.patch +add makeDBPage, getShow to AxiomServer domain
+20071215.01.tpd.patch +rewrite boot code to lisp and add to bookvol5
+20071215.02.tpd.patch +fix Makefile typos for bookvol11 and axbook
+20071215.03.gxv.patch +7023 fix memory leak in makegraph.c (Greg Vanuxem)
+20071216.01.tpd.patch +put Aldor into build process
+20071216.02.tpd.patch +put Aldor into build process
+20071216.03.acr.patch +7019 fix F,3 mathml rendering (Arthur Ralfs)
+20071217.01.acr.patch +fix hex(10) mathml rendering (Arthur Ralfs)
+20071217.02.tpd.patch +7041, 7042 ignore regression gensyms
+20071218.01.acr.patch +fix lastType output (Arthur Ralfs)
+20071225.01.sxw.patch +fix top_level throw target typo (Steve Wilson)
+20071228.01.tpd.patch +create graphics for complex gamma function
+20071229.01.jap.patch +fix hardcoded firefox pathnames (Jose Portes)
+20071230.01.acr.patch +fix ambiguous mathml output (Arthur Ralfs)
+20071230.02.tpd.patch +update summation.input with new mathml output
+20071230.03.tpd.patch +prevent spurious remake of axbook
+20080102.01.tpd.patch +fix Makefile typo for axbook
+20080103.01.tpd.patch +7090/355 handle besselK
+20080104.01.tpd.patch +function renames in regression files
+20080104.02.tpd.patch +correct authorship of besselk patches
+20080106.01.tpd.patch +revert dgamma change
+20080107.01.tpd.patch +add multiple platforms to Makefile
+20080107.02.tpd.patch +regression test gamma and polygamma
+20080107.03.tpd.patch +regression test besselk
+20080116.01.tpd.patch +regression test special fns against Abramowitz & Stegun
+20080119.01.tpd.patch +add E1 special function
+20080119.02.tpd.patch +add Franz Lehner
+20080119.03.tpd.patch +handle E1(0.0) correctly
+20080120.01.gxv.patch +handle numlock in hyperdoc (Greg Vanuxem)
+20080120.02.tpd.patch +7101/204 fix MoreSystemCommand
+20080120.03.tpd.patch +7102/412 fix equality in TBAGG
+20080125.01.tpd.patch +add En special function
+20080125.02.tpd.patch +change VERSION variable
+ +
+

March 2008 Release

+ +March 2008 Release Notes +

+
+20080127.01.tpd.patch +refcard
+20080130.01.tpd.patch +Ei
+20080201.01.tpd.patch +7016 LexTriangularPackage
+20080209.01.tpd.patch +add Exponential Integral to book
+20080210.01.tpd.patch +powerpc-macosx defs patch
+20080215.01.tpd.patch +++E examples
+20080216.01.wxh.patch +hash tables to speed compiles
+20080216.02.tpd.patch +add function examples
+20080217.01.wxh.patch +fix hash tables to speed compiles
+20080218.01.tpd.patch +add function examples
+20080218.02.tpd.patch +add function examples
+20080219.01.tpd.patch +add additional firefox hyperdoc pages
+20080220.01.tpd.patch +add additional firefox hyperdoc pages
+20080221.01.tpd.patch +add additional firefox hyperdoc pages
+20080221.02.wxh.patch +7099 complex gamma function investigation
+20080222.01.tpd.patch +7099 logGamma vs log(Gamma)
+20080222.02.tpd.patch +add additional hyperdoc page translations
+20080222.03.tpd.patch +move hyperdoc bitmaps location
+20080229.01.tpd.patch +add additional hyperdoc page translations
+20080301.01.tpd.patch +add additional hyperdoc page translations
+20080302.01.tpd.patch +add additional hyperdoc page translations
+20080303.01.tpd.patch +add additional hyperdoc page translations
+20080304.01.tpd.patch +add additional hyperdoc page translations
+20080305.01.tpd.patch +add additional hyperdoc page translations
+20080312.01.tpd.patch +BasicSieve, primes, intfact documentation
+20080313.01.pab.patch +hashcode for Aldor
+20080314.01.wxh.patch +heugcd fricas rev 256
+20080316.01.acr.patch +bug 7113 invisible times
+20080316.02.tpd.patch +CATS verification
+20080317.01.tpd.patch +CATS verification
+20080318.01.tpd.patch +CATS verification
+20080323.01.tpd.patch +add menus to firefox axiom console
+20080325.01.tpd.patch +mathml invisibletimes regression testing
+20080325.02.tpd.patch +move display function to bookvol5
+20080325.03.tpd.patch +handle firefox operations page requests
+ +
+

May 2008 Release

+ +May 2008 Release Notes +

+
+20080328.01.tpd.patch +CATS integration regression testing
+20080330.01.tpd.patch +CATS integration regression testing
+20080331.01.tpd.patch +CATS integration regression testing
+20080401.01.tpd.patch +CATS integration regression testing
+20080402.01.tpd.patch +CATS integration regression testing
+20080403.01.tpd.patch +CATS integration regression testing
+20080404.01.tpd.patch +faq 45, faq 46
+20080406.01.tpd.patch +CATS integration regression testing
+20080408.01.tpd.patch +mapleok cleanup
+20080409.01.tpd.patch +CATS integration regression testing
+20080409.02.tpd.patch +add src/doc/toe.gif by Max Tegmark
+20080409.03.tpd.patch +CATS integration regression testing
+20080413.01.tpd.patch +CATS integration regression testing
+20080414.01.tpd.patch +CATS integration test suite
+20080415.01.tpd.patch +CATS Schaums-Axiom equivalence testing (1)
+20080416.01.tpd.patch +CATS Schaums-Axiom equivalence testing (2-7)
+20080417.01.tpd.patch +fixed 14:150
+20080417.02.tpd.patch +CATS Schaums-Axiom equivalence testing (8-10)
+20080418.01.tpd.patch +CATS Schaums-Axiom equivalence testing (1-11)
+20080419.01.tpd.patch +CATS Schaums-Axiom equivalence testing (12)
+20080420.01.tpd.patch +CATS Schaums-Axiom equivalence testing (13)
+20080421.01.tpd.patch +CATS Schaums-Axiom equivalence testing (14-16)
+20080423.01.tpd.patch +CATS Schaums-Axiom equivalence testing (17-22)
+20080424.01.tpd.patch +CATS Schaums-Axiom equivalence testing (23)
+20080425.01.tpd.patch +CATS Schaums-Axiom equivalence testing (24-25)
+20080426.01.tpd.patch +CATS Schaums-Axiom equivalence testing (26-28)
+20080427.01.tpd.patch +CATS Schaums-Axiom equivalence testing (29-34)
+20080428.01.tpd.patch +CATS Schaums-Axiom post-mortem fixex
+20080429.01.tpd.patch +CATS post-mortem typo fixes for schaum12
+20080430.01.tpd.patch +CATS schaum13 post-mortem fixes
+20080501.01.tpd.patch +CATS Schaums-Axiom post-mortem fixes
+20080502.01.tpd.patch +CATS Schaums-Axiom post-mortem fixes
+20080504.01.tpd.patch +CATS Schaums-Axiom post-mortem fixes
+20080504.02.tpd.patch +CATS Schaums-Axiom post-mortem fixes
+20080505.01.tpd.patch +CATS Schaums-Axiom post-mortem fixes
+20080505.02.tpd.patch +CATS Schaums-Axiom branch cut analysis
+20080506.01.tpd.patch +CATS Schaums-Axiom post-mortem fixups
+20080508.01.wxh.patch +intef.spad integrate(asech(x)/x,x) bug
+20080508.02.tpd.patch +ignore gensyms in schaums regression tests
+20080511.01.tpd.patch +CATS Schaums-Axiom post-mortem fixup
+20080523.01.tpd.patch +MIT integration tests
+20080526.01.tpd.patch +add fedora8-64 to Makefile
+ +
+

July 2008 Release

+ +July 2008 Release Notes +

+
+20080527.01.tpd.patch +Fedora 9 updates
+20080528.01.tpd.patch +configure/readme rewrite
+20080529.01.dxh.patch +fix hyperdoc on interrupted rebuilds
+20080530.01.tpd.patch +books creation
+20080531.01.tpd.patch +construct book PDFs
+20080601.01.tpd.patch +bookvol8/graphics documentation
+20080603.01.tpd.patch +remove src/graph/view2d
+20080604.01.tpd.patch +bookvol8 extract view3d
+20080605.01.tpd.patch +remove view3d directory and files
+20080606.01.tpd.patch +make viewman from bookvol8
+20080606.02.tpd.patch +remove view
+20080606.03.tpd.patch +correct pathnames in changelog
+20080607.01.tpd.patch +make viewalone from bookvol8
+20080607.02.tpd.patch +Remove src/graph/viewalone directory
+20080607.03.tpd.patch +FAQ 48: Getting Axiom source from git
+20080608.01.tpd.patch +make gdraws from bookvol8
+20080608.02.tpd.patch +remove gdraws directory, use bookvol8
+20080608.03.tpd.patch +remove src/graph, use bookvol8
+20080609.01.tpd.patch +move hypertex into bookvol7
+20080609.02.tpd.patch +remove unused files in src/hyper
+20080609.03.tpd.patch +src/hyper directory removed, use bookvol7
+20080610.01.tpd.patch +build bookvol11
+20080610.02.tpd.patch +stop redundant builds
+20080611.01.tpd.patch +general form updates for books
+20080613.01.tpd.patch +compress viewman.c to a single file
+20080614.01.tpd.patch +compress viewalone to a single C file
+20080618.01.tpd.patch +use dvipdfm for hyperlinking (Anatoly Raportirenko)
+20080619.01.tpd.patch +add Anatoly Raportirenko
+20080619.02.tpd.patch +systematically index chunks
+20080619.03.tpd.patch +add Ralf Hemmecke documentation to ax.boot
+20080620.01.wxh.patch +fix direct product multiply in Monoid
+20080621.01.wxh.patch +default to close on failed read
+20080701.01.tpd.patch +update faq for X11 libs
+20080704.01.tpd.patch +add Samantha Goldrich to credits
+20080707.01.tpd.patch +construct hypertex from bookvol7
+20080715.01.tpd.patch +bookvol7.1 hyperdoc pages
+20080715.02.tpd.patch +remove src/hyper/pages/*.ht
+20080715.03.gxv.patch +mousewheel handling by Greg Vanuxem
+20080715.04.tpd.patch +remove ht.db from bookvol7
+20080717.01.tpd.patch +remove src/graph
+20080718.01.tpd.patch +bookvol7.1 hyperdoc additional documentation
+20080719.01.tpd.patch +bookvol7.1 more documentation
+20080720.01.tpd.patch +bookvol7.1 hyperdoc documentation added
+20080721.01.tpd.patch +bookvol7.1 hyperdoc specific macros
+20080722.01.tpd.patch +change VERSION number
+ +
+

September 2008 Release

+ +September 2008 Release Notes +

+
+20080724.01.tpd.patch +document lisp calls from hypertex
+20080725.01.tpd.patch +reflow for line latex overflows
+20080727.01.tpd.patch +update What's New pages
+20080728.01.tpd.patch +expand RootPage->Topics->Numbers
+20080729.01.tpd.patch +expand next tree level
+20080729.02.tpd.patch +more hyperdoc pages 1 of 2
+20080729.03.tpd.patch +bookvol7.1 part 2 of 2
+20080802.01.tpd.patch +Build ht.db from bookvol7.1
+20080809.01.tpd.patch +Remove pages directory
+20080814.01.tpd.patch +Use uncompress at build time
+20080815.01.tpd.patch +make firefox pages before input tests
+20080816.01.tpd.patch +comment out long running test
+20080816.02.tpd.patch +add additional regression tests
+20080817.01.tpd.patch +fix uncompress
+20080817.02.tpd.patch +recover function.input
+20080817.03.tpd.patch +exported function documentation
+20080818.01.tpd.patch +demo Axiom type towers (Hemmecke)
+20080818.02.tpd.patch +new input files (Stumbo, Cyganski, Hemmecke)
+20080819.01.tpd.patch +overload.input added (Cyganski)
+20080820.01.tpd.patch +fix typos in latex code
+20080821.01.tpd.patch +add MappingPackage4
+20080822.01.tpd.patch +add linalg,overload regressions
+20080823.01.tpd.patch +UnaryRecursiveAggregate API examples
+20080823.02.tpd.patch +stream API examples
+20080823.03.tpd.patch +++CapitalLetter syntax change
+20080824.01.tpd.patch +use ++X for example lines
+20080824.02.tpd.patch +expose difference between ^ and ** (Xaiojun)
+20080827.01.wsp.patch +replace \over with \frac (Page)
+20080828.01.mxr.patch +add cost to bottomUp output (Page)
+20080829.01.tpd.patch +graphviz dotfile decoration
+20080830.01.tpd.patch +graphviz dotfile decoration
+20080831.01.tpd.patch +graphviz dotfile decoration
+20080901.01.tpd.patch +add start of multivar poly test suite
+20080904.01.tst.patch +add reduce example (Tsikas)
+20080904.02.tpd.patch +graphviz dotfile decoration
+20080905.01.tpd.patch +graphviz dotfile decoration
+20080906.01.tpd.patch +move aggcat.spad to bookvol10
+20080908.01.tpd.patch +bookvol10 latex cleanup
+20080909.01.tpd.patch +bookvol0 change \over to \frac
+20080911.01.tpd.patch +bookvol11 fix firefox background image
+20080911.02.tpd.patch +bookvol10 merge coerce.spad
+20080912.01.tpd.patch +split bookvol10 into 10, 10.1..4
+20080916.01.tpd.patch +bookvol10.2 document additional categories
+20080917.01.tpd.patch +bookvol10.2 remove dup function defn in FLAGG
+20080917.02.wsp.patch +mkfunc.spad add parse function
+20080918.01.tpd.patch +bookvol10.2 add BASTYPE, SETCAT, ABELSG
+20080918.02.tpd.patch +bookvol10.2 add FINITE, ORDSET, SGROUP
+20080919.01.tpd.patch +bookvol10.2 add more categories
+20080920.01.tpd.patch +bookvol10.2 add more categories
+20080921.01.tpd.patch +September 2008 release fixups
+ +
+

November 2008 Release

+ +November 2008 Release Notes +

+
+20080925.01.tpd.patch +bookvol10.2 new categories added
+20080926.01.tpd.patch +bookvol10.2 new categories, attributes added
+20080927.01.tpd.patch +bookvol10.2 new categories, absorb naalgc
+20080928.01.tpd.patch +bookvol10.2 new categories, absorb trigcat
+20080930.01.tpd.patch +bookvol10.2 move XF, FPC from ffcat.spad
+20081001.01.tpd.patch +bookvol10.2 add more categores
+20081002.01.tpd.patch +bookvol10.2 add more categores
+20081003.01.tpd.patch +bookvol10.2 add POLYCAT
+20081004.01.tpd.patch +bookvol10.2 add UPOLYC, update attributes
+20081005.01.tpd.patch +bookvol10.2 add categories
+20081006.01.tpd.patch +bookvol10.2 add categories
+20081007.01.tpd.patch +bookvol10.2 add categories, update ATTREG
+20081008.01.tpd.patch +bookvol10.2 add ACF, fix MONOGEN, FFCAT
+20081023.01.tpd.patch +bookvol10.2 add categories
+20081026.01.tpd.patch +bookvol10.2 add categories
+20081027.01.tpd.patch +add sae.input regression test
+20081029.01.tpd.patch +bookvol10.2 add categories
+20081101.01.tpd.patch +bookvol10.2 add categories
+20081105.01.axj.patch +add reclos2.input
+20081106.01.tpd.patch +remove matcat.spad from src/algebra/Makefile
+20081107.01.tpd.patch +ignore probabilistic results
+20081108.01.tpd.patch +bookvol10.2 add categories
+20081109.01.tpd.patch +bookvol10.2 add categories
+20081110.01.tpd.patch +bookvol10.2 add categories
+20081112.01.tpd.patch +bookvol10.2 add categories
+20081114.01.tpd.patch +bookvol10.3 add dhmatrix
+20081115.01.tpd.patch +bookvol10.2 remove duplicates in export lists
+20081116.01.tpd.patch +rosetta.pamphlet fix magnus URL
+20081118.01.tpd.patch +make parallel test work
+20081119.01.tpd.patch +november 2008 release
+20081119.02.tpd.patch +november 2008 fixups
+ +
+

January 2009 Release

+ +In process, not yet released

+
+20081122.02.tpd.patch +ubuntu64 parallel core support
+ + + \ No newline at end of file diff --git a/src/axiom-website/releasenotes.html b/src/axiom-website/releasenotes.html new file mode 100644 index 0000000..fab2ec8 --- /dev/null +++ b/src/axiom-website/releasenotes.html @@ -0,0 +1,1280 @@ + + + + Axiom Computer Algebra System + + + + +
+ [ + + Home + + ] +   + [ + + Screenshots + + ] +   + [ + + FAQ + + ] +   + [ + + Download + + ] +   + [ + + Documentation + + ] +   + [ + + Current State + + ] +   + [ + + Community + + ] +   + [ + + Developers + + ] +   + [ + + Patches + + ] +
+ +
+
+ +RELEASES + + + +

November 23, 2008 Release Notes

+patch list +
+November 2008 Release Notes
+
+Axiom website
+
+  New patch tracking has been added:
+     * Patches
+
+  Volumes have been significantly updated:
+     * Book Volume 10.2
+     * Book Volume 10.3
+
+Book Volume 10.2 Axiom Categories completed
+
+   The effort here is to create fully indexed, cross-referenced,
+   graphical documentation for Axiom categories in a standalone
+   form. This is a "live" literate document which contains the 
+   actual source code used to build the system.
+
+   * Book Volume 10.2
+
+Book Volume 10.3 Axiom Domains started
+
+   This volume will contain the Axiom domains.
+
+   * Book Volume 10.3
+
+Rosetta documentation
+
+  * Fix the Magnus URL
+
+Input Files
+
+  * New input files (Jakubi, Maltey, Rubey, Page, Daly}
+     dhmatrix, reclos2
+  * Changed input files (Hebisch, Daly)
+     sae, r20bugs
+
+Build changes
+
+  * Testing is now run in parallel which will significantly
+    speed up the final test phase
+ 
+
+ + +

September 23, 2008 Release Notes

+patch list +
+September 2008 Release Notes
+
+
+Axiom website
+   The effort here is to improve the support for offline literate
+   documentation. The primary changes are the inclusion of graphs
+   and additional book volumes.
+
+* 
+  Contains the new algebra volumes and subvolumes.
+* 
+  Contains a "clickable" graph that indexes into the algebra.
+
+
+
+
+Graphviz, PDF, and HTML integration
+   The effort here is to unify these three technologies in a way
+   that simplifies the user interface and improves documentation
+
+* Graphviz is used if available but not required
+* Algebra graphs are automatically generated at build time
+  from algebra source files
+* Graphviz graphs now properly hyperlink into PDF files allowing
+  any node in a graph to link to any document page
+
+
+
+
+Book volume 0 (Jenks and Sutor)
+
+* 
+* replace \over with \frac
+
+
+
+Book volume 7.1 (Hyperdoc pages)
+   The effort here is to create a literate document that contains
+   all of the "live" pages used in hyperdoc. The PDF is being
+   constructed so that a user can effectively "browse" the static
+   hyperdoc pages, which are included, without a running Axiom.
+
+* 
+* The source for all of the pages is now contained in this book.
+* Hyperdoc now fetches the pages directly from the book.
+* The hyper page directory and all files are gone.
+* Some of the static page images are now inside the PDF
+* Pages have href links allowing "in-pdf" navigation of pages
+
+
+
+
+Book volume 10 (Algebra)
+   The effort here is to create a way to describe and deeply
+   document the algebra. This volume was split to better handle
+   the structure of Axiom's information. 
+
+* Split into 5 volumes
+   - 10   Implementation
+   - 10.1 Theory
+   - 10.2 Categories
+   - 10.3 Domains
+   - 10.4 Packages
+
+
+
+
+Book volume 10.2 (Algebra Categories)
+   The effort here is to create fully indexed, cross-referenced,
+   graphical documentation for Axiom categories in a standalone
+   form. This is a "live" literate document which contains the 
+   actual source code used to build the system.
+
+* 
+* Contains 60 categories so far
+* Has partial graphs for each category
+* Has list of exported functions 
+* Has information about source of functions
+* Has index cross reference by function and category
+* Has PDF href links so that URLs work:
+   
+* Has forward/backward links between categories
+* Automatically generates "clickable" graphs:
+   
+* Graph clicking automatically opens to the proper source code
+
+
+
+
+New algebra examples (Daly, Tsikas)
+   The effort here is to create "real time" documentation that
+   gives the end user an example of how to construct the proper
+   arguments and call a function. This puts examples into the
+   system so users don't need to consult other documents.
+
+* )d op someop shows examples of function usage
+* about 100 new function examples were added 
+* new comment syntax added to allow automatic API testing
+
+
+
+
+Input Files
+   There is a new effort to automatically extract the algebra
+   examples in order to regression test the user API to the
+   algebra. In addition there is ongoing test work.
+
+* New input files (Hemmecke, Stumbo, Cyganski, Daly)
+     bini, biquat, ifthenelse, liu, overload, sqrt3, typetower
+* Changed input files (Hemmecke, Stumbo, Cyganski, Daly)
+     bern, function, linalg, regset, test, tutchap2
+
+
+
+
+Build changes
+
+* graphics does not depend on compress, done at build time
+* firefox html pages are now built before tests are run
+
+
+
+Algebra changes
+
+* FLAGG (FiniteLinearAggregate) -- removed a duplicate function
+
+
+
+Interpreter changes (Page)
+
+* add cost function to bottomUp output
+
+
+
+ 
+
+ + +

July 23, 2008 Release Notes

+patch list +
+The July 2008 release marks the second large-scale change toward a
+literate Axiom distribution. The original change was to make every
+file into a pamphlet document. This second change draws Axiom into
+tighter collections, called books (for obvious reasons). 
+
+There is a new "books" directory which contains 14 books, most of
+which are new:
+
+ bookvol0:  The reconstructed Jenks and Sutor book
+ bookvol1:  The published tutorial volume
+ bookvol2:  Users Guide
+ bookvol3:  Programmers Guide
+ bookvol4:  Developers Guide
+ bookvol5:  Interpreter
+ bookvol6:  Command
+ bookvol7:  Hyperdoc
+ bookvol7.1 Hyperdoc Pages
+ bookvol8:  Graphics
+ bookvol9:  Compiler
+ bookvol10: Algebra
+ bookvol11: Browser
+ bookvol12: Crystal
+
+All of these books now exist. Portions of the current system have
+been moved completely into book form (Graphics and Hyperdoc) so
+the old files have been removed from the src tree. Both Graphics
+and Hyperdoc are now built directly from their book. Work will 
+follow on other parts of the system.
+
+These books are online at:
+
+
+
+
+
+There is an interesting side-effect of using literate technology.
+Once you combine C and .h files into a single document so that each
+file is a separate chunk it becomes obvious that there is no need for
+local include files. The lines that read:
+   #include "foo.h"
+become
+   <>
+and get expanded inline. Once you do this it also becomes obvious
+that many include files get included multiple times (a clear waste
+of disk I/O and preparser time). Further it becomes clear that there
+is no need for creating tiny .o files since all of the source can
+be combined into one C file using chunks. 
+
+These approaches were used to reduce the compile-time overhead
+for both the graphics and the hyperdoc functions.
+
+
+
+
+In addition to consolidating the source files for Graphics into
+bookvol8 there were several other changes.
+
+  * the functions for graphics, that is, viewman, view2d, view3d
+    and viewalone are now built as single C files. 
+  * the graphics code was reorganized into chapters of related code
+  * there is the beginnings of documentation
+  * there is the beginnings of a test suite based on the 
+    CRC Handbook of Curves and Surfaces
+  * the book is hyperlinked (Bill Page, Frederic Lehobey, Anatoly Raportirenko)
+  * redundant code was eliminated
+  * the code is fully indexed
+  * the src/graph subdirectory is gone
+
+
+
+In addition to consolidating the source files for Hyperdoc into
+bookvol7 there were several other changes.
+
+  * the functions for hyperdoc, that is, spadbuf, ex2ht, htadd, hthits
+    and hypertex are now built as single C files
+  * the hyperdoc code was reorganized into chapters of related code
+  * there is the beginnngs of documentation
+  * redundant code was eliminated
+  * scroll wheel handling was added (Gregory Vanuxem)
+  * the book is hyperlinked (Bill Page, Frederic Lehobey, Anatoly Raportirenko)
+  * the code is fully indexed
+  * )hd now works to start or restart Hyperdoc (Jose Alfredo Portes)
+  * the src/hyper subdirectory is gone
+
+
+
+In addition to consolidating the hyperdoc pages into bookvol7.1
+there were several other changes:
+
+  * the pages are hyperlinked
+  * new latex macros were written to simplify page handling
+  * images of hyperdoc pages were added
+  * forward and backward hyperlinks between pages images works
+    making it possible to "browse" the hyperdoc pages using only
+    the single PDF file (Bill Page, Frederic Lehobey, Anatoly Raportirenko)
+  * the pages are fully indexed
+  * the src/hyper/pages subdirectory is gone
+
+
+
+A combined table of contents PDF is now automatically generated
+which covers all of the volumes. This makes it easier to find the
+topic of interest.
+
+
+In addition there were several other changes.
+
+  * Some fixes were made for different platforms
+       In particular, Fedora9 breaks the build and needs work
+
+  * The configure script was replaced by instructions
+       The standard (export AXIOM) works everywhere so the configure
+       script is useless for guessing.
+
+  * the FAQ was updated about git
+       Since the May 2008 release the primary Gold source code platform
+       is github. The FAQ was updated to reflect this.
+
+  * the FAQ was updated about X11
+       X11 has moved yet again so more notes are needed
+
+  * move axbook to books
+       The hyperlinked version of Jenks is now built from the books dir
+
+  * add Ralf Hemmecke's documentation to ax.boot
+  * add Monoid multiply to DirectProduct (Ralf Hemmecke)
+  * add Monoid multiply regression test
+
+
+ + +

May 27, 2008 Release Notes

+patch list +
+
+COMPUTER ALGEBRA TEST SUITE
+
+A large part of the effort for these two months has involved detailed
+test cases of Axiom's integration routines against Schaum's Handbook
+of Mathematical Formulas. Of the 619 integrals, the detailed results are:
+
+419 Schaums and Axiom agree
+137 No closed form solution
+ 60 Cannot simplify
+
+  2 Typos found in Schaums
+  1 Axiom bug
+
+The Axiom bug was in src/algebra/intef.spad.pamphlet. There was a
+fix applied to this code for a previously identified bug but the
+previous fix was incorrect.
+
+In addition, there were
+ src/input/danzwill2.input.pamphlet added for the MIT Integration tests
+ src/inputmapleok.input.pamphlet to fix typos
+ src/input/kamke1.input.pamphlet had ode97 removed due to running time
+ src/input/kamke2.input.pamphlet ode104, ode105 removed for running time
+
+DOCUMENTATION
+	Max Tegmark's 'toe' diagram, src/doc/toe.gif
+
+PORTING
+        GCLOPTS-CUSTRELOC disable-locbfd for MACOSXPPC
+
+ + +

March 25, 2008 Release Notes

+patch list +
+Summary: March 2008 release
+
+Axiom is now available at github, a git-based code repository.
+This site will have the Gold version of Axiom, that is, only
+code that changes at each two-month release. To get a clone type:
+
+   git-clone git://github.com/daly/axiom.git
+
+
+USER VISIBLE CHANGES
+
+  One primary focus of this release has been extending the Firefox
+  toward being a full Axiom user interface (as opposed to a simple
+  hyperdoc replacement). The Firefox console page has new, AJAX
+  based, dropdown menus which are planned to be dynamically updated
+  to display available functions for the last computed type. This 
+  should make it much easier to find the applicable functions by
+  category and type. They are currently static in this release.
+
+* Firefox Pages
+
+ o Dropdown menus were added to the Axiom console page
+ o More hyperdoc pages were translated to Firefox/html
+ o Bitmaps and graphics are now properly handled in pages
+ o A minor mathml fix was applied (for invisible times)
+
+* Refcard
+
+ o An Axiom reference card of Axiom commands was created (src/doc/refcard)
+
+* Examples
+
+ o It is often difficult to figure the exact arguments required to call
+   any given function in Axiom. The )display operation command used to
+   only show the available modemaps. This command has now been changed.
+    )display operation foo
+   now shows examples of function calls for foo.
+
+* Help
+
+ o The plot routines have new help files and documentation
+
+PORTING
+
+ o Axiom was ported to MAC-OSX
+
+ o The binary download page now has binaries for 
+   Ubuntu, OpenSUsE, Redhat9, Redhat72, Debian, MACOSX at 
+    
+
+ o Binaries for the this release will be available shortly.
+
+
+INTERNALS
+
+* Compiler changes
+ o  hashtables were used to speed up compiles
+
+* Algebra changes
+ o  There are new special functions, Ei,En,Ei1,Ei2,Ei3,Ei4,Ei5,Ei6
+ o  The prime and BasicSieve functions are faster
+ o  The Brent/Pollard algorithm was documented
+ o  Bad gcd reductions are checked (heugcd regression test file added)
+ o  The plot routines have new help files and documentation
+
+* Makefile changes
+ o  Bi-capital SVN copies are no longer made
+
+* Interpreter changes
+ o  Book Volume 5 has new documentation on the display function
+ o  The display function code has been translated and moved to book volume 5
+ o  PI has a higher internal precision
+ o  Mappings are now properly hashed for Aldor
+
+CATS (Computer Algebra Test Suite)
+
+ o The differential equations regression tests are being checked against
+   Mathematica, Maple, and Maxima. This has happended for the kamke2.input
+   regression test file and will happen for the other regression tests.
+
+o  Complex Gamma, logGamma, and log(Gamma) have additional tests and
+   documentation. 
+
+ + +

January 25, 2008 Release Notes

+patch list +
+Summary: January 2008 release
+
+There have been two major concentrations of effort in this release.
+
+The first concentration is on the new Firefox Hyperdoc and the
+second concentration was the verification of Axiom against
+published standards.
+
+
+
+
+Firefox Hyperdoc
+
+The Firefox Hyperdoc has been integrated with the rest of the
+interpreter. The new )browse command causes Axiom to listen and
+serve hyperdoc pages on port 8085. 
+
+The interpreter was changed to add the )browse command. As a 
+side-effect new documentation was added to the interpreter
+volume (bookvol5) to explain top-level command handling. In
+addition, lisp and boot code was rewritten as part of the
+literate change.
+
+New sections were added to cover the beginning of the Computer
+Algebra Test Suite (CATS) subsection which brings a focus on
+compliance with published standards.
+
+Arthur Ralf's mathml-enabled version of the Jenks book is fully
+integrated into the Firefox Hyperdoc. Arthur also fixed some
+rendering and ambiguity issues.
+
+  axbook.tgz              fix the user/group settings
+  axserver.spad           fix lastType output re: errors
+  bookvol5                browse and top-level command handling 
+  bookvol11               add standards compliance for gamma
+  gammacomplexinverse.png added
+  gammacomplex.png        added
+  gammareal3.png          added
+  loggamma.png            added
+  mathml.spad             fix ambiguity bug in mathml output
+  mathml.spad             fix hex(10) mathml rendering 
+  mathml.spad             fix F,3 mathml rendering 
+  mathml.spad             remove code to eat %% 
+  psi.png                 added 
+
+
+
+
+Standards Verification
+
+The Computer Algebra Test Suite (CATS) effort checks the results
+that Axiom generates against published results. Axiom has an
+extensive set of regression tests (the KAMKE suite) for ordinary
+differential equations, for integration (the SCHAUM suite), and
+for numeric special functions (the ABRAMOWITZ suite). In addition,
+results have been checked against Mathematica, Maple, and Maxima.
+
+  asinatan.input   regression for the functions asin and atan
+  asinhatanh.input regression for the functions asinh and atanh
+  besselk.input    regression for the function besselK
+  e1.input         regression for the function E1
+  en.input         regression for the function En
+  exp.input        regression for the function exp
+  gamma.input      regression for the function gamma
+  log.input        regression for the functions log
+  pfaffian.input   regression for the function pfaffian
+  seccsc.input     regression for the functions sec and csc
+  sincos.input     regression for the functions sin and cos
+  sinhcosh.input   regression for the functions sinh and cosh
+  tancot.input     regression for the functions tan and cot
+  tanhcoth.input   regression for the functions tanh and coth
+
+
+
+
+New Functions Added
+
+Axiom is missing various special functions found in other computer
+algebra systems. This release adds two new ones, the Exponential
+Integral E1 and the higher order Exponential Integral En.  These have
+been tested against the published results.
+
+  special.spad E1 added
+  special.spad En added
+
+
+
+
+Bugs Fixed
+
+There are 15 bug fixes in this release:
+
+  bug 7015: fix hex(10) mathml rendering
+  bug 7016: remove code to eat %% 
+  bug 7019: fix F,3 mathml rendering 
+  bug 7023: discardGraph free corrected 
+  bug 7042: ignore regression test gensym
+  bug 7045: wrong Makefile for Xpm fix 
+  bug 7052: spurious remake of axbook
+  bug 7054: /home/silver path in bookvol11
+  bug 7057: ambiguity in mathml
+  bug 7089/343: FreeAbelianGroup order 
+  bug 7090/355 handle besselK
+  bug 7093: Function name fix
+  bug 7100/149: numlock in hyperdoc
+  bug 7101/204: MoreSystemCommand unnecessary loading 
+  bug 7102/412: Equality testing in TableAggregate 
+
+
+
+
+Regression test fixes
+
+As changes happen in the system the regression tests are updated
+to reflect the new conditions. Changing the category of PositiveInteger
+caused (a + -bi) to print as (a - bi). New builds raised gensym faults
+which were fixed. And new builds change random numbers so the tests
+that depend on them are marked "ok" despite failures due to randomness.
+The FreeAbelianGroup bug is tested.
+
+  acplot.spad     fix output form of negative numbers
+  calculus2.input fix function names 
+  classtalk.input ignore gensyms
+  collect.input   fix function names 
+  dfloat.input    handle negative number output
+  easter.input    fix function names 
+  elemnum.input   handle negative number output
+  exlap.input     fix function names 
+  exsum.input     fix function names 
+  free.input      added to test bug
+  grpthry.input   mark random generation failures ok
+  grpthry.input   fix function names 
+  ico.input       mark random generation failures ok
+  intg0.input     ignore gensyms
+  is.input        type declare function
+  kamke3.input    mark random generation failures ok
+  knot2.input     fix function names 
+  lodo.spad       ignore regression test gensym 
+  mapleok.input   ignore gensyms
+  mathml.input    handle new mathml sub/sup change
+  ndftip.input    fix missing blank lines
+  pmint.input     rewritten
+  repa6.input     fix function names 
+  r20bugs.input   change spacing
+  sf.spad         fix output form of negative numbers
+  tbagg.input     regression for equality testing in TableAggregate
+
+
+
+
+Algebra file changes
+
+The fundamental change was a supposedly transparent move of
+the category for PositiveInteger. This had the effect of changing
+the output form and broke several regression tests. Some mathml
+issues were fixed. The new functions E1 and En were added. The
+FreeAbelianGroup bug was fixed.
+
+  axserver.spad  fix lastType output re: errors
+  acplot.spad    fix output form of negative numbers
+  combfunc.spad  fix bold font handling
+  integer.spad   category change for PositiveInteger
+  sf.spad fix    output form of negative numbers
+  sf.spad        handle besselK
+  op.spad        handle besselK 
+  combfunc.spad  handle besselK 
+  free.spad      fix FreeAbelianGroup bug 
+  special.spad   add E1
+  special.spad   add En
+  mathml.spad    fix ambiguity bug in mathml output
+  mathml.spad    fix hex(10) mathml rendering 
+  mathml.spad    fix F,3 mathml rendering 
+  mathml.spad    remove code to eat %% 
+
+
+
+Interpreter changes
+
+The primary changes are the addition of bookvol11 for the Firefox
+Hyperdoc and the literate documentation of the top level command
+handling in bookvol5 (the interpreter) along with rewrites of the
+lisp/boot code.
+
+  bootfuns.lisp move $systemCommands to bookvol5
+  bookvol5      browse and top-level command handling
+  bookvol11     added
+  http.lisp     mathObject2String for hex(10) 
+  incl.boot     move incBiteOff to bookvol5
+  intint.lisp   move setCurrentLine to bookvol5
+  int-top.boot  move ncloopCommand, etc. to bookvol5
+  i-syscmd.boot move $SYSCOMMANDS to bookvol5
+  Makefile      wrong Makefile for Xpm fix 
+  makegraph.c   discardGraph free corrected 
+  nci.lisp      move ncloopInclude to bookvol5
+  setq.lisp     move command initialization to bookvol5
+
+
+
+Documentation changes
+
+  bookvol5      explain top level input handling (lisp/boot rewrite)
+  combfunc.spad fix bold font handling
+  axiom.sty     add binom 
+
+
+
+Patches released
+
+  20071129.01.tpd.patch
+  20071129.02.tpd.patch
+  20071205.01.tpd.patch
+  20071206.01.tpd.patch
+  20071208.01.tpd.patch
+  20071215.01.tpd.patch
+  20071215.02.tpd.patch
+  20071215.03.gxv.patch
+  20071216.01.tpd.patch
+  20071216.02.tpd.patch
+  20071216.03.acr.patch
+  20071217.01.acr.patch
+  20071217.02.tpd.patch
+  20071218.01.acr.patch
+  20071225.01.sxw.patch
+  20071228.01.tpd.patch
+  20071229.01.jap.patch
+  20071230.01.acr.patch
+  20071230.02.tpd.patch
+  20071230.03.tpd.patch
+  20080102.01.tpd.patch
+  20080103.01.tpd.patch
+  20080104.01.tpd.patch
+  20080104.02.tpd.patch
+  20080106.01.tpd.patch
+  20080107.01.tpd.patch
+  20080107.02.tpd.patch
+  20080107.03.tpd.patch
+  20080116.01.tpd.patch
+  20080119.01.tpd.patch
+  20080119.02.tpd.patch
+  20080119.03.tpd.patch
+  20080120.01.gxv.patch
+  20080120.02.tpd.patch
+  20080120.03.tpd.patch
+
+
+
+ + +

November 23, 2007 Release Notes

+patch list +
+Summary: November 2007 release
+
+All of the golden sources are up to date.
+  savannah.nongnu.org/projects/axiom CVS
+  sourceforge.net/projects/axiom CVS
+  arch@axiom-developer.org ARCH (axiom--main--1--patch-54)
+  git@axiom-developer.org GIT
+  
+
+ADD NEW CREDITS
+  New patches were posted by Arthur and Alfredo so their tlas were
+  added to the changelog
+
+20071001 acr Arthur C. Ralfs 
+20070914 jap Jose Alfredo Portes 
+
+
+PORT TO DIFFERENT SYSTEMS
+  As part of the new axiom website there is a binary release page.
+  The stanzas for these supported systems were added.
+
+20071119 tpd Makefile.pamphlet add fedora6,7,8 stanzas
+
+
+NEW Axiom WEBSITE: http://axiom-developer.org STARTED.
+  The new Axiom website (currently at axiom.axiom-developer.org)
+  has been started. It will include the binary release page.
+
+
+REMOVE OLD REGRESSION SYSTEM
+  The previous regression test system was removed. A new combined
+  regression test and help documentation system was built to replace
+  this mechanism.
+
+20070901 tpd src/input/Makefile remove ALGEBRA variable
+20070901 tpd src/algebra/perm.spad remove TEST mechanism
+20070901 tpd src/algebra/view2d.spad remove TEST mechanism
+20070901 tpd src/algebra/fr.spad remove TEST mechanism
+
+
+FIX BOOK DOCUMENTATION
+  Minor typos have been discovered in the book during documentation.
+
+20070905 tpd src/doc/book remove duplicate upperCase, lowerCase typo
+20070903 tpd src/doc/bookvol4 fix typos
+20070902 tpd src/doc/book MultiSet -> Multiset
+20080829 tpd src/doc/book.pamphlet correct typo
+
+
+FIX BUGS
+  Various bugs have been found and fixed.
+
+20071101 tpd src/interp/i-output.boot fix bugs 7010 (209), 7011
+20070920 tpd src/input/Makefile add bug101.input regression test
+20070920 tpd src/input/bug101.input test laplace(log(z),z,w) bug 101
+20070920 wxh src/algebra/laplace.spad fix laplace(log(z),z,w) bug 101
+20070916 tpd src/input/Makefile add bug103.input regression test
+20070916 tpd src/input/bug103.input test solve(z=z,z) bug fix
+20070916 tpd src/algebra/polycat.spad solve(z=z,z) bug fix
+20070916 tpd src/algebra/catdef.spad add zero? to exquo
+20070915 tpd merge bug100 branch
+20070915 tpd src/input/Makefile add bug100.input regression test
+20070915 tpd src/input/bug100.input test integrate((z^a+1)^b,z) infinite loop
+20070915 wxh src/algebra/intef.spad fix integrate((z^a+1)^b,z) infinite loop
+20070915 tpd src/algebra/carten minor edit for regression cleanup
+20070914 wxh src/hyper/hyper fix bad bracing of )hd change
+20070914 tpd src/algebra/fraction.spad remove double )spool command
+20070914 tpd src/algebra/kl.spad remove double )spool command
+20070914 tpd src/algebra/lindep.spad remove double )spool command
+20070914 tpd src/algebra/radix.spad remove double )spool command
+
+
+ENABLE DYNAMIC RESTART OF HYPERDOC
+  Hyperdoc can now be started dynamically or restarted if killed.
+
+20070914 jap adapt changes for )hd restart to Axiom sources
+20070914 wxh src/sman/bookvol6 enable restart of hyperdoc with )hd
+20070914 wxh src/include/sman.h1 enable restart of hyperdoc with )hd
+20070914 wxh src/hyper/hyper enable restart of hyperdoc with )hd
+
+
+SET UP THE NEW FIREFOX BASED HYPERDOC
+  Hyperdoc is going away. A new version of hyperdoc is being built
+  which uses html/javascript/mathml. These files change the interpreter
+  and algebra to support the new hyperdoc machinery.
+
+20071019 acr src/interp/http.lisp use new return values
+20071019 acr src/algebra/axserver.spad use new return values
+20071014 acr src/algebra/axserver.spad use getContentType(pathvar)
+20071013 acr license/license.ralfs license rewrite
+20071013 acr src/interp/http.lisp faster page service
+20071013 acr src/algebra/axserver.spad faster page service
+20071001 tpd src/algebra/exposed.lisp add (|AxiomServer| . AXSERV) to basic
+20071001 tpd src/algebra/Makefile add axserver.spad
+20071001 acr src/algebra/axserver.spad axserver socket connection code
+20071001 tpd src/interp/Makefile add http.lisp
+20071001 acr src/interp/http.lisp axserver socket connection code
+20071001 acr license/license.ralfs added
+
+
+REGRESSION TEST CALCULUS
+  A new regression test suite for calculus is being built. The first
+  of these files has been added to the system.
+
+20070913 tpd src/input/Makefile schaum1.input added
+20070913 tpd src/input/schaum1.input added
+
+
+REGRESSION TEST ORDINARY DIFFERENTIAL EQUATIONS
+  A regression test suite for ordinary differential equations was built. 
+
+20071005 tpd src/input/Makefile kamke7.input regression test added
+20071005 tpd src/input/kamke7.input ODE regression test added
+20071005 tpd src/input/Makefile kamke6.input regression test added
+20071005 tpd src/input/kamke6.input ODE regression test added
+20071005 tpd src/input/Makefile kamke5.input regression test added
+20071005 tpd src/input/kamke5.input ODE regression test added
+20071005 tpd src/input/Makefile kamke4.input regression test added
+20071005 tpd src/input/kamke4.input ODE regression test added
+20071005 tpd src/input/Makefile kamke3.input regression test added
+20071005 tpd src/input/kamke3.input ODE regression test added
+20071004 tpd src/input/Makefile kamke2.input regression test added
+20071004 tpd src/input/kamke2.input ODE regression test added
+20071004 tpd src/input/Makefile kamke1.input regression test added
+20071004 tpd src/input/kamke1.input ODE regression test added
+20071004 tpd src/input/Makefile kamke0.input regression test added
+20071004 tpd src/input/kamke0.input ODE regression test added
+
+
+REGRESSION TEST PFAFFIAN
+  Martin added the pfaffian regression test. It was added and removed
+  due to documentation license issues. New documentation is being written.
+
+20070929 tpd src/input/Makefile pfaffian regression test removed
+20070929 tpd src/input/pfaffian.input regression test removed
+20070927 tpd src/input/Makefile pfaffian regression test added 
+20070927 mxr src/input/pfaffian.input regression test added 
+
+
+ADD PORTIONS OF THE GUESS PACKAGE
+  The newton.spad file is actually part of the fffg.spad file so it
+  was removed. The very top level spad functions in GUESS still do
+  not work properly.
+
+20070909 tpd src/algebra/newton.spad included in fffg.spad
+20070909 tpd src/algebra/Makefile remove newton.spad (duplicate)
+
+
+FIX BUILD PROCESS
+  The build process was not properly suppressing output by default.
+
+20070907 tpd src/algebra/acplot.spad fix PlaneAlgebraicCurvePlot.help NOISE
+20070907 tpd src/algebra/Makefile make regression respect NOISE variable
+20070907 tpd src/input/Makefile make regression respect NOISE variable
+
+
+FIX INSTALL PROCESS
+  The install process had a bug.
+
+20070906 tpd Makefile int/sman/axiom command to target command for install
+20070906 tpd src/sman/Makefile copy axiom command to int
+
+
+ADD ALDOR RELEASE
+  The aldor release has been added to zips. It will soon be part of
+  the build mechanism but separately maintained like GCL.
+
+20070901 tpd zips/aldor.20070901.tgz add pdf documentation
+20070901 tpd zips/aldor.20070901.tgz added
+
+
+ADD )HELP FACILITY
+  The )help facility was recovered. The documentation is now integrated
+  into the spad files and used both for help documentation and algebra
+  regression testing. 
+
+20070902 tpd src/doc/Makefile document how to add help files
+20070902 tpd src/algebra/Makefile document how to add help files
+20070906 tpd merge help files branch
+20070906 tpd src/doc/spadhelp add ZeroDimensionalSolvePackage
+20070906 tpd src/algebra/Makefile add ZeroDimensionalSolvePackage.help 
+20070906 tpd src/algebra/zerodim.spad add ZeroDimensionalSolvePackage.help
+20070906 tpd src/algebra/zerodim.spad ZeroDimensionalSolvePackage.input
+20070906 tpd src/doc/spadhelp add XPolynomialRing
+20070906 tpd src/algebra/Makefile add XPolynomialRing.help 
+20070906 tpd src/algebra/xpoly.spad add XPolynomialRing.help (XPR)
+20070906 tpd src/algebra/xpoly.spad XPolynomialRing.input
+20070906 tpd src/doc/spadhelp add XPolynomial
+20070906 tpd src/algebra/Makefile add XPolynomial.help 
+20070906 tpd src/algebra/xpoly.spad add XPolynomial.help (XPOLY)
+20070906 tpd src/algebra/xpoly.spad XPolynomial.input
+20070906 tpd src/doc/spadhelp add XPBWPolynomial
+20070906 tpd src/algebra/Makefile add XPBWPolynomial.help 
+20070906 tpd src/algebra/xlpoly.spad add XPBWPolynomial.help (XPBWPOLY)
+20070906 tpd src/algebra/xlpoly.spad XPBWPolynomial.input
+20070905 tpd src/doc/spadhelp add WuWenTsunTriangularSet
+20070905 tpd src/algebra/Makefile add WuWenTsunTriangularSet.help 
+20070905 tpd src/algebra/triset.spad add WuWenTsunTriangularSet.help (WUTSET)
+20070905 tpd src/algebra/triset.spad WuWenTsunTriangularSet.input
+20070905 tpd src/doc/spadhelp add Void
+20070905 tpd src/algebra/Makefile add Void.help 
+20070905 tpd src/algebra/void.spad add Void.help (VOID)
+20070905 tpd src/algebra/void.spad Void.input
+20070905 tpd src/doc/spadhelp add Vector
+20070905 tpd src/algebra/Makefile add Vector.help 
+20070905 tpd src/algebra/vector.spad add Vector.help (Vector)
+20070905 tpd src/algebra/vector.spad Vector.input
+20070905 tpd src/doc/spadhelp add UniversalSegment
+20070905 tpd src/algebra/Makefile add UniversalSegment.help 
+20070905 tpd src/algebra/seg.spad add UniversalSegment.help (UNISEG)
+20070905 tpd src/algebra/seg.spad UniversalSegment.input
+20070905 tpd src/doc/spadhelp add UnivariatePolynomial
+20070905 tpd src/algebra/Makefile add UnivariatePolynomial.help 
+20070905 tpd src/algebra/poly.spad add UnivariatePolynomial.help (UP)
+20070905 tpd src/algebra/poly.spad UnivariatePolynomial.input
+20070905 tpd src/doc/spadhelp add TwoDimensionalArray
+20070905 tpd src/algebra/Makefile add TwoDimensionalArray.help 
+20070905 tpd src/algebra/array2.spad add TwoDimensionalArray.help (ARRAY2)
+20070905 tpd src/algebra/array2.spad TwoDimensionalArray.input
+20070905 tpd src/doc/spadhelp add TextFile
+20070905 tpd src/algebra/Makefile add TextFile.help 
+20070905 tpd src/algebra/files.spad add TextFile.help (TEXTFILE)
+20070905 tpd src/algebra/files.spad TextFile.input
+20070905 tpd src/doc/spadhelp add Table
+20070905 tpd src/algebra/Makefile add Table.help 
+20070905 tpd src/algebra/table.spad add Table.help (TABLE)
+20070905 tpd src/algebra/table.spad Table.input
+20070905 tpd src/doc/spadhelp add StringTable
+20070905 tpd src/algebra/Makefile add StringTable.help 
+20070905 tpd src/algebra/table.spad add StringTable.help (STRTBL)
+20070905 tpd src/algebra/table.spad StringTable.input
+20070905 tpd src/doc/spadhelp add String
+20070905 tpd src/algebra/Makefile add String.help 
+20070905 tpd src/algebra/stream.spad add String.help (STRING)
+20070905 tpd src/algebra/stream.spad String.input
+20070905 tpd src/doc/spadhelp add Stream
+20070905 tpd src/algebra/Makefile add Stream.help 
+20070905 tpd src/algebra/stream.spad add Stream.help (STREAM)
+20070905 tpd src/algebra/stream.spad Stream.input
+20070905 tpd src/doc/spadhelp add SquareFreeRegularTriangularSet
+20070905 tpd src/algebra/Makefile add SquareFreeRegularTriangularSet.help 
+20070905 tpd src/algebra/sregset.spad add SquareFreeRegularTriangularSet.help
+20070905 tpd src/algebra/sregset.spad SquareFreeRegularTriangularSet.input
+20070905 tpd src/doc/spadhelp add SquareMatrix
+20070905 tpd src/algebra/Makefile add SquareMatrix.help 
+20070905 tpd src/algebra/matrix.spad add SquareMatrix.help (SQMATRIX)
+20070905 tpd src/algebra/matrix.spad SquareMatrix.input
+20070905 tpd src/doc/spadhelp add SparseTable
+20070905 tpd src/algebra/Makefile add SparseTable.help 
+20070905 tpd src/algebra/table.spad add SparseTable.help (STBL)
+20070905 tpd src/algebra/table.spad SparseTable.input
+20070905 tpd src/doc/spadhelp add SingleInteger
+20070905 tpd src/algebra/Makefile add SingleInteger.help 
+20070905 tpd src/algebra/si.spad add SingleInteger.help (SINT)
+20070905 tpd src/algebra/si.spad SingleInteger.input
+20070905 tpd src/doc/spadhelp add Set
+20070905 tpd src/algebra/Makefile add Set.help 
+20070905 tpd src/algebra/sets.spad add Set.help (SET)
+20070905 tpd src/algebra/sets.spad Set.input
+20070905 tpd src/doc/spadhelp add SegmentBinding
+20070905 tpd src/algebra/Makefile add SegmentBinding.help 
+20070905 tpd src/algebra/seg.spad add SegmentBinding.help (SEGBIND)
+20070905 tpd src/algebra/seg.spad SegmentBinding.input
+20070905 tpd src/doc/spadhelp add Segment
+20070905 tpd src/algebra/Makefile add Segment.help 
+20070905 tpd src/algebra/seg.spad add Segment.help (SEG)
+20070905 tpd src/algebra/seg.spad Segment.input
+20070905 tpd src/algebra/integer.spad update RomanNumeral.help (ROMAN)
+20070905 tpd src/algebra/integer.spad update RomanNumeral.input
+20070904 tpd src/doc/spadhelp add RegularTriangularSet
+20070904 tpd src/algebra/Makefile add RegularTriangularSet.help 
+20070904 tpd src/algebra/regset.spad add RegularTriangularSet.help (REGSET)
+20070904 tpd src/algebra/regset.spad RegularTriangularSet.input
+20070904 tpd src/doc/spadhelp add RealClosure
+20070904 tpd src/algebra/Makefile add RealClosure.help 
+20070904 tpd src/algebra/reclos.spad add RealClosure.help (RECLOS)
+20070904 tpd src/algebra/reclos.spad RealClosure.input
+20070904 tpd src/doc/spadhelp add RadixExpansion
+20070904 tpd src/algebra/Makefile add RadixExpansion.help 
+20070904 tpd src/algebra/radix.spad add RadixExpansion.help (RADIX)
+20070904 tpd src/algebra/radix.spad RadixExpansion.input
+20070903 tpd src/doc/spadhelp add Polynomial
+20070903 tpd src/algebra/Makefile add Polynomial.help 
+20070903 tpd src/algebra/multpoly.spad add Polynomial.help (POLY)
+20070903 tpd src/algebra/multpoly.spad Polynomial.input
+20070903 tpd src/doc/spadhelp add Permanent
+20070903 tpd src/algebra/Makefile add Permanent.help 
+20070903 tpd src/algebra/perman.spad add Permanent.help (PERMAN)
+20070903 tpd src/algebra/perman.spad Permanent.input
+20070903 tpd src/doc/spadhelp add PartialFraction
+20070903 tpd src/algebra/Makefile add PartialFraction.help 
+20070903 tpd src/algebra/pfr.spad add PartialFraction.help (PFR)
+20070903 tpd src/algebra/pfr.spad PartialFraction.input
+20070903 tpd src/doc/spadhelp add OrderlyDifferentialPolynomial
+20070903 tpd src/algebra/Makefile add OrderlyDifferentialPolynomial.help 
+20070903 tpd src/algebra/dpolcat.spad add OrderlyDifferentialPolynomial (ODPOL)
+20070903 tpd src/algebra/dpolcat.spad OrderlyDifferentialPolynomial.input
+20070903 tpd src/doc/spadhelp add OrderedVariableList
+20070903 tpd src/algebra/Makefile add OrderedVariableList.help 
+20070903 tpd src/algebra/variable.spad add OrderedVariableList.help (OVAR)
+20070903 tpd src/algebra/variable.spad OrderedVariableList.input
+20070903 tpd src/doc/spadhelp add Operator
+20070903 tpd src/algebra/Makefile add Operator.help 
+20070903 tpd src/algebra/opalg.spad add Operator.help (OP)
+20070903 tpd src/algebra/opalg.spad Operator.input
+20070903 tpd src/algebra/radix.spad fix typos in help file
+20070903 tpd src/algebra/integer.spad fix typos in help file
+20070902 tpd src/doc/spadhelp add OneDimensionalArray
+20070902 tpd src/algebra/Makefile add OneDimensionalArray.help 
+20070902 tpd src/algebra/array1.spad add OneDimensionalArray.help (ARRAY1)
+20070902 tpd src/algebra/array1.spad OneDimensionalArray.input
+20070902 tpd src/doc/spadhelp add Octonion
+20070902 tpd src/algebra/Makefile add Octonion.help 
+20070902 tpd src/algebra/oct.spad add Octonion.help (OCT)
+20070902 tpd src/algebra/oct.spad Octonion.input
+20070902 tpd src/doc/spadhelp add None
+20070902 tpd src/algebra/Makefile add None.help 
+20070902 tpd src/algebra/any.spad add None.help (NONE)
+20070902 tpd src/algebra/any.spad None.input
+20070902 tpd src/doc/spadhelp add MultivariatePolynomial
+20070902 tpd src/algebra/Makefile add MultivariatePolynomial.help 
+20070902 tpd src/algebra/multpoly.spad add MultivariatePolynomial.help (MPOLY)
+20070902 tpd src/algebra/multpoly.spad MultivariatePolynomial.input
+20070902 tpd src/doc/spadhelp add Multiset
+20070902 tpd src/algebra/Makefile add Multiset.help 
+20070902 tpd src/algebra/mset.spad add Multiset.help (MSET)
+20070902 tpd src/algebra/mset.spad Multiset.input
+20070902 tpd src/doc/spadhelp add Matrix
+20070902 tpd src/algebra/Makefile add Matrix.help 
+20070902 tpd src/algebra/matrix.spad add Matrix.help (MATRIX)
+20070902 tpd src/algebra/matrix.spad Matrix.input
+20070902 tpd src/doc/spadhelp add MappingPackage3
+20070902 tpd src/algebra/Makefile add MappingPackage3.help 
+20070902 tpd src/algebra/mappkg.spad add MappingPackage3.help (MAPPKG3)
+20070902 tpd src/algebra/mappkg.spad MappingPackage3.input
+20070902 tpd src/doc/spadhelp add MappingPackage2
+20070902 tpd src/algebra/Makefile add MappingPackage2.help 
+20070902 tpd src/algebra/mappkg.spad add MappingPackage2.help (MAPPKG2)
+20070902 tpd src/algebra/mappkg.spad MappingPackage2.input
+20070902 tpd src/doc/spadhelp add MappingPackage1
+20070902 tpd src/algebra/Makefile add MappingPackage1.help 
+20070902 tpd src/algebra/mappkg.spad add MappingPackage1.help (MAPPKG1)
+20070902 tpd src/algebra/mappkg.spad MappingPackage1.input
+20070902 tpd src/doc/spadhelp add MakeFunction
+20070902 tpd src/algebra/Makefile add MakeFunction.help 
+20070902 tpd src/algebra/mkfunc.spad add MakeFunction.help (MKFUNC)
+20070902 tpd src/algebra/mkfunc.spad MakeFunction.input
+20070902 tpd src/algebra/view2d.spad fix typos in help file
+20070902 tpd src/algebra/files.spad fix typos in help file
+20070902 tpd src/algebra/acplot.spad fix typos in help file
+20070902 tpd src/doc/spadhelp add Magma
+20070902 tpd src/algebra/Makefile add Magma.help 
+20070902 tpd src/algebra/xlpoly.spad add Magma.help (MAGMA)
+20070902 tpd src/algebra/xlpoly.spad Magma.input
+20070902 tpd src/doc/spadhelp add LyndonWord
+20070902 tpd src/algebra/Makefile add LyndonWord.help 
+20070902 tpd src/algebra/xlpoly.spad add LyndonWord.help (LWORD)
+20070902 tpd src/algebra/xlpoly.spad LyndonWord.input
+20070902 tpd src/doc/spadhelp add List
+20070902 tpd src/algebra/Makefile add List.help 
+20070902 tpd src/algebra/list.spad add List.help (LIST)
+20070902 tpd src/algebra/list.spad List.input
+20070901 tpd src/doc/spadhelp add Permutation
+20070901 tpd src/algebra/Makefile add Permutation.help 
+20070901 tpd src/algebra/acplot.spad add Permutation.help (PERM)
+20070901 tpd src/algebra/acplot.spad Permutation.input
+20070901 tpd src/doc/spadhelp add TwoDimensionalViewport
+20070901 tpd src/algebra/Makefile add TwoDimensionalViewport.help
+20070901 tpd src/algebra/view2d.spad add TwoDimensionalViewport.help (VIEW2D)
+20070901 tpd src/doc/spadhelp add PlaneAlgebraicCurvePlot
+20070901 tpd src/algebra/Makefile add PlaneAlgebraicCurvePlot.help 
+20070901 tpd src/algebra/acplot.spad add PlaneAlgebraicCurvePlot.help (ACPLOT)
+20070901 tpd src/algebra/acplot.spad PlaneAlgebraicCurvePlot.input
+20070901 tpd src/doc/spadhelp add RealSolvePackage
+20070901 tpd src/algebra/Makefile add RealSolvePackage.help 
+20070901 tpd src/algebra/acplot.spad add RealSolvePackage.help (REALSOLV)
+20070901 tpd src/algebra/acplot.spad RealSolvePackage.input
+20070830 tpd src/doc/spadhelp add LinearOrdinaryDifferentialOperator2
+20070830 tpd src/algebra/Makefile add LinearOrdinaryDifferentialOperator2.help 
+20070830 tpd src/algebra/lodo.spad add LinearOrdinaryDifferentialOperator2.help
+20070830 tpd src/algebra/lodo.spad LinearOrdinaryDifferentialOperator2.input
+20070830 tpd src/doc/spadhelp add LinearOrdinaryDifferentialOperator1
+20070830 tpd src/algebra/Makefile add LinearOrdinaryDifferentialOperator1.help 
+20070830 tpd src/algebra/lodo.spad add LinearOrdinaryDifferentialOperator1.help
+20070830 tpd src/algebra/lodo.spad LinearOrdinaryDifferentialOperator1.input
+20070830 tpd src/doc/spadhelp add LinearOrdinaryDifferentialOperator
+20070830 tpd src/algebra/Makefile add LinearOrdinaryDifferentialOperator.help 
+20070830 tpd src/algebra/lodo.spad add LinearOrdinaryDifferentialOperator.help
+20070830 tpd src/algebra/lodo.spad add LinearOrdinaryDifferentialOperator.input
+20070830 tpd src/doc/spadhelp add LiePolynomial
+20070830 tpd src/algebra/Makefile add LiePolynomial.help 
+20070830 tpd src/algebra/xlpoly.spad add LiePolynomial.help (LPOLY)
+20070830 tpd src/algebra/xlpoly.spad add LiePolynomial.input
+20070830 tpd src/doc/spadhelp add LieExponentials
+20070830 tpd src/algebra/Makefile add LieExponentials.help 
+20070830 tpd src/algebra/xlpoly.spad add LieExponentials.help (LEXP)
+20070830 tpd src/algebra/xlpoly.spad add LieExponentials.input
+20070830 tpd src/doc/spadhelp add Library
+20070830 tpd src/algebra/Makefile add Library.help 
+20070830 tpd src/algebra/files.spad add Library.help (LIB)
+20070830 tpd src/algebra/files.spad add Library.input
+20070830 tpd src/doc/spadhelp add LexTriangularPackage
+20070830 tpd src/algebra/Makefile add LexTriangularPackage.help 
+20070830 tpd src/algebra/zerodim.spad add LexTriangularPackage.help (LEXTRIPK)
+20070830 tpd src/algebra/zerodim.spad add LexTriangularPackage.input
+20070830 tpd src/doc/spadhelp add KeyedAccessFile
+20070830 tpd src/algebra/Makefile add KeyedAccessFile.help 
+20070830 tpd src/algebra/numtheor.spad add KeyedAccessFile.help (KAFILE)
+20070830 tpd src/algebra/numtheor.spad add KeyedAccessFile.input
+20070830 tpd src/doc/spadhelp add Kernel
+20070830 tpd src/algebra/Makefile add Kernel.help 
+20070830 tpd src/algebra/numtheor.spad add Kernel.help (KERNEL)
+20070830 tpd src/algebra/numtheor.spad add Kernel.input
+20070830 tpd src/doc/spadhelp add IntegerNumberTheoryFunctions
+20070830 tpd src/algebra/Makefile add IntegerNumberTheoryFunctions.help 
+20070830 tpd src/algebra/numtheor.spad add IntegerNumberTheoryFunctions.help
+20070830 tpd src/algebra/numtheor.spad add IntegerNumberTheoryFunctions.input
+20070830 tpd src/doc/spadhelp add IntegerLinearDependence
+20070830 tpd src/algebra/Makefile add IntegerLinearDependence.help 
+20070830 tpd src/algebra/lindep.spad add IntegerLinearDependence.help (ZLINDEP)
+20070830 tpd src/algebra/lindep.spad add IntegerLinearDependence.input
+20070830 tpd src/doc/spadhelp add RomanNumeral
+20070830 tpd src/algebra/Makefile add RomanNumeral.help 
+20070830 tpd src/algebra/integer.spad add RomanNumeral.help (ROMAN)
+20070830 tpd src/algebra/integer.spad add RomanNumeral.input
+20070830 tpd src/doc/spadhelp add Integer
+20070830 tpd src/algebra/Makefile add Integer.help 
+20070830 tpd src/algebra/integer.spad add Integer.help (INT)
+20070830 tpd src/algebra/integer.spad add Integer.input
+20070830 tpd src/doc/spadhelp add HexadecimalExpansion
+20070830 tpd src/algebra/Makefile add HexadecimalExpansion.help 
+20070830 tpd src/algebra/radix.spad add HexadecimalExpansion.help (HEXADEC)
+20070830 tpd src/algebra/radix.spad add HexadecimalExpansion.input
+20070829 tpd src/doc/spadhelp add Heap
+20070829 tpd src/algebra/Makefile add Heap.help 
+20070829 tpd src/algebra/bags.spad add Heap.help (FR)
+20070829 tpd src/algebra/bags.spad add Heap.input
+20070829 tpd src/doc/spadhelp add GroebnerFactorizationPackage
+20070829 tpd src/algebra/Makefile add GroebnerFactorizationPackage.help 
+20070829 tpd src/algebra/fparfrac.spad add GroebnerFactorizationPackage.help
+20070829 tpd src/algebra/fparfrac.spad add GroebnerFactorizationPackage.input
+20070829 tpd src/doc/spadhelp add GeneralSparseTable
+20070829 tpd src/algebra/Makefile add GeneralSparseTable.help 
+20070829 tpd src/algebra/table.spad add GeneralSparseTable.help (GSTBL)
+20070829 tpd src/algebra/table.spad add GeneralSparseTable.input 
+20070829 tpd src/doc/spadhelp add FullPartialFractionExpansion
+20070829 tpd src/algebra/Makefile add FullPartialFractionExpansion.help 
+20070829 tpd src/algebra/fparfrac.spad add FullPartialFractionExpansion.help
+20070829 tpd src/algebra/fparfrac.spad add FullPartialFractionExpansion.input
+20070829 tpd src/doc/spadhelp add Fraction
+20070829 tpd src/algebra/Makefile add Fraction.help 
+20070829 tpd src/algebra/fraction.spad add Fraction.help (FR)
+20070829 tpd src/algebra/fraction.spad add Fraction.input
+20070829 tpd src/doc/spadhelp add Float
+20070829 tpd src/algebra/Makefile add Float.help 
+20070829 tpd src/algebra/float.spad add Float.help (FLOAT)
+20070829 tpd src/algebra/float.spad add Float.input
+20070828 tpd src/doc/spadhelp add FlexibleArray
+20070828 tpd src/algebra/Makefile add FlexibleArray.help 
+20070828 tpd src/algebra/array1.spad add FlexibleArray.help (FARRAY)
+20070828 tpd src/algebra/array1.spad add FlexibleArray.input
+20070827 tpd src/doc/spadhelp add FileName
+20070827 tpd src/algebra/Makefile add FileName.help 
+20070827 tpd src/algebra/fname.spad add FileName.help (FNAME)
+20070827 tpd src/algebra/fname.spad add FileName.input
+20070827 tpd src/doc/spadhelp add File
+20070827 tpd src/algebra/Makefile add File.help 
+20070827 tpd src/algebra/files.spad add File.help (FILE)
+20070827 tpd src/algebra/files.spad add File.input
+20070827 tpd src/doc/spadhelp add FactoredFunctions2
+20070827 tpd src/algebra/Makefile add FactoredFunctions2.help 
+20070827 tpd src/algebra/fr.spad add FactoredFunctions2.help (FR2)
+20070827 tpd src/algebra/fr.spad add FactoredFunctions2.input
+20070827 tpd src/doc/spadhelp add Factored
+20070827 tpd src/algebra/Makefile add Factored.help 
+20070827 tpd src/algebra/fr.spad add Factored.help (FR)
+20070827 tpd src/algebra/fr.spad add Factored.input
+
+
+ + diff --git a/src/axiom-website/rosetta.html b/src/axiom-website/rosetta.html new file mode 100644 index 0000000..0e0bd55 --- /dev/null +++ b/src/axiom-website/rosetta.html @@ -0,0 +1,5781 @@ + + + + Axiom Computer Algebra System + + + + +
+ [ + + Home + + ] +   + [ + + Screenshots + + ] +   + [ + + FAQ + + ] +   + [ + + Download + + ] +   + [ + + Documentation + + ] +   + [ + + Current State + + ] +   + [ + + Community + + ] +   + [ + + Developers + + ] +   + [ + + Patches + + ] +
+ +
+
+
+

+ Rosetta Stone +

+

+ +The following is a collection of synonyms for various operations in +the computer algebra systems Axiom, Derive, GAP, DoCon, +Macsyma, Magnus, Maxima, Maple, Mathematica, MuPAD, Octave, +Pari, Reduce, Scilab, Sumit and Yacas. This collection does not +attempt to be comprehensive, but hopefully it will be useful in giving +an indication of how to translate between the syntaxes used by the +different systems in many common situations. Note that a blank entry +means either (a) that there may be an exact translation of a +particular operation for the indicated system, but we don't know what +it is or (b) there is no exact translation but it may still be +possible to work around this lack with a related functionality. + +While commercial systems are not provided on this CD the intent of the +Rosetta effort is to make it possible for experienced Computer Algebra +users to experiment with other systems. Thus the commands for +commercial systems are included to allow users of those systems to +translate. + +Some of these systems are special purpose and do not support a lot of +the functionality of the more general purpose systems. Where they do +support an interpreter the commands are provided. + +Originally written by Michael Wester. +Modified for Rosetta by Timothy Daly, Alexander Hulpke (GAP). + +

System availability

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
SystemLicenseStatus (2002)URL
AldorNon-freeavailablehttp://www.aldor.org
AxiomModifiedBSDavailablehttp://axiom.axiom-developer.org
Derivecommercialavailablehttp://www.mathware.com
DoConopen sourceavailablehttp://www.haskell.org/docon
GAPGPLRosettahttp://www.gap-system.org/~gap
Macsymacommercialdeadunavailable
MagnusGPLRosettahttp://sourceforge.net/projects/magnus
MaximaGPLRosettahttp://www.ma.utexas.edu/maxima.html
Maplecommercialavailablehttp://www.maplesoft.com
Mathematicacommercialavailablehttp://www.wolfram.com
MuPADcommercialavailablehttp://www.mupad.de
OctaveGPLRosettahttp://www.octave.org
PariGPLRosettahttp://www.parigp-home.de
Reducecommercialavailablehttp://www.zib.de/Symbolik/reduce
ScilabScilabavailablehttp://www-rocq.inria.fr/scilab
Sumitavailablehttp://www-sop.inria.fr/cafe/soft-f.html
YacasGPLavailablehttp://yacas.sourceforge.net
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
SystemTypeInterpreted or Compiled +
AxiomGeneral Purposeboth
DeriveGeneral Purpose
DoConGeneral PurposeInterpreted in Haskell
GAPGroup Theory
MacsymaGeneral Purpose
MagnusInfinite Group Theory
MaximaGeneral Purpose
MapleGeneral Purpose
MathematicaGeneral Purpose
MuPADGeneral Purpose
OctaveNumerical Computing
PariNumber Theory
ReduceGeneral Purpose
ScilabGeneral Purpose
SumitFunctional Equations
YacasGeneral Purpose
+
+ +

Programming and Miscellaneous

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
SystemUnix/Microsoft user initialization file
Axiom.axiom.input
GAP.gaprcGAP.RC
DoCon
Derivederive.ini
Macsymamacsyma-init.macsymamac-init.mac
Magnus
Maximamacsyma-init.macsymamac-init.mac
Maple.mapleinitmaplev5.ini
Mathematicainit.minit.m
MuPAD.mupadinit\mupad\bin\userinit.mu
Octave
Pari
Reduce.reducercreduce.rc
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
SystemDescribe keywordFind keywords containing pattern
Axiom)what operations pattern
Derive
DoCon
GAP?keyword??keyword
Macsymadescribe("keyword")\$apropos("pattern");
Magnus
Maximadescribe("keyword")\$apropos("pattern");
Maple?keyword?pattern\,\fnm
Mathematica?keyword?*pattern*
MuPAD?keyword?*pattern*
Octavehelp -i keyword
Pari
Reduce
Scilab
Sumit
Yacas
+
+ +Only if the pattern is not a keyword and then the matches are simplistic. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
SystemCommentLine Cont.Prev. Expr.Case sensitiveVariables assumed
Axiom-- commentinput _CR input%Yesreal
Derive"comment"input CRinputNoreal
DoCon
GAP# commentinput\CRinputlastYesno assumption
Macsyma/* comment */input CR input;%Noreal
Magnus
Maxima/* comment */input CR input;%Noreal
Maple# commentinput CR input;%Yescomplex
Mathematica(* comment *)input CR input%Yescomplex
MuPAD# comment #input CR input;%Yescomplex
Octave##Yes
Pari
Reduce% commentinput CR input;wsNocomplex
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
SystemLoad a fileTime a commandQuit
Axiom)read "file" )quiet)set messages time on)quit
Derive[Transfer Load Derive][Quit]
DoCon
GAPRead("file");time; \h{(also see {\tt Runtime();})}quit;
Macsymaload("file")\$showtime: all\$quit();
Magnus
Maximaload("file")\$showtime: all\$quit();
Mapleread("file"):readlib(showtime): on;quit
Mathematica<< fileTiming[command]Quit[]
MuPADread("file"):time(command);quit
Octaveload filetic(); cmd ; toc()quit \OR\ exit
Pari
Reducein "file"\$on time;quit;
Scilabquit
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Display}\h{Suppress}
\h{output}\h{output}\h{Substitution: $f(x, y) \rightarrow f(z, w)$}
Axiominputinput;subst(f(x, y), [x = z, y = w])
Deriveinputvar:= input[Manage Substitute]
DoCon
GAPinput;input;;Value(f,[x,y],[z,w]);\fnm
Macsymainput;input\$subst([x = z, y = w], f(x, y));
Magnus
Maximainput;input\$subst([x = z, y = w], f(x, y));
Mapleinput;input:subs(\{x = z, y = w\}, f(x, y));
Mathematicainputinput;f[x, y] /. \{x -> z, y -> w\}
MuPADinput;input:subs(f(x, y), [x = z, y = w]);
Octaveinputinput;
Pari
Reduceinput;input\$sub(\{x = z, y = w\}, f(x, y));
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Set}\h{List}\h{Matrix}
Axiomset [1, 2][1, 2]matrix([[1, 2],[3, 4]])
Derive\{1, 2\}[1, 2][[1,2], [3,4]]
DoCon
GAPSet([1,2])[1, 2][[1,2], [3,4]]\fnm
Macsyma[1, 2][1, 2]matrix([1, 2], [3, 4])
Magnus
Maxima[1, 2][1, 2]matrix([1, 2], [3, 4])
Maple\{1, 2\}[1, 2]matrix([[1, 2], [3, 4]])
Mathematica\{1, 2\}\{1, 2\}\{\{1, 2\}, \{3, 4\}\}
MuPAD\{1, 2\}[1, 2]export(Dom): \q export(linalg):
matrix:= ExpressionField(normal)):
matrix([[1, 2], [3, 4]])
Octave
Pari
Reduce\{1, 2\}\{1, 2\}mat((1, 2), (3, 4))
Scilablist(1,2)A=[1,2;3,4]
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Equation}\h{List element}\h{Matrix element}\h{Length of a list}
Axiomx = 0l . 2m(2, 3)\#l
Derivex = 0l SUB 2m SUB 2 SUB 3DIMENSION(l)
DoCon
GAPx=0l[2]m[2][3]Length(l)
Macsymax = 0l[2]m[2, 3]length(l)
Magnus
Maximax = 0l[2]m[2, 3]length(l)
Maplex = 0l[2]m[2, 3]nops(l)
Mathematicax == 0l[[2]]m[[2, 3]]Length[l]
MuPADx = 0l[2]m[2, 3]nops(l)
Octave
Pari
Reducex = 0part(l, 2)m(2, 3)length(l)
Scilabl(2)
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\m{2}{\rm Prepend/append an element to a list}\h{Append two lists}
Axiomcons(e, l)concat(l, e)append(l1, l2)
DeriveAPPEND([e], l)APPEND(l, [e])APPEND(l1, l2)
DoCon
GAPConcatenation([e],l)Add(l,e)Append(l1, l2)
Macsymacons(e, l)endcons(e, l)append(l1, l2)
Magnus
Maximacons(e, l)endcons(e, l)append(l1, l2)
Maple[e, op(l)][op(l), e][op(l1), op(l2)]
MathematicaPrepend[l, e]Append[l, e]Join[l1, l2]
MuPAD[e, op(l)]append(l, e)l1 . l2
Octave
Pari
Reducee . lappend(l, {e})append(l1, l2)
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Matrix column dimension}\h{Convert a list into a column vector}
Axiomncols(m)transpose(matrix([l]))
DeriveDIMENSION(m SUB 1)[l]\`{}
DoCon
GAPLength(mat[1])\h{objects are identical}
Macsymamat\_\,ncols(m)transpose(matrix(l))
Magnus
Maximamat\_\,ncols(m)transpose(matrix(l))
Maplelinalg[coldim](m)linalg[transpose](matrix([l]))
MathematicaDimensions[m][[2]]Transpose[\{l\}]
MuPADlinalg::ncols(m)transpose(matrix([l]))\,\fnm
Octave
Pari
Reduceload\_\,package(linalg)\$matrix v(length(l), 1)\$
column\_dim(m)for i:=1:length(l) do
\q\q v(i, 1):= part(l, i)
Scilab
Sumit
Yacas
+
+ +See the definition of matrix above. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Convert a column vector into a list}
Axiom[v(i, 1) for i in 1..nrows(v)]
Derivev\`{} SUB 1
DoCon
GAP\h{objects are identical}
Macsymapart(transpose(v), 1)
Magnus
Maximapart(transpose(v), 1)
Mapleop(convert(linalg[transpose](v), listlist))
MathematicaFlatten[v]
MuPAD[op(v)]
Octave
Pari
Reduceload\_\,package(linalg)\$
for i:=1:row\_\,dim(v) collect(v(i, 1))
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{True}\h{False}\h{And}\h{Or}\h{Not}\h{Equal}\h{Not equal} +
Axiomtruefalseandornot==
DeriveTRUEFALSEANDORNOT=/=
DoCon
GAPtruefalse\fnmandornot=<>
Macsymatruefalseandornot=\#
Magnus
Maximatruefalseandornot=\#
Mapletruefalseandornot=<>
MathematicaTrueFalse\\||!==!=
MuPADtruefalseandornot=<>
Octave
Pari
Reducetnilandornot=neq
Scilab\%t\%f
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{If+then+else statements}\h{Strings (concatenated)}
Axiomif \_ then \_ else if \_ then \_ else \_concat(["x", "y"])
DeriveIF(\_, \_, IF(\_, \_, \_))"xy"
DoCon
GAPif \_ then \_ elif \_ then \_ else \_ fiConcatenation("x","y")
Macsymaif \_ then \_ else if \_ then \_ else \_concat("x", "y")
Magnus
Maximaif \_ then \_ else if \_ then \_ else \_concat("x", "y")
Mapleif \_ then \_ elif \_ then \_ else \_ fi"x" . "y"
MathematicaIf[\_, \_, If[\_, \_, \_]]"x" <> "y"
MuPADif \_ then \_ elif \_ then \_ else \_"x" . "y"
\q\q end\_if
Octave
Pari
Reduceif \_ then \_ else if \_ then \_ else \_"xy" \OR\ mkid(x, y)
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Simple loop and Block}\h{Generate the list $[1, 2, \ldots, n]$}
Axiomfor i in 1..n repeat ( x; y )[f(i) for i in 1..n]
DeriveVECTOR([x, y], i, 1, n)VECTOR(f(i), i, 1, n)
DoCon
GAPfor i in [1..n] do \_ od;[1..n] {\rm or} [1,2..n]
Macsymafor i:1 thru n do (x, y);makelist(f(i), i, 1, n);
Magnus
Maximafor i:1 thru n do (x, y);makelist(f(i), i, 1, n);
Maplefor i from 1 to n do x; y od;[f(i) \$ i = 1..n];
MathematicaDo[x; y, \{i, 1, n\}]Table[f[i], \{i, 1, n\}]
MuPADfor i from 1 to n do x; y[f(i) \$ i = 1..n];
\q\q end\_for;
Octave
Pari
Reducefor i:=1:n do <>;for i:=1:n collect f(i);
Scilab
Sumit
Yacas
\end{tabular}
[10pt] + +\begin{tabular}{l|l} +\h{Complex loop iterating on a list}
Axiomfor x in [2, 3, 5] while x**2 < 10 repeat output(x)
Derive
DoCon
GAPfor x in [2, 3, 5] do while x\^{}2<10 do Print(x);od;od;
Macsymafor x in [2, 3, 5] while x\^{}2 < 10 do print(x)\$
Magnus
Maximafor x in [2, 3, 5] while x\^{}2 < 10 do print(x)\$
Maplefor x in [2, 3, 5] while x\^{}2 < 10 do print(x) od:
MathematicaFor[l = \{2, 3, 5\}, l != \{\} \\l[[1]]\^{}2 < 10,
\q l = Rest[l], Print[l[[1]]] ]
MuPADfor x in [2, 3, 5] do if x\^{}2 < 10 then print(x) end\_if
\q end\_for:
Octave
Pari
Reducefor each x in \{2, 3, 5\} do if x\^{}2 < 10 then write(x)\$
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{\small Assignment}\h{Function definition}\h{Clear vars and funs}
Axiomy:= f(x)f(x, y) == x*y {\small\tt )clear properties y f}
Derivey:= f(x)f(x, y):= x*yy:= f:=
DoCon
GAPy:= f(x);f:=function(x, y) return x*y; end;\h{There are +no symbolic variables}
Macsymay: f(x);f(x, y):= x*y;remvalue(y)\$
remfunction(f)\$
Magnus
Maximay: f(x);f(x, y):= x*y;remvalue(y)\$
remfunction(f)\$
Mapley:= f(x);f:= proc(x, y) x*y end;y:= 'y': f:= 'f':
Mathematicay = f[x]f[x\_, y\_\,]:= x*yClear[y, f]
MuPADy:= f(x);f:= proc(x, y)y:= NIL: f:= NIL:
\q\q begin x*y end\_\,proc;
Octave
Pari
Reducey:= f(x);procedure f(x, y); x*y;clear y, f;
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Function definition with a local variable}
Axiomf(x) == (local n; n:= 2; n*x)
Derive
DoCon
GAPf:=function(x) local n; n:=2;return n*x; end;
Macsymaf(x):= block([n], n: 2, n*x);
Magnus
Maximaf(x):= block([n], n: 2, n*x);
Maplef:= proc(x) local n; n:= 2; n*x end;
Mathematicaf[x\_\,]:= Module[\{n\}, n = 2; n*x]
MuPADf:= proc(x) local n; begin n:= 2; n*x end\_\,proc;
Octave
Pari
Reduceprocedure f(x); begin scalar n; n:= 2; return(n*x) end;
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Return unevaluated symbol}\h{Define a function from an expression}
Axiome:= x*y;\q 'efunction(e, f, x, y)
Derivee:= x*y\q 'ef(x, y):== e
DoCon
GAP\h{No unevaluated symbols}\fnm
Macsymae: x*y\$\q 'e;define(f(x, y), e);
Magnus
Maximae: x*y\$\q 'e;define(f(x, y), e);
Maplee:= x*y:\q 'e';f:= unapply(e, x, y);
Mathematicae = x*y;\q HoldForm[e]f[x\_, y\_\,] = e
MuPADe:= x*y:\q hold(e);f:= hold(func)(e, x, y);
Octave
Pari
Reducee:= x*y\$for all x, y let f(x, y):= e;
Scilab
Sumit
Yacas
+
+ +Variables can be assigned to generators of a suitable free +object, for example x:=X(Rationals,"x"); or +f:=FreeGroup(2);x:=f.1;}. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Fun.\ of an indefinite number of args}\h{Apply ``+'' to sum a list}
Axiomreduce(+, [1, 2])
DeriveLST l:= l
DoCon
GAPlst:=function(args) \_ end;Sum([1,2])
Macsymalst([l]):= l;apply("+", [1, 2])
Magnus
Maximalst([l]):= l;apply("+", [1, 2])
Maplelst:=proc() [args[1..nargs]] end;convert([1, 2], \`{}+\`{})
Mathematicalst[l\_\,\_\,\_\,]:= \{l\}Apply[Plus, \{1, 2\}]
MuPADlst:= proc(l) begin [args()]\_\,plus(op([1, 2]))
\q\q\q end\_\,proc;
Octave
Pari
Reducexapply(+, \{1, 2\})\,\fnm
Scilab
Sumit
Yacas
+
+ +procedure xapply(f, lst); lisp(f . cdr(lst)) + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Apply a fun.\ to a}
\h{list of its args}\h{Map an anonymous function onto a list}
Axiomreduce(f, l)map(x +-> x + y, [1, 2])
Derivex:= [1, 2]
VECTOR(x SUB i + y, i, 1, DIMENSION(x))
DoCon
GAPList(l,f)List([1,2],x->x+y)
Macsymaapply(f, l)map(lambda([x], x + y), [1, 2])
Magnus
Maximaapply(f, l)map(lambda([x], x + y), [1, 2])
Maplef(op(l))map(x -> x + y, [1, 2])
MathematicaApply[f, l]Map[\# + y \, \{1, 2\}]
MuPADf(op(l))map([1, 2], func(x + y, x))
Octave
Pari
Reducexapply(f, l)\,\fnmfor each x in \{1, 2\} collect x + y
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Pattern matching: $f(3 y) + f(z y) \rightarrow 3 f(y) + f(z y)$}
Axiomf:= operator('f);
( rule f((n | integer?(n)) * x) == n*f(x) )( \_
\q\q f(3*y) + f(z*y))
Derive
DoCon
GAP
Macsymamatchdeclare(n, integerp, x, true)\$
defrule(fnx, f(n*x), n*f(x))\$
apply1(f(3*y) + f(z*y), fnx);
Magnus
Maximamatchdeclare(n, integerp, x, true)\$
defrule(fnx, f(n*x), n*f(x))\$
apply1(f(3*y) + f(z*y), fnx);
Maplemap(proc(q) local m;
\q\q\q if match(q = f(n*y), y, 'm') and
\q\q\q\q\q type(rhs(op(m)), integer) then
\q\q\q\q subs(m, n * f(y)) else q fi
\q\q end,
\q\q f(3*y) + f(z*y));
Mathematicaf[3*y] + f[z*y] /. f[n\_Integer * x\_\,] -> n*f[x]
MuPADd:= domain("match"): \q d::FREEVARIABLE:= TRUE:
n:= new(d, "n", func(testtype(m, DOM\_INT), m)):
x:= new(d, "x", TRUE):
map(f(3*y) + f(z*y),
\q\q proc(q) local m; begin m:= match(q, f(n*x));
\q\q\q if m = FAIL then q
\q\q\q else subs(hold("n" * f("x")), m) end\_if
\q\q end\_\,proc);
Octave
Pari
Reduceoperator f;
f(3*y) + f(z*y)
\q\q where \{f(n * x) => n*f(x) when fixp(n)\};
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Define a new infix operator and then use it}
Axiom
Derive
DoCon
GAP\h{One can overload existing infix operators for ones own purposes}
Macsymainfix("")\$ \q ""(x, y):= sqrt(x\^{}2 + y\^{}2)\$ \q + 3 4;
Magnus
Maximainfix("")\$ \q ""(x, y):= sqrt(x\^{}2 + y\^{}2)\$ \q + 3 4;
Maple\`{}\\`{}:= (x, y) -> sqrt(x\^{}2 + y\^{}2): \q 3 \ 4; +
Mathematicax\_ $\backslash$[Tilde] y\_:= Sqrt[x\^{}2 + y\^{}2]; \q + 3 $\backslash$[Tilde] 4
MuPADtilde:= proc(x, y) begin sqrt(x\^{}2 + y\^{}2) end\_\,proc:
\q 3 \tilde 4;
Octave
Pari
Reduceinfix |\$ \q procedure |(x, y); sqrt(x\^{}2 + y\^{}2)\$ \q + 3 | 4;
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Main expression}
\h{operator}\h{\nth{1} operand}\h{List of expression operands}
Axiom\fnmkernels(e) . 1kernels(e)
Derive{\em various}\fnm
DoCon
GAP\m{3}{\rm There are no formal unevaluated expressions}
Macsymapart(e, 0)part(e, 1)args(e)
Magnus
Maximapart(e, 0)part(e, 1)args(e)
Mapleop(0, e)op(1, e)[op(e)]
MathematicaHead[e]e[[1]]ReplacePart[e, List, 0]
MuPADop(e, 0)op(e, 1)[op(e)]
Octave
Pari
Reducepart(e, 0)part(e, 1)for i:=1:arglength(e)
\q\q collect part(e, i)
Scilab
Sumit
Yacas
+
+ +The following commands work only on expressions that consist of a +single level (e.g., $x + y + z$ but not $a/b + c/d$). +TERMS, FACTORS, NUMERATOR, LHS, etc. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Print text and results}
Axiomoutput(concat(["sin(", string(0), ") = ",
\q string(sin(0))]));
Derive"sin(0)" = sin(0)
DoCon
GAPPrint("There is no sin, but factors(10)= ",Factors(10), +"$\backslash$n")
Macsymaprint("sin(", 0, ") =", sin(0))\$
Magnus
Maximaprint("sin(", 0, ") =", sin(0))\$
Mapleprintf("sin(\%a) = \%a$\backslash$n", 0, sin(0)):
MathematicaPrint[StringForm["sin(\`{}\`{}) = \`{}\`{}", 0, Sin[0]]];
MuPADprint(Unquoted, "sin(".0.")" = sin(0)):
Octave
Pari
Reducewrite("sin(", 0, ") = ", sin(0))\$
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Generate FORTRAN}\h{Generate \TeX/\LaTeX}
AxiomoutputAsFortran(e)outputAsTex(e)
Derive[Transfer Save Fortran]
DoCon
GAPPrint(LaTeX(e));
Macsymafortran(e)\$ \OR gentran(eval(e))\$tex(e);
Magnus
Maximafortran(e)\$ \OR gentran(eval(e))\$tex(e);
Maplefortran([e]);latex(e);
MathematicaFortranForm[e]TexForm[e]
MuPADgenerate::fortran(e);generate::TeX(e);
Octave
Pari
Reduceon fort; \q e; \q off fort; \ORload\_\,package(tri)\$
load\_\,package(gentran)\$ gentran e;on TeX; e; off TeX;
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Import two space separated columns of integers from {\tt file}}
Axiom
Derive[Transfer Load daTa] ({\rm from} file.dat)
DoCon
GAP
Macsymaxy: read\_num\_data\_to\_matrix("file", nrows, 2)\$
Magnus
Maximaxy: read\_num\_data\_to\_matrix("file", nrows, 2)\$
Maplexy:= readdata("file", integer, 2):
Mathematicaxy = ReadList["file", Number, RecordLists -> True]
MuPAD
Octave
Pari
Reduce
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Export two space separated columns of integers to {\tt file}\fnm}
Axiom)set output algebra "file" \q ({\rm creates} file.spout)
for i in 1..n repeat output( \_
\q concat([string(xy(i, 1)), " ", string(xy(i, 2))]) )
)set output algebra console
Derivexy [Transfer Print Expressions File]\q({\rm creates} file.prt)
DoCon
GAPPrintTo("file");for i in [1..n] do
\q AppendTo("file",xy[i][1]," ",xy[i][2],"$\backslash$n");od;
Macsymawritefile("file")\$ \q for i:1 thru n do
\q print(xy[i, 1], xy[i, 2])\$ \q closefile()\$
Magnus
Maximawritefile("file")\$ \q for i:1 thru n do
\q print(xy[i, 1], xy[i, 2])\$ \q closefile()\$
Maplewritedata("file", xy);
Mathematicaoutfile = OpenWrite["file"];
Do[WriteString[outfile,
\q xy[[i, 1]], " ", xy[[i, 2]], "$\backslash$n"], \{i, 1, n\}]
Close[outfile];
MuPADfprint(Unquoted, Text, "file",
\q ("$\backslash$n", xy[i, 1], xy[i, 2]) \$ i = 1..n):
Octave
Pari
Reduceout "file"; \q for i:=1:n do
\q write(xy(i, 1), " ", xy(i, 2)); \q shut "file";
Scilab
Sumit
Sumit
Yacas
+
+ +

+Some editing of file will be necessary for all systems but +Maple and Mathematica. +

+ +

Mathematics and Graphics

+ +Since GAP aims at discrete mathematics, it does not provide much of +the calculus functionality listed in the following section. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
$e$$\pi$$i$$+\infty$$\sqrt{2}$$2^{1/3}$
Axiom\%e\%pi\%i\%plusInfinitysqrt(2)2**(1/3)
Derive\#epi\#iinfSQRT(2)2\^{}(1/3)
DoCon
GAPE(4)infinityER(2)\fnm
Macsyma\%e\%pi\%iinfsqrt(2)2\^{}(1/3)
Magnus
Maxima\%e\%pi\%iinfsqrt(2)2\^{}(1/3)
Mapleexp(1)PiIinfinitysqrt(2)2\^{}(1/3)
MathematicaEPiIInfinitySqrt[2]2\^{}(1/3)
MuPADEPIIinfinitysqrt(2)2\^{}(1/3)
Octave
Pari
Reduceepiiinfinitysqrt(2)2\^{}(1/3)
Scilab
Sumit
Yacas
+
+ER represents special cyclotomic numbers and is not a +root function.} + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Euler's constant}\h{Natural log}\h{Arctangent}$n!$
Axiomlog(x)atan(x)factorial(n)
Deriveeuler\_\,gammaLOG(x)ATAN(x)n!
DoCon
GAPLogInt(x,base)Factorial(n)
Macsyma\%gammalog(x)atan(x)n!
Magnus
Maxima\%gammalog(x)atan(x)n!
Maplegammalog(x)arctan(x)n!
MathematicaEulerGammaLog[x]ArcTan[x]n!
MuPADEULERln(x)atan(x)n!
Octave
Pari
ReduceEuler\_\,Gammalog(x)atan(x)factorial(n)
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Legendre polynomial}\h{Chebyshev poly.\ of the \nth{1} kind}
AxiomlegendreP(n, x)chebyshevT(n, x)
DeriveLEGENDRE\_\,P(n, x)CHEBYCHEV\_\,T(n, x)
DoCon
GAP
Macsymalegendre\_\,p(n, x)chebyshev\_\,t(n, x)
Magnus
Maximalegendre\_\,p(n, x)chebyshev\_\,t(n, x)
Mapleorthopoly[P](n, x)orthopoly[T](n, x)
MathematicaLegendreP[n, x]ChebyshevT[n, x]
MuPADorthpoly::legendre(n, x)orthpoly::chebyshev1(n, x)
Octave
Pari
ReduceLegendreP(n, x)ChebyshevT(n, x)
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Fibonacci number}\h{Elliptic integral of the \nth{1} kind}
Axiomfibonacci(n)
DeriveFIBONACCI(n)ELLIPTIC\_\,E(phi, k\^{}2)
DoCon
GAPFibonacci(n)
Macsymafib(n)elliptic\_\,e(phi, k\^{}2)
Magnus
Maximafib(n)elliptic\_\,e(phi, k\^{}2)
Maplecombinat[fibonacci](n)EllipticE(sin(phi), k)
MathematicaFibonacci[n]EllipticE[phi, k\^{}2]
MuPADnumlib::fibonacci(n)
Octave
Pari
ReduceEllipticE(phi, k\^{}2)
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
$\Gamma(x)$$\psi(x)$\h{Cosine integral}\h{Bessel fun.\ (\nth{1})}
AxiomGamma(x)psi(x)real(Ei(\%i*x))besselJ(n, x)
DeriveGAMMA(x)PSI(x)CI(x)BESSEL\_\,J(n, x)
DoCon
GAP
Macsymagamma(x)psi[0](x)cos\_\,int(x)bessel\_j[n](x)
Magnus
Maximagamma(x)psi[0](x)cos\_\,int(x)bessel\_j[n](x)
MapleGAMMA(x)Psi(x)Ci(x)BesselJ(n, x)
MathematicaGamma[x]PolyGamma[x]CosIntegral[x]BesselJ[n, x]
MuPADgamma(x)psi(x)besselJ(n, x)
Octave
Pari
ReduceGamma(x)Psi(x)Ci(x)BesselJ(n, x)
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Hypergeometric fun.\ ${}_2F_1(a, b; c; x)$}\h{Dirac delta} \h{Unit step fun.}
Axiom
DeriveGAUSS(a, b, c, x)STEP(x)
DoCon
GAP
Macsymahgfred([a, b], [c], x)delta(x)unit\_\,step(x) +
Magnus
Maximahgfred([a, b], [c], x)delta(x)unit\_\,step(x) +
Maplehypergeom([a, b], [c], x)Dirac(x)Heaviside(x)
MathematicaHypergeometricPFQ[\{a,b\},\{c\},x] \m{2}{<< Calculus\`{}DiracDelta\`{}}
MuPADdirac(x)heaviside(x)
Octave
Pari
Reducehypergeometric(\{a, b\}, \{c\}, x)
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Define $|x|$ via a piecewise function}
Axiom
Derivea(x):= -x*CHI(-inf, x, 0) + x*CHI(0, x, inf)
DoCon
GAP
Macsymaa(x):= -x*unit\_\,step(-x) + x*unit\_\,step(x)\$
Magnus
Maximaa(x):= -x*unit\_\,step(-x) + x*unit\_\,step(x)\$
Maplea:= x -> piecewise(x < 0, -x, x):
Mathematica<< Calculus\`{}DiracDelta\`{}
a[x\_]:= -x*UnitStep[-x] + x*UnitStep[x]
MuPADa:= proc(x) begin -x*heaviside(-x) + x*heaviside(x)
\q\q end\_\,proc:
Octave
Pari
Reduce
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Assume $x$ is real}\h{Remove that assumption}
Axiom
Derivex :epsilon Realx:=
DoCon
GAP
Macsymadeclare(x, real)\$remove(x, real)\$
Magnus
Maximadeclare(x, real)\$remove(x, real)\$
Mapleassume(x, real);x:= 'x':
Mathematicax/: Im[x] = 0;Clear[x]
MuPADassume(x, Type::RealNum):unassume(x, Type::RealNum):
Octave
Pari
Reduce
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Assume $0 < x \le 1$}\h{Remove that assumption}
Axiom
Derivex :epsilon (0, 1]x:=
DoCon
GAP
Macsymaassume(x > 0, x <= 1)\$forget(x > 0, x <= 1)\$
Magnus
Maximaassume(x > 0, x <= 1)\$forget(x > 0, x <= 1)\$
Mapleassume(x > 0);x:= 'x':
additionally(x <= 1);
MathematicaAssumptions -> 0 < x <= 1\,\fnm
MuPADassume(x > 0): assume(x <= 1):unassume(x):
Octave
Pari
Reduce
Scilab
Sumit
Yacas
+
+ +Note: This is an option for {\tt Integrate}.} + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Basic simplification of an expression $e$}
Axiomsimplify(e) \OR\ normalize(e) \OR\ complexNormalize(e)
Derivee
DoCon
GAPe
Macsymaratsimp(e) \OR\ radcan(e)
Magnus
Maximaratsimp(e) \OR\ radcan(e)
Maplesimplify(e)
MathematicaSimplify[e] \OR\ FullSimplify[e]
MuPADsimplify(e) \OR\ normal(e)
Octave
Pari
Reducee
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Use an unknown function}\h{Numerically evaluate an expr.}
Axiomf:= operator('f); \q f(x)exp(1) :: Complex Float
Derivef(x):=Precision:= Approximate
f(x)APPROX(EXP(1))
Precision:= Exact
DoCon
GAPEvalF(123/456)
Macsymaf(x)sfloat(exp(1));
Magnus
Maximaf(x)sfloat(exp(1));
Maplef(x)evalf(exp(1));
Mathematicaf[x]N[Exp[1]]
MuPADf(x)float(exp(1));
Octave
Pari
Reduceoperator f; \q f(x)on rounded; \q exp(1);
off rounded;
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
$ n \bmod m$\h{Solve $e \equiv 0 \bmod m$ for $x$}
Axiomrem(n, m)solve(e = 0 :: PrimeField(m), x)
DeriveMOD(n, m)SOLVE\_\,MOD(e = 0, x, m)
DoCon
GAPn mod m\h{solve using finite fields}
Macsymamod(n, m)modulus: m\$ \q solve(e = 0, x)
Magnus
Maximamod(n, m)modulus: m\$ \q solve(e = 0, x)
Maplen mod mmsolve(e = 0, m)
MathematicaMod[n, m]Solve[\{e == 0, Modulus == m\}, x]
MuPADn mod msolve(poly(e = 0, [x], IntMod(m)), x)
Octave
Pari
Reduceon modular;load\_\,package(modsr)\$ \q on modular;
setmod m\$ \q nsetmod m\$ \q m\_solve(e = 0, x)
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Put over common denominator}\h{Expand into separate fractions}
Axioma/b + c/d(a*d + b*c)/(b*d) :: \_
\q MPOLY([a], FRAC POLY INT)
DeriveFACTOR(a/b + c/d, Trivial)EXPAND((a*d + b*c)/(b*d))
DoCon
GAPa/b+c/d
Macsymaxthru(a/b + c/d)expand((a*d + b*c)/(b*d))
Magnus
Maximaxthru(a/b + c/d)expand((a*d + b*c)/(b*d))
Maplenormal(a/b + c/d)expand((a*d + b*c)/(b*d))
MathematicaTogether[a/b + c/d]Apart[(a*d + b*c)/(b*d)]
MuPADnormal(a/b + c/d)expand((a*d + b*c)/(b*d))
Octave
Pari
Reducea/b + c/don div; (a*d + b*c)/(b*d)
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Manipulate the root of a polynomial}
Axioma:= rootOf(x**2 - 2); \q a**2
Derive
DoCon
GAPx:=X(Rationals,"x");
\q a:=RootOfDefiningPolynomial(AlgebraicExtension(Rationals,x\^{}2-2)); +a\^{}2
Macsymaalgebraic:true\$ \q tellrat(a\^{}2 - 2)\$ \q rat(a\^{}2);
Magnus
Maximaalgebraic:true\$ \q tellrat(a\^{}2 - 2)\$ \q rat(a\^{}2);
Maplea:= RootOf(x\^{}2 - 2): \q simplify(a\^{}2);
Mathematicaa = Root[\#\^{}2 - 2 \, 2] \q a\^{}2
MuPAD
Octave
Pari
Reduceload\_\,package(arnum)\$ \q defpoly(a\^{}2 - 2); \q a\^{}2;
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Noncommutative multiplication}\h{Solve a pair of equations}
Axiomsolve([eqn1, eqn2], [x, y])
Derivex :epsilon NonscalarSOLVE([eqn1, eqn2], [x, y])
y :epsilon Nonscalar
x . y
DoCon
GAP*
Macsymax . ysolve([eqn1, eqn2], [x, y])
Magnus
Maximax . ysolve([eqn1, eqn2], [x, y])
Maplex \* ysolve(\{eqn1, eqn2\}, \{x, y\})
Mathematicax ** ySolve[\{eqn1, eqn2\}, \{x, y\}]
MuPADsolve(\{eqn1, eqn2\}, \{x, y\})
Octave
Pari
Reduceoperator x, y;solve(\{eqn1, eqn2\}, \{x, y\})
noncom x, y;
x() * y()
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\m{2}{\rm Decrease/increase angles in trigonometric functions}
Axiom\m{2}{simplify(normalize(sin(2*x)))}
DeriveTrigonometry:= ExpandTrigonometry:= Collect
sin(2*x)2*sin(x)*cos(x)
DoCon
GAP
Macsymatrigexpand(sin(2*x))trigreduce(2*sin(x)*cos(x))
Magnus
Maximatrigexpand(sin(2*x))trigreduce(2*sin(x)*cos(x))
Mapleexpand(sin(2*x))combine(2*sin(x)*cos(x))
MathematicaTrigExpand[Sin[2*x]]TrigReduce[2*Sin[x]*Cos[x]]
MuPADexpand(sin(2*x))combine(2*sin(x)*cos(x), sincos) +
Octave
Pari
Reduceload\_\,package(assist)\$
trigexpand(sin(2*x))trigreduce(2*sin(x)*cos(x))
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Gr\"obner basis}
Axiomgroebner([p1, p2, ...])
Derive
DoCon
GAP
Macsymagrobner([p1, p2, ...])
Magnus
Maximagrobner([p1, p2, ...])
MapleGroebner[gbasis]([p1, p2, ...], plex(x1, x2, ...))
MathematicaGroebnerBasis[\{p1, p2, ...\}, \{x1, x2, ...\}]
MuPADgroebner::gbasis([p1, p2, ...])
Octave
Pari
Reduceload\_\,package(groebner)\$ \q groebner(\{p1, p2, ...\})
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Factorization of $e$ over $i = \sqrt{-1}$}
Axiomfactor(e, [rootOf(i**2 + 1)])
DeriveFACTOR(e, Complex)
DoCon
GAPFactors(GaussianIntegers,e)
Macsymagfactor(e); \OR\ factor(e, i\^{}2 + 1);
Magnus
Maximagfactor(e); \OR\ factor(e, i\^{}2 + 1);
Maplefactor(e, I);
MathematicaFactor[e, Extension -> I]
MuPADQI:= Dom::AlgebraicExtension(Dom::Rational, i\^{}2 + 1);
QI::name:= "QI": \q Factor(poly(e, QI));
Octave
Pari
Reduceon complex, factor; \q e; \q off complex, factor;
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Real part}\h{Convert a complex expr.\ to rectangular form}
Axiomreal(f(z))complexForm(f(z))
DeriveRE(f(z))f(z)
DoCon
GAP(f(z)+GaloisCyc(f(z),-1))/2
Macsymarealpart(f(z))rectform(f(z))
Magnus
Maximarealpart(f(z))rectform(f(z))
MapleRe(f(z))evalc(f(z))
MathematicaRe[f[z]]ComplexExpand[f[z]]
MuPADRe(f(z))rectform(f(z))
Octave
Pari
Reducerepart(f(z))repart(f(z)) + i*impart(f(z))
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Matrix addition}\h{Matrix multiplication}\h{Matrix transpose}
AxiomA + BA * Btranspose(A)
DeriveA + BA . BA\`{}
DoCon
GAPA + BA * BTransposedMat(A)
MacsymaA + BA . Btranspose(A)
Magnus
MaximaA + BA . Btranspose(A)
Mapleevalm(A + B)evalm(A \* B)linalg[transpose](A)
MathematicaA + BA . BTranspose[A]
MuPADA + BA * Btranspose(A)
Octave
Pari
ReduceA + BA * Btp(A)
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Solve the matrix equation $A x = b$}
Axiomsolve(A, transpose(b)) . 1 . particular :: Matrix \_\_\_
Derive
DoCon
GAPSolutionMat(TransposedMat(A),b)
Macsymaxx: genvector('x, mat\_nrows(b))\$
x: part(matlinsolve(A . xx = b, xx), 1, 2)
Magnus
Maximaxx: genvector('x, mat\_nrows(b))\$
x: part(matlinsolve(A . xx = b, xx), 1, 2)
Maplex:= linalg[linsolve](A, b)
Mathematicax = LinearSolve[A, b]
MuPAD
Octave
Pari
Reduce
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Sum: $\sum_{i = 1}^n f(i)$}\h{Product: $\prod_{i = 1}^n f(i)$}
Axiomsum(f(i), i = 1..n)product(f(i), i = 1..n)
DeriveSUM(f(i), i, 1, n)PRODUCT(f(i), i, 1, n)
DoCon
GAPSum([1..n],f)Product([1..n],f)
Macsymaclosedform(closedform(
\q sum(f(i), i, 1, n))\q product(f(i), i, 1, n))
Magnus
Maximaclosedform(closedform(
\q sum(f(i), i, 1, n))\q product(f(i), i, 1, n))
Maplesum(f(i), i = 1..n)product(f(i), i = 1..n)
MathematicaSum[f[i], \{i, 1, n\}]Product[f[i], \{i, 1, n\}]
MuPADsum(f(i), i = 1..n)product(f(i), i = 1..n)
Octave
Pari
Reducesum(f(i), i, 1, n)prod(f(i), i, 1, n)
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Limit: $\lim_{x \rightarrow 0-} f(x)$}\h{Taylor/Laurent/etc.\ series}
Axiomlimit(f(x), x = 0, "left")series(f(x), x = 0, 3)
DeriveLIM(f(x), x, 0, -1)TAYLOR(f(x), x, 0, 3)
DoCon
GAP
Macsymalimit(f(x), x, 0, minus)taylor(f(x), x, 0, 3)
Magnus
Maximalimit(f(x), x, 0, minus)taylor(f(x), x, 0, 3)
Maplelimit(f(x), x = 0, left)series(f(x), x = 0, 4)
MathematicaLimit[f[x], x->0, Direction->1]Series[f[x],\{x, 0, 3\}]
MuPADlimit(f(x), x = 0, Left)series(f(x), x = 0, 4)
Octave
Pari
Reducelimit!-(f(x), x, 0)taylor(f(x), x, 0, 3)
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Differentiate: $\frac{d^3 f(x, y)}{dx \, dy^2}$} \h{Integrate: $\int_0^1 f(x) \, dx$}
AxiomD(f(x, y), [x, y], [1, 2])integrate(f(x), x = 0..1)
DeriveDIF(DIF(f(x, y), x), y, 2)INT(f(x), x, 0, 1)
DoCon
GAP
Macsymadiff(f(x, y), x, 1, y, 2)integrate(f(x), x, 0, 1)
Magnus
Maximadiff(f(x, y), x, 1, y, 2)integrate(f(x), x, 0, 1)
Maplediff(f(x, y), x, y\$2)int(f(x), x = 0..1)
MathematicaD[f[x, y], x, \{y, 2\}]Integrate[f[x], \{x, 0, 1\}]
MuPADdiff(f(x, y), x, y\$2)int(f(x), x = 0..1)
Octave
Pari
Reducedf(f(x, y), x, y, 2)int(f(x), x, 0, 1)
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Laplace transform}\h{Inverse Laplace transform}
Axiomlaplace(e, t, s)inverseLaplace(e, s, t)
DeriveLAPLACE(e, t, s)
DoCon
GAP
Macsymalaplace(e, t, s)ilt(e, s, t)
Magnus
Maximalaplace(e, t, s)ilt(e, s, t)
Mapleinttrans[laplace](e,t,s)inttrans[invlaplace](e,s,t)
Mathematica\m{2}{\q << Calculus\`{}LaplaceTransform\`{}}
LaplaceTransform[e, t, s]{\st InverseLaplaceTransform[e,s,t]} +
MuPADtransform::laplace(e,t,s)transform::ilaplace(e, s, t)
Octave
Pari
Reduce\m{2}{\q load\_\,package(laplace)\$ \q load\_\,package(defint)\$} +
laplace(e, t, s)invlap(e, t, s)
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Solve an ODE (with the initial condition $y'(0) = 1$)}
Axiomsolve(eqn, y, x)
DeriveAPPLY\_\,IC(RHS(ODE(eqn, x, y, y\_)), [x, 0], [y, 1])
DoCon
GAP
Macsymaode\_ibc(ode(eqn, y(x), x), x = 0, diff(y(x), x) = 1)
Magnus
Maximaode\_ibc(ode(eqn, y(x), x), x = 0, diff(y(x), x) = 1)
Mapledsolve(\{eqn, D(y)(0) = 1\}, y(x))
MathematicaDSolve[\{eqn, y'[0] == 1\}, y[x], x]
MuPADsolve(ode(\{eqn, D(y)(0) = 1\}, y(x)))
Octave
Pari
Reduceodesolve(eqn, y(x), x)
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{Define the differential operator $L = D_x + I$ and apply it to $\sin x$}
AxiomDD : LODO(Expression Integer, e +-> D(e, x)) := D();
L:= DD + 1; \q L(sin(x))
Derive
DoCon
GAP
Macsymaload(opalg)\$ \q L: (diffop(x) - 1)\$ \q L(sin(x));
Magnus
Maximaload(opalg)\$ \q L: (diffop(x) - 1)\$ \q L(sin(x));
Mapleid:= x -> x: \q L:= (D + id): \q L(sin)(x);
MathematicaL = D[\#, x]\+ Identity; \q Through[L[Sin[x]]]
MuPADL:= (D + id): \q L(sin)(x);
Octave
Pari
Reduce
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\h{2D plot of two separate curves overlayed}
Axiomdraw(x, x = 0..1); \q draw(acsch(x), x = 0..1);
Derive[Plot Overlay]
DoCon
GAP
Macsymaplot(x, x, 0, 1)\$ \q plot(acsch(x), x, 0, 1)\$
Magnus
Maximaplot(x, x, 0, 1)\$ \q plot(acsch(x), x, 0, 1)\$
Mapleplot(\{x, arccsch(x)\}, x = 0..1):
MathematicaPlot[\{x, ArcCsch[x]\}, \{x, 0, 1\}];
MuPADplotfunc(x, acsch(x), x = 0..1):
Octave
Pari
Reduceload\_\,package(gnuplot)\$ \q plot(y = x, x = (0 .. 1))\$
plot(y = acsch(x), x = (0 .. 1))\$
Scilab
Sumit
Yacas
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
SystemSimple 3D plotting
Axiomdraw(abs(x*y), x = 0..1, y = 0..1);
Derive[Plot Overlay]
DoCon
GAP
Macsymaplot3d(abs(x*y), x, 0, 1, y, 0, 1)\$
Magnus
Maximaplot3d(abs(x*y), x, 0, 1, y, 0, 1)\$
Mapleplot3d(abs(x*y), x = 0..1, y = 0..1):
MathematicaPlot3D[Abs[x*y], \{x, 0, 1\}, \{y, 0, 1\}];
MuPADplotfunc(abs(x*y), x = 0..1, y = 0..1):
Octave
Pari
Reduceload\_\,package(gnuplot)\$
plot(z = abs(x*y), x = (0 .. 1), y = (0 .. 1))\$
Scilab
Sumit
Yacas
+
+ + + + + diff --git a/src/axiom-website/rosetta.pdf b/src/axiom-website/rosetta.pdf new file mode 100644 index 0000000..7a9bfdd Binary files /dev/null and b/src/axiom-website/rosetta.pdf differ diff --git a/src/axiom-website/rosetta.tex b/src/axiom-website/rosetta.tex new file mode 100644 index 0000000..9e14ccc --- /dev/null +++ b/src/axiom-website/rosetta.tex @@ -0,0 +1,1511 @@ +\documentclass{article} +\normalsize\baselineskip=12pt +\parskip=0pt +\parindent=10pt +\tolerance=5000 +\pretolerance=5000 +\frenchspacing +\hangindent=10pt +\skip\footins=18pt +\global\textwidth 31pc \global\textheight 47pc +\headsep 12pt +\oddsidemargin 0pt +\evensidemargin 0pt +% +\renewcommand{\textfraction}{.1} +\renewcommand{\floatpagefraction}{.75} +% +\catcode`@=11 +\def\ps@plain{\let\@mkboth\@gobbletwo% + \let\@oddhead\@empty\def\@oddfoot{\sysdetails} + \let\@evenhead\@empty\let\@evenfoot\@oddfoot} +\def\ps@empty{\let\@mkboth\@gobbletwo% + \let\@oddhead\@empty\def\@oddfoot{\sysdetails} + \let\@evenhead\@empty\let\@evenfoot\@oddfoot} +\catcode`@=12 +% +\def\sysdetails{\parbox{\textwidth}{% +Based on material originally published in {\sl Computer Algebra Systems: A +Practical Guide\/} edited by Michael J.\ Wester, John Wiley \& Sons, +Chichester, United Kingdom, ISBN 0-471-98353-5, xvi+436 pages, 1999.}} +% +\pagestyle{plain} + +\begin{document} +% +% \nth{n} produces 1^{st}, 2^{nd}, 3^{rd}, 4^{th}, etc. +% +\def\nth#1{$#1^{\rm \ifcase#1 th\or st\or nd\or rd\else th\fi}$} +% +% Abbreviations +% +\newcommand{\Axiom}{{\sf Axiom}} +\newcommand{\Derive}{{\sf Derive}} +\newcommand{\DoCon}{{\sf DoCon}} +\newcommand{\GAP}{{\sf GAP}} +\newcommand{\Gmp}{{\sf Gmp}} +\newcommand{\Macsyma}{{\sf Macsyma}} +\newcommand{\Magnus}{{\sf Magnus}} +\newcommand{\Maxima}{{\sf Maxima}} +\newcommand{\Maple}{{\sf Maple}} +\newcommand{\Mathematica}{{\sf Mathematica}} +\newcommand{\MuPAD}{{\sf MuPAD}} +\newcommand{\Octave}{{\sf Octave}} +\newcommand{\Pari}{{\sf Pari}} +\newcommand{\Reduce}{{\sf Reduce}} +\newcommand{\Scilab}{{\sf Scilab}} +\newcommand{\Sumit}{{\sf Sumit}} +\newcommand{\Yacas}{{\sf Yacas}} + +\section{Introduction} + +The following is a collection of synonyms for various operations in +the computer algebra systems \Axiom, \Derive, \GAP, \Gmp, \DoCon, +\Macsyma, \Magnus, \Maxima, \Maple, \Mathematica, \MuPAD, \Octave, +\Pari, \Reduce, \Scilab, \Sumit\ and \Yacas. This collection does not +attempt to be comprehensive, but hopefully it will be useful in giving +an indication of how to translate between the syntaxes used by the +different systems in many common situations. Note that a blank entry +means either (a) that there may be an exact translation of a +particular operation for the indicated system, but we don't know what +it is or (b) there is no exact translation but it may still be +possible to work around this lack with a related functionality. + +While commercial systems are not provided on this CD the intent of the +Rosetta effort is to make it possible for experienced Computer Algebra +users to experiment with other systems. Thus the commands for +commercial systems are included to allow users of those systems to +translate. + +Some of these systems are special purpose and do not support a lot of +the functionality of the more general purpose systems. Where they do +support an interpreter the commands are provided. + +Originally written by Michael Wester. +Modified for Rosetta by Timothy Daly, Alexander Hulpke (GAP). + +\section{System availability} + +\begin{tabular}{l|lll} +System & \rm{License} & \rm{Status (May 2002)} & \rm{Web Location} \\ +\hline +\Axiom & BSD & available & http://www.aldor.org \\ +\Axiom & open source & pending & http://home.earthlink.net/~jgg964/axiom.html \\ +\Derive & commercial & available & http://www.mathware.com \\ +\DoCon & open source & available & http://www.haskell.org/docon \\ +\GAP & GPL & Rosetta & http://www.gap-system.org/~gap \\ +\Gmp & GPL & Rosetta & http://www.swox.com/gmp \\ +\Macsyma & commercial & dead & unavailable \\ +\Magnus & GPL & Rosetta & http://sourceforge.net/projects/magnus \\ +\Maxima & GPL & Rosetta & http://www.ma.utexas.edu/maxima.html\\ +\Maple & commercial & available & http://www.maplesoft.com \\ +\Mathematica & commercial & available & http://www.wolfram.com \\ +\MuPAD & commercial & available & http://www.mupad.de \\ +\Octave & GPL & Rosetta & http://www.octave.org \\ +\Pari & GPL & Rosetta & http://www.parigp-home.de \\ +\Reduce & commercial & available & http://www.zib.de/Symbolik/reduce \\ +\Scilab & Scilab & available & http://www-rocq.inria.fr/scilab \\ +\Sumit & & available & http://www-sop.inria.fr/cafe/soft-f.html \\ +\Yacas & GPL & available & http://yacas.sourceforge.net \\ +\end{tabular} \\[10pt] +\\ +\begin{tabular}{l|ll} +System & \rm{Type} & \rm{Interpreted or Compiled}\\ +\hline +\Axiom & General Purpose & both \\ +\Derive & General Purpose & \\ +\DoCon & General Purpose & Interpreted in Haskell \\ +\GAP & Group Theory & \\ +\Gmp & arb. prec. arithmetic & \\ +\Macsyma & General Purpose & \\ +\Magnus & Infinite Group Theory & \\ +\Maxima & General Purpose & \\ +\Maple & General Purpose & \\ +\Mathematica & General Purpose & \\ +\MuPAD & General Purpose & \\ +\Octave & Numerical Computing & \\ +\Pari & Number Theory & \\ +\Reduce & General Purpose & \\ +\Scilab & General Purpose & \\ +\Sumit & Functional Equations & \\ +\Yacas & General Purpose & \\ +\end{tabular} \\[10pt] + +\section{Programming and Miscellaneous} + +\begingroup +\newcommand{\OR}{{\em or }} +\newcommand{\fnm}{\footnotemark} +\newcommand{\h}[1]{{\rm #1}} +\newcommand{\m}[2]{\multicolumn{#1}{l}{#2}} +\newcommand{\q}{\quad} +\newcommand{\st}{\small\tt} +\parindent=0pt +\hfuzz=1pt +\begin{tt} + +\begin{tabular}{l|ll} +& \m{2}{\rm Unix/Microsoft user initialization file} \\ +\hline +\Axiom & \~{}/axiom.input & \\ +\GAP & \~{}/.gaprc & GAP.RC \\ +\Gmp & & \\ +\DoCon & & \\ +\Derive & & derive.ini \\ +\Macsyma & \~{}/macsyma-init.macsyma & mac-init.mac \\ +\Magnus & & \\ +\Maxima & \~{}/macsyma-init.macsyma & mac-init.mac \\ +\Maple & \~{}/.mapleinit & maplev5.ini \\ +\Mathematica & \~{}/init.m & init.m \\ +\MuPAD & \~{}/.mupadinit & + $\backslash$mupad$\backslash$bin$\backslash$userinit.mu \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & \~{}/.reducerc & reduce.rc \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& \h{Describe {\em keyword}} & \h{Find keywords containing {\em pattern}} \\ +\hline +\Axiom & & )what operations pattern \\ +\Derive & & \\ +\DoCon & & \\ +\GAP & ?keyword & ??keyword\\ +\Gmp & & \\ +\Macsyma & describe("keyword")\$ & apropos("pattern"); \\ +\Magnus & & \\ +\Maxima & describe("keyword")\$ & apropos("pattern"); \\ +\Maple & ?keyword & ?pattern\,\fnm \\ +\Mathematica & ?keyword & ?*pattern* \\ +\MuPAD & ?keyword & ?*pattern* \\ +\Octave & help -i keyword & \\ +\Pari & & \\ +\Reduce & & \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\footnotetext{Only if the pattern is not a keyword and then the matches are +simplistic.} + +\begin{tabular}{l|l@{ }llll} +& & & \h{Prev.} & \h{Case} & \h{Variables} \\ +& \h{Comment} & \h{Line continuation} & \h{expr.} & \h{sensitive} & \h{assumed} + \\ +\hline +\Axiom & -- comment & input \_input & \% & Yes & real \\ +\Derive & "comment" & input \~{}input & & No & real \\ +\DoCon & & & & & \\ +\GAP & \# comment & input$\backslash$input&last&Yes&no assumption\\ +\Gmp & & & & & \\ +\Macsyma & /* comment */ & inputinput; & \% & No & real \\ +\Magnus & & & & & \\ +\Maxima & /* comment */ & inputinput; & \% & No & real \\ +\Maple & \# comment & inputinput; & \% & Yes & complex \\ +\Mathematica & (* comment *) & inputinput & \% & Yes & complex \\ +\MuPAD & \# comment \# & inputinput; & \% & Yes & complex \\ +\Octave & \#\# & & & Yes & \\ +\Pari & & & & & \\ +\Reduce & \% comment & inputinput; & ws & No & complex \\ +\Scilab & & & & & \\ +\Sumit & & & & & \\ +\Yacas & & & & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|lll} +& \h{Load a file} & \h{Time a command} & \h{Quit} \\ +\hline +\Axiom & )read "file" )quiet & )set messages time on & )quit \\ +\Derive & [Transfer Load Derive] & & [Quit] \\ +\DoCon & & & \\ +\GAP & Read("file"); & time; \h{(also see {\tt Runtime();})}&quit;\\ +\Gmp & & & \\ +\Macsyma & load("file")\$ & showtime: all\$ & quit(); \\ +\Magnus & & & \\ +\Maxima & load("file")\$ & showtime: all\$ & quit(); \\ +\Maple & read("file"): & readlib(showtime): on; & quit \\ +\Mathematica & @<< file & Timing[command] & Quit[] \\ +\MuPAD & read("file"): & time(command); & quit \\ +\Octave & load file & tic(); cmd ; toc() & quit \OR\ exit\\ +\Pari & & & \\ +\Reduce & in "file"\$ & on time; & quit; \\ +\Scilab & & & quit \\ +\Sumit & & & \\ +\Yacas & & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|lll} +& \h{Display} & \h{Suppress} & \\ +& \h{output} & \h{output} & \h{Substitution: $f(x, y) \rightarrow f(z, w)$} \\ +\hline +\Axiom & input & input; & subst(f(x, y), [x = z, y = w]) \\ +\Derive & input & var:= input & [Manage Substitute] \\ +\DoCon & & & \\ +\GAP & input; & input;; & Value(f,[x,y],[z,w]);\fnm \\ +\Gmp & & & \\ +\Macsyma & input; & input\$ & subst([x = z, y = w], f(x, y)); \\ +\Magnus & & & \\ +\Maxima & input; & input\$ & subst([x = z, y = w], f(x, y)); \\ +\Maple & input; & input: & subs(\{x = z, y = w\}, f(x, y)); \\ +\Mathematica & input & input; & f[x, y] /. \{x -> z, y -> w\} \\ +\MuPAD & input; & input: & subs(f(x, y), [x = z, y = w]); \\ +\Octave & input & input; & \\ +\Pari & & & \\ +\Reduce & input; & input\$ & sub(\{x = z, y = w\}, f(x, y)); \\ +\Scilab & & & \\ +\Sumit & & & \\ +\Yacas & & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|lll} +& \h{Set} & \h{List} & \h{Matrix} \\ +\hline +\Axiom & set [1, 2] & [1, 2] & matrix(@[[1, 2],[3, 4]]) \\ +\Derive & \{1, 2\} & [1, 2] & @[[1,2], [3,4]] \\ +\DoCon & & & \\ +\GAP & Set([1,2]) & [1, 2] & @[[1,2], [3,4]]\fnm \\ +\Gmp & & & \\ +\Macsyma & [1, 2] & [1, 2] & matrix([1, 2], [3, 4]) \\ +\Magnus & & & \\ +\Maxima & [1, 2] & [1, 2] & matrix([1, 2], [3, 4]) \\ +\Maple & \{1, 2\} & [1, 2] & matrix(@[[1, 2], [3, 4]]) \\ +\Mathematica & \{1, 2\} & \{1, 2\} & \{\{1, 2\}, \{3, 4\}\} \\ +\MuPAD & \{1, 2\} & [1, 2] & export(Dom): \q export(linalg): \\ + & & & matrix:= ExpressionField(normal)): \\ + & & & matrix(@[[1, 2], [3, 4]]) \\ +\Octave & & & \\ +\Pari & & & \\ +\Reduce & \{1, 2\} & \{1, 2\} & mat((1, 2), (3, 4)) \\ +\Scilab & & list(1,2) & A=[1,2;3,4]\\ +\Sumit & & & \\ +\Yacas & & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|llll} +& \h{Equation} & \h{List element} & \h{Matrix element} & \h{Length of a list} \\ +\hline +\Axiom & x = 0 & l . 2 & m(2, 3) & \#l \\ +\Derive & x = 0 & l SUB 2 & m SUB 2 SUB 3 & DIMENSION(l) \\ +\DoCon & & & & \\ +\GAP & x=0 & l[2] & m[2][3] & Length(l) \\ +\Gmp & & & & \\ +\Macsyma & x = 0 & l[2] & m[2, 3] & length(l) \\ +\Magnus & & & & \\ +\Maxima & x = 0 & l[2] & m[2, 3] & length(l) \\ +\Maple & x = 0 & l[2] & m[2, 3] & nops(l) \\ +\Mathematica & x == 0 & l@[[2]] & m@[[2, 3]] & Length[l] \\ +\MuPAD & x = 0 & l[2] & m[2, 3] & nops(l) \\ +\Octave & & & & \\ +\Pari & & & & \\ +\Reduce & x = 0 & part(l, 2) & m(2, 3) & length(l) \\ +\Scilab & & l(2) & & \\ +\Sumit & & & & \\ +\Yacas & & & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|lll} +& \m{2}{\rm Prepend/append an element to a list} & \h{Append two lists} \\ +\hline +\Axiom & cons(e, l) & concat(l, e) & append(l1, l2) \\ +\Derive & APPEND([e], l) & APPEND(l, [e]) & APPEND(l1, l2) \\ +\DoCon & & & \\ +\GAP & Concatenation([e],l) & Add(l,e) & Append(l1, l2) \\ +\Gmp & & & \\ +\Macsyma & cons(e, l) & endcons(e, l) & append(l1, l2) \\ +\Magnus & & & \\ +\Maxima & cons(e, l) & endcons(e, l) & append(l1, l2) \\ +\Maple & [e, op(l)] & [op(l), e] & [op(l1), op(l2)] \\ +\Mathematica & Prepend[l, e] & Append[l, e] & Join[l1, l2] \\ +\MuPAD & [e, op(l)] & append(l, e) & l1 . l2 \\ +\Octave & & & \\ +\Pari & & & \\ +\Reduce & e . l & append(l, {e}) & append(l1, l2) \\ +\Scilab & & & \\ +\Sumit & & & \\ +\Yacas & & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& \h{Matrix column dimension} & \h{Convert a list into a column vector} \\ +\hline +\Axiom & ncols(m) & transpose(matrix([l])) \\ +\Derive & DIMENSION(m SUB 1) & [l]\`{} \\ +\DoCon & & \\ +\GAP & Length(mat[1]) & \h{objects are identical} \\ +\Gmp & & \\ +\Macsyma & mat\_\,ncols(m) & transpose(matrix(l)) \\ +\Magnus & & \\ +\Maxima & mat\_\,ncols(m) & transpose(matrix(l)) \\ +\Maple & linalg[coldim](m) & linalg[transpose](matrix([l])) \\ +\Mathematica & Dimensions[m]@[[2]] & Transpose[\{l\}] \\ +\MuPAD & linalg::ncols(m) & transpose(matrix([l]))\,\fnm \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & load\_\,package(linalg)\$ & matrix v(length(l), 1)\$ \\ + & column\_dim(m) & for i:=1:length(l) do \\ + & & \q\q v(i, 1):= part(l, i) \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\footnotetext{See the definition of {\tt matrix} above.} + +\begin{tabular}{l|l} +& \h{Convert a column vector into a list} \\ +\hline +\Axiom & [v(i, 1) for i in 1..nrows(v)] \\ +\Derive & v\`{} SUB 1 \\ +\DoCon & \\ +\GAP & \h{objects are identical} \\ +\Gmp & \\ +\Macsyma & part(transpose(v), 1) \\ +\Magnus & \\ +\Maxima & part(transpose(v), 1) \\ +\Maple & op(convert(linalg[transpose](v), listlist)) \\ +\Mathematica & Flatten[v] \\ +\MuPAD & [op(v)] \\ +\Octave & \\ +\Pari & \\ +\Reduce & load\_\,package(linalg)\$ \\ + & for i:=1:row\_\,dim(v) collect(v(i, 1)) \\ +\Scilab & \\ +\Sumit & \\ +\Yacas & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|lllllll} +& \h{True} & \h{False} & \h{And} & \h{Or} & \h{Not} & \h{Equal} & \h{Not equal} + \\ +\hline +\Axiom & true & false & and & or & not & = & \~{}= \\ +\Derive & TRUE & FALSE & AND & OR & NOT & = & /= \\ +\DoCon & & & & & & & \\ +\GAP & true & false\fnm & and & or & not & = & <> \\ +\Gmp & & & & & & & \\ +\Macsyma & true & false & and & or & not & = & \# \\ +\Magnus & & & & & & & \\ +\Maxima & true & false & and & or & not & = & \# \\ +\Maple & true & false & and & or & not & = & <> \\ +\Mathematica & True & False & \&\& & || & ! & == & != \\ +\MuPAD & true & false & and & or & not & = & <> \\ +\Octave & & & & & & & \\ +\Pari & & & & & & & \\ +\Reduce & t & nil & and & or & not & = & neq \\ +\Scilab & \%t & \%f & & & & & \\ +\Sumit & & & & & & & \\ +\Yacas & & & & & & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& \h{If+then+else statements} & \h{Strings (concatenated)} \\ +\hline +\Axiom & if \_ then \_ else if \_ then \_ else \_ & concat(["x", "y"]) \\ +\Derive & IF(\_, \_, IF(\_, \_, \_)) & "xy" \\ +\DoCon & & \\ +\GAP & if \_ then \_ elif \_ then \_ else \_ fi & Concatenation("x","y")\\ +\Gmp & & \\ +\Macsyma & if \_ then \_ else if \_ then \_ else \_ & concat("x", "y") \\ +\Magnus & & \\ +\Maxima & if \_ then \_ else if \_ then \_ else \_ & concat("x", "y") \\ +\Maple & if \_ then \_ elif \_ then \_ else \_ fi & "x" . "y" \\ +\Mathematica & If[\_, \_, If[\_, \_, \_]] & "x" <> "y" \\ +\MuPAD & if \_ then \_ elif \_ then \_ else \_ & "x" . "y" \\ + & \q\q end\_if & \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & if \_ then \_ else if \_ then \_ else \_ & "xy" \OR\ mkid(x, y)\\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& \h{Simple loop and Block} & \h{Generate the list $[1, 2, \ldots, n]$} \\ +\hline +\Axiom & for i in 1..n repeat ( x; y ) & [f(i) for i in 1..n] \\ +\Derive & VECTOR([x, y], i, 1, n) & VECTOR(f(i), i, 1, n) \\ +\DoCon & & \\ +\GAP & for i in [1..n] do \_ od; & [1..n] {\rm or} [1,2..n]\\ +\Gmp & & \\ +\Macsyma & for i:1 thru n do (x, y); & makelist(f(i), i, 1, n); \\ +\Magnus & & \\ +\Maxima & for i:1 thru n do (x, y); & makelist(f(i), i, 1, n); \\ +\Maple & for i from 1 to n do x; y od; & [f(i) \$ i = 1..n]; \\ +\Mathematica & Do[x; y, \{i, 1, n\}] & Table[f[i], \{i, 1, n\}] \\ +\MuPAD & for i from 1 to n do x; y & [f(i) \$ i = 1..n]; \\ + & \q\q end\_for; & \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & for i:=1:n do @<>; & for i:=1:n collect f(i); \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|l} +& \h{Complex loop iterating on a list} \\ +\hline +\Axiom & for x in [2, 3, 5] while x**2 < 10 repeat output(x) \\ +\Derive & \\ +\DoCon & \\ +\GAP & for x in [2, 3, 5] do while x\^{}2<10 do Print(x);od;od; \\ +\Gmp & \\ +\Macsyma & for x in [2, 3, 5] while x\^{}2 < 10 do print(x)\$ \\ +\Magnus & \\ +\Maxima & for x in [2, 3, 5] while x\^{}2 < 10 do print(x)\$ \\ +\Maple & for x in [2, 3, 5] while x\^{}2 < 10 do print(x) od: \\ +\Mathematica & For[l = \{2, 3, 5\}, l != \{\} \&\& l@[[1]]\^{}2 < 10, \\ + & \q l = Rest[l], Print[l@[[1]]] ] \\ +\MuPAD & for x in [2, 3, 5] do if x\^{}2 < 10 then print(x) end\_if \\ + & \q end\_for: \\ +\Octave & \\ +\Pari & \\ +\Reduce & for each x in \{2, 3, 5\} do if x\^{}2 < 10 then write(x)\$ \\ +\Scilab & \\ +\Sumit & \\ +\Yacas & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|lll} +& \h{\small Assignment} & \h{Function definition} & \h{Clear vars and funs} \\ +\hline +\Axiom & y:= f(x) & f(x, y) == x*y & + {\small\tt )clear properties y f} \\ +\Derive & y:= f(x) & f(x, y):= x*y & y:= f:= \\ +\DoCon & & & \\ +\GAP & y:= f(x); & f:=function(x, y) return x*y; end; & \h{There are +no symbolic variables}\\ +\Gmp & & & \\ +\Macsyma & y: f(x); & f(x, y):= x*y; & remvalue(y)\$ \\ + & & & remfunction(f)\$ \\ +\Magnus & & & \\ +\Maxima & y: f(x); & f(x, y):= x*y; & remvalue(y)\$ \\ + & & & remfunction(f)\$ \\ +\Maple & y:= f(x); & f:= proc(x, y) x*y end; & y:= 'y': f:= 'f': \\ +\Mathematica & y = f[x] & f[x\_, y\_\,]:= x*y & Clear[y, f] \\ +\MuPAD & y:= f(x); & f:= proc(x, y) & y:= NIL: f:= NIL: \\ + & & \q\q begin x*y end\_\,proc; & \\ +\Octave & & & \\ +\Pari & & & \\ +\Reduce & y:= f(x); & procedure f(x, y); x*y; & clear y, f; \\ +\Scilab & & & \\ +\Sumit & & & \\ +\Yacas & & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|l} +& \h{Function definition with a local variable} \\ +\hline +\Axiom & f(x) == (local n; n:= 2; n*x) \\ +\Derive & \\ +\DoCon & \\ +\GAP & f:=function(x) local n; n:=2;return n*x; end; \\ +\Gmp & \\ +\Macsyma & f(x):= block([n], n: 2, n*x); \\ +\Magnus & \\ +\Maxima & f(x):= block([n], n: 2, n*x); \\ +\Maple & f:= proc(x) local n; n:= 2; n*x end; \\ +\Mathematica & f[x\_\,]:= Module[\{n\}, n = 2; n*x] \\ +\MuPAD & f:= proc(x) local n; begin n:= 2; n*x end\_\,proc; \\ +\Octave & \\ +\Pari & \\ +\Reduce & procedure f(x); begin scalar n; n:= 2; return(n*x) end; \\ +\Scilab & \\ +\Sumit & \\ +\Yacas & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& \h{Return unevaluated symbol} & \h{Define a function from an expression} \\ +\hline +\Axiom & e:= x*y;\q 'e & function(e, f, x, y) \\ +\Derive & e:= x*y\q 'e & f(x, y):== e \\ +\DoCon & & \\ +\GAP & \h{No unevaluated symbols}\fnm&\\ +\Gmp & & \\ +\Macsyma & e: x*y\$\q 'e; & define(f(x, y), e); \\ +\Magnus & & \\ +\Maxima & e: x*y\$\q 'e; & define(f(x, y), e); \\ +\Maple & e:= x*y:\q 'e'; & f:= unapply(e, x, y); \\ +\Mathematica & e = x*y;\q HoldForm[e] & f[x\_, y\_\,] = e \\ +\MuPAD & e:= x*y:\q hold(e); & f:= hold(func)(e, x, y); \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & e:= x*y\$ & for all x, y let f(x, y):= e; \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] +\footnotetext{Variables can be assigned to generators of a suitable free +object, for example {\tt x:=X(Rationals,"x");} or {\tt +f:=FreeGroup(2);x:=f.1;}.} +\addtocounter{footnote}{-1}% + +\begin{tabular}{l|ll} +& \h{Fun.\ of an indefinite number of args} & \h{Apply ``+'' to sum a list} \\ +\hline +\Axiom & & reduce(+, [1, 2]) \\ +\Derive & LST l:= l & \\ +\DoCon & & \\ +\GAP & lst:=function(args) \_ end; & Sum([1,2])\\ +\Gmp & & \\ +\Macsyma & lst([l]):= l; & apply("+", [1, 2]) \\ +\Magnus & & \\ +\Maxima & lst([l]):= l; & apply("+", [1, 2]) \\ +\Maple & lst:=proc() [args[1..nargs]] end; & convert([1, 2], \`{}+\`{}) \\ +\Mathematica & lst[l\_\,\_\,\_\,]:= \{l\} & Apply[Plus, \{1, 2\}] \\ +\MuPAD & lst:= proc(l) begin [args()] & \_\,plus(op([1, 2])) \\ + & \q\q\q end\_\,proc; & \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & & xapply(+, \{1, 2\})\,\fnm \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\footnotetext{\tt procedure xapply(f, lst); lisp(f . cdr(lst))\$} +\addtocounter{footnote}{-1}% + +\begin{tabular}{l|ll} +& \h{Apply a fun.\ to a} & \\ +& \h{list of its args} & \h{Map an anonymous function onto a list} \\ +\hline +\Axiom & reduce(f, l) & map(x +-> x + y, [1, 2]) \\ +\Derive & & x:= [1, 2] \\ + & & VECTOR(x SUB i + y, i, 1, DIMENSION(x)) \\ +\DoCon & & \\ +\GAP & List(l,f) & List([1,2],x->x+y) \\ +\Gmp & & \\ +\Macsyma & apply(f, l) & map(lambda([x], x + y), [1, 2]) \\ +\Magnus & & \\ +\Maxima & apply(f, l) & map(lambda([x], x + y), [1, 2]) \\ +\Maple & f(op(l)) & map(x -> x + y, [1, 2]) \\ +\Mathematica & Apply[f, l] & Map[\# + y \&, \{1, 2\}] \\ +\MuPAD & f(op(l)) & map([1, 2], func(x + y, x)) \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & xapply(f, l)\,\fnm & for each x in \{1, 2\} collect x + y \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|l} +& \h{Pattern matching: $f(3 y) + f(z y) \rightarrow 3 f(y) + f(z y)$} \\ +\hline +\Axiom & f:= operator('f); \\ + & ( rule f((n | integer?(n)) * x) == n*f(x) )( \_ \\ + & \q\q f(3*y) + f(z*y)) \\ +\Derive & \\ +\DoCon & \\ +\GAP & \\ +\Gmp & \\ +\Macsyma & matchdeclare(n, integerp, x, true)\$ \\ + & defrule(fnx, f(n*x), n*f(x))\$ \\ + & apply1(f(3*y) + f(z*y), fnx); \\ +\Magnus & \\ +\Maxima & matchdeclare(n, integerp, x, true)\$ \\ + & defrule(fnx, f(n*x), n*f(x))\$ \\ + & apply1(f(3*y) + f(z*y), fnx); \\ +\Maple & map(proc(q) local m; \\ + & \q\q\q if match(q = f(n*y), y, 'm') and \\ + & \q\q\q\q\q type(rhs(op(m)), integer) then \\ + & \q\q\q\q subs(m, n * f(y)) else q fi \\ + & \q\q end, \\ + & \q\q f(3*y) + f(z*y)); \\ +\Mathematica & f[3*y] + f[z*y] /. f[n\_Integer * x\_\,] -> n*f[x] \\ +\MuPAD & d:= domain("match"): \q d::FREEVARIABLE:= TRUE: \\ + & n:= new(d, "n", func(testtype(m, DOM\_INT), m)): \\ + & x:= new(d, "x", TRUE): \\ + & map(f(3*y) + f(z*y), \\ + & \q\q proc(q) local m; begin m:= match(q, f(n*x)); \\ + & \q\q\q if m = FAIL then q \\ + & \q\q\q else subs(hold("n" * f("x")), m) end\_if \\ + & \q\q end\_\,proc); \\ +\Octave & \\ +\Pari & \\ +\Reduce & operator f; \\ + & f(3*y) + f(z*y) \\ + & \q\q where \{f(\~{}n * \~{}x) => n*f(x) when fixp(n)\}; \\ +\Scilab & \\ +\Sumit & \\ +\Yacas & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|l} +& \h{Define a new infix operator and then use it} \\ +\hline +\Axiom & \\ +\Derive & \\ +\DoCon & \\ +\GAP &\h{One can overload existing infix operators for ones own purposes}\\ +\Gmp & \\ +\Macsyma & infix("\~{}")\$ \q "\~{}"(x, y):= sqrt(x\^{}2 + y\^{}2)\$ \q + 3 \~{} 4; \\ +\Magnus & \\ +\Maxima & infix("\~{}")\$ \q "\~{}"(x, y):= sqrt(x\^{}2 + y\^{}2)\$ \q + 3 \~{} 4; \\ +\Maple & \`{}\&\~{}\`{}:= (x, y) -> sqrt(x\^{}2 + y\^{}2): \q 3 \&\~{} 4; + \\ +\Mathematica & x\_ $\backslash$[Tilde] y\_:= Sqrt[x\^{}2 + y\^{}2]; \q + 3 $\backslash$[Tilde] 4 \\ +\MuPAD & tilde:= proc(x, y) begin sqrt(x\^{}2 + y\^{}2) end\_\,proc: \\ + & \q 3 \&tilde 4; \\ +\Octave & \\ +\Pari & \\ +\Reduce & infix |\$ \q procedure |(x, y); sqrt(x\^{}2 + y\^{}2)\$ \q + 3 | 4; \\ +\Scilab & \\ +\Sumit & \\ +\Yacas & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|lll} +& \h{Main expression} & & \\ +& \h{operator} & \h{\nth{1} operand} & \h{List of expression operands} \\ +\hline +\Axiom\fnm & & kernels(e) . 1 & kernels(e) \\ +\Derive & & & {\em various}\fnm \\ +\DoCon & & & \\ +\GAP &\m{3}{\rm There are no formal unevaluated expressions}\\ +\Gmp & & & \\ +\Macsyma & part(e, 0) & part(e, 1) & args(e) \\ +\Magnus & & & \\ +\Maxima & part(e, 0) & part(e, 1) & args(e) \\ +\Maple & op(0, e) & op(1, e) & [op(e)] \\ +\Mathematica & Head[e] & e@[[1]] & ReplacePart[e, List, 0] \\ +\MuPAD & op(e, 0) & op(e, 1) & [op(e)] \\ +\Octave & & & \\ +\Pari & & & \\ +\Reduce & part(e, 0) & part(e, 1) & for i:=1:arglength(e) \\ + & & & \q\q collect part(e, i) \\ +\Scilab & & & \\ +\Sumit & & & \\ +\Yacas & & & \\ +\end{tabular} \\[10pt] + +\addtocounter{footnote}{-1}% +\footnotetext{The following commands work only on expressions that consist of a +single level (e.g., $x + y + z$ but not $a/b + c/d$).} +\addtocounter{footnote}{-1}% +\footnotetext{{\tt TERMS}, {\tt FACTORS}, {\tt NUMERATOR}, {\tt LHS}, etc.} + +\begin{tabular}{l|l} +& \h{Print text and results} \\ +\hline +\Axiom & output(concat(["sin(", string(0), ") = ", \\ + & \q string(sin(0))])); \\ +\Derive & "sin(0)" = sin(0) \\ +\DoCon & \\ +\GAP & Print("There is no sin, but factors(10)= ",Factors(10), +"$\backslash$n")\\ +\Gmp & \\ +\Macsyma & print("sin(", 0, ") =", sin(0))\$ \\ +\Magnus & \\ +\Maxima & print("sin(", 0, ") =", sin(0))\$ \\ +\Maple & printf("sin(\%a) = \%a$\backslash$n", 0, sin(0)): \\ +\Mathematica & Print[StringForm["sin(\`{}\`{}) = \`{}\`{}", 0, Sin[0]]]; \\ +\MuPAD & print(Unquoted, "sin(".0.")" = sin(0)): \\ +\Octave & \\ +\Pari & \\ +\Reduce & write("sin(", 0, ") = ", sin(0))\$ \\ +\Scilab & \\ +\Sumit & \\ +\Yacas & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& \h{Generate FORTRAN} & \h{Generate \TeX/\LaTeX} \\ +\hline +\Axiom & outputAsFortran(e) & outputAsTex(e) \\ +\Derive & [Transfer Save Fortran] & \\ +\DoCon & & \\ +\GAP &&Print(LaTeX(e));\\ +\Gmp & & \\ +\Macsyma & fortran(e)\$ \OR gentran(eval(e))\$ & tex(e); \\ +\Magnus & & \\ +\Maxima & fortran(e)\$ \OR gentran(eval(e))\$ & tex(e); \\ +\Maple & fortran([e]); & latex(e); \\ +\Mathematica & FortranForm[e] & TexForm[e] \\ +\MuPAD & generate::fortran(e); & generate::TeX(e); \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & on fort; \q e; \q off fort; \OR & load\_\,package(tri)\$ \\ + & load\_\,package(gentran)\$ gentran e; & on TeX; e; off TeX; \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|l} +& \h{Import two space separated columns of integers from {\tt file}} \\ +\hline +\Axiom & \\ +\Derive & [Transfer Load daTa] ({\rm from} file.dat) \\ +\DoCon & \\ +\GAP & \\ +\Gmp & \\ +\Macsyma & xy: read\_num\_data\_to\_matrix("file", nrows, 2)\$ \\ +\Magnus & \\ +\Maxima & xy: read\_num\_data\_to\_matrix("file", nrows, 2)\$ \\ +\Maple & xy:= readdata("file", integer, 2): \\ +\Mathematica & xy = ReadList["file", Number, RecordLists -> True] \\ +\MuPAD & \\ +\Octave & \\ +\Pari & \\ +\Reduce & \\ +\Scilab & \\ +\Sumit & \\ +\Yacas & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|l} +& \h{Export two space separated columns of integers to {\tt file}\fnm} \\ +\hline +\Axiom & )set output algebra "file" \q ({\rm creates} file.spout) \\ + & for i in 1..n repeat output( \_ \\ + & \q concat([string(xy(i, 1)), " ", string(xy(i, 2))]) ) \\ + & )set output algebra console \\ +\Derive & xy [Transfer Print Expressions File]\q({\rm creates} file.prt)\\ +\DoCon & \\ +\GAP & PrintTo("file");for i in [1..n] do\\ + &\q AppendTo("file",xy[i][1]," ",xy[i][2],"$\backslash$n");od;\\ +\Gmp & \\ +\Macsyma & writefile("file")\$ \q for i:1 thru n do \\ + & \q print(xy[i, 1], xy[i, 2])\$ \q closefile()\$ \\ +\Magnus & \\ +\Maxima & writefile("file")\$ \q for i:1 thru n do \\ + & \q print(xy[i, 1], xy[i, 2])\$ \q closefile()\$ \\ +\Maple & writedata("file", xy); \\ +\Mathematica & outfile = OpenWrite["file"]; \\ + & Do[WriteString[outfile, \\ + & \q xy@[[i, 1]], " ", xy@[[i, 2]], "$\backslash$n"], \{i, 1, n\}] \\ + & Close[outfile]; \\ +\MuPAD & fprint(Unquoted, Text, "file", \\ + & \q ("$\backslash$n", xy[i, 1], xy[i, 2]) \$ i = 1..n): \\ +\Octave & \\ +\Pari & \\ +\Reduce & out "file"; \q for i:=1:n do \\ + & \q write(xy(i, 1), " ", xy(i, 2)); \q shut "file"; \\ +\Scilab & \\ +\Sumit & \\ +\Sumit & \\ +\Yacas & \\ +\end{tabular} \\[10pt] + +\footnotetext{Some editing of {\tt file} will be necessary for all systems but +\Maple\ and \Mathematica.} + +\section{Mathematics and Graphics} + +{\rm Since {\GAP} aims at discrete mathematics, it does not provide much of +the calculus functionality listed in the following section.} + +\begin{tabular}{l|llllll} +& $e$ & $\pi$ & $i$ & $+\infty$ & $\sqrt{2}$ & $2^{1/3}$ \\ +\hline +\Axiom & \%e & \%pi & \%i & \%plusInfinity & sqrt(2) & 2**(1/3) \\ +\Derive & \#e & pi & \#i & inf & SQRT(2) & 2\^{}(1/3) \\ +\DoCon & & & & & & \\ +\GAP & & & E(4) & infinity & ER(2)\fnm &\\ +\Gmp & & & & & & \\ +\Macsyma & \%e & \%pi & \%i & inf & sqrt(2) & 2\^{}(1/3) \\ +\Magnus & & & & & & \\ +\Maxima & \%e & \%pi & \%i & inf & sqrt(2) & 2\^{}(1/3) \\ +\Maple & exp(1) & Pi & I & infinity & sqrt(2) & 2\^{}(1/3) \\ +\Mathematica & E & Pi & I & Infinity & Sqrt[2] & 2\^{}(1/3) \\ +\MuPAD & E & PI & I & infinity & sqrt(2) & 2\^{}(1/3) \\ +\Octave & & & & & & \\ +\Pari & & & & & & \\ +\Reduce & e & pi & i & infinity & sqrt(2) & 2\^{}(1/3) \\ +\Scilab & & & & & & \\ +\Sumit & & & & & & \\ +\Yacas & & & & & & \\ +\end{tabular} \\[10pt] +\footnotetext{{\tt ER} represents special cyclotomic numbers and is not a +root function.} +\addtocounter{footnote}{-1}% + +\begin{tabular}{l|llll} +& \h{Euler's constant} & \h{Natural log} & \h{Arctangent} & $n!$ \\ +\hline +\Axiom & & log(x) & atan(x) & factorial(n) \\ +\Derive & euler\_\,gamma & LOG(x) & ATAN(x) & n! \\ +\DoCon & & & & \\ +\GAP & & LogInt(x,base) && Factorial(n) \\ +\Gmp & & & & \\ +\Macsyma & \%gamma & log(x) & atan(x) & n! \\ +\Magnus & & & & \\ +\Maxima & \%gamma & log(x) & atan(x) & n! \\ +\Maple & gamma & log(x) & arctan(x) & n! \\ +\Mathematica & EulerGamma & Log[x] & ArcTan[x] & n! \\ +\MuPAD & EULER & ln(x) & atan(x) & n! \\ +\Octave & & & & \\ +\Pari & & & & \\ +\Reduce & Euler\_\,Gamma & log(x) & atan(x) & factorial(n) \\ +\Scilab & & & & \\ +\Sumit & & & & \\ +\Yacas & & & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& \h{Legendre polynomial} & \h{Chebyshev poly.\ of the \nth{1} kind} \\ +\hline +\Axiom & legendreP(n, x) & chebyshevT(n, x) \\ +\Derive & LEGENDRE\_\,P(n, x) & CHEBYCHEV\_\,T(n, x) \\ +\DoCon & & \\ +\GAP & & \\ +\Gmp & & \\ +\Macsyma & legendre\_\,p(n, x) & chebyshev\_\,t(n, x) \\ +\Magnus & & \\ +\Maxima & legendre\_\,p(n, x) & chebyshev\_\,t(n, x) \\ +\Maple & orthopoly[P](n, x) & orthopoly[T](n, x) \\ +\Mathematica & LegendreP[n, x] & ChebyshevT[n, x] \\ +\MuPAD & orthpoly::legendre(n, x) & orthpoly::chebyshev1(n, x) \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & LegendreP(n, x) & ChebyshevT(n, x) \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& \h{Fibonacci number} & \h{Elliptic integral of the \nth{1} kind} \\ +\hline +\Axiom & fibonacci(n) & \\ +\Derive & FIBONACCI(n) & ELLIPTIC\_\,E(phi, k\^{}2) \\ +\DoCon & & \\ +\GAP & Fibonacci(n) & \\ +\Gmp & & \\ +\Macsyma & fib(n) & elliptic\_\,e(phi, k\^{}2) \\ +\Magnus & & \\ +\Maxima & fib(n) & elliptic\_\,e(phi, k\^{}2) \\ +\Maple & combinat[fibonacci](n) & EllipticE(sin(phi), k) \\ +\Mathematica & Fibonacci[n] & EllipticE[phi, k\^{}2] \\ +\MuPAD & numlib::fibonacci(n) & \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & & EllipticE(phi, k\^{}2) \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|llll} +& $\Gamma(x)$ & $\psi(x)$ & \h{Cosine integral} & \h{Bessel fun.\ (\nth{1})} \\ +\hline +\Axiom & Gamma(x) & psi(x) & real(Ei(\%i*x)) & besselJ(n, x) \\ +\Derive & GAMMA(x) & PSI(x) & CI(x) & BESSEL\_\,J(n, x) \\ +\DoCon & & & & \\ +\GAP & & & & \\ +\Gmp & & & & \\ +\Macsyma & gamma(x) & psi[0](x) & cos\_\,int(x) & bessel\_j[n](x) \\ +\Magnus & & & & \\ +\Maxima & gamma(x) & psi[0](x) & cos\_\,int(x) & bessel\_j[n](x) \\ +\Maple & GAMMA(x) & Psi(x) & Ci(x) & BesselJ(n, x) \\ +\Mathematica & Gamma[x] & PolyGamma[x] & CosIntegral[x] & BesselJ[n, x] \\ +\MuPAD & gamma(x) & psi(x) & & besselJ(n, x) \\ +\Octave & & & & \\ +\Pari & & & & \\ +\Reduce & Gamma(x) & Psi(x) & Ci(x) & BesselJ(n, x) \\ +\Scilab & & & & \\ +\Sumit & & & & \\ +\Yacas & & & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|lll} +& \h{Hypergeometric fun.\ ${}_2F_1(a, b; c; x)$} & \h{Dirac delta} & + \h{Unit step fun.} \\ +\hline +\Axiom & & & \\ +\Derive & GAUSS(a, b, c, x) & & STEP(x) \\ +\DoCon & & & \\ +\GAP & & & \\ +\Gmp & & & \\ +\Macsyma & hgfred([a, b], [c], x) & delta(x) & unit\_\,step(x) + \\ +\Magnus & & & \\ +\Maxima & hgfred([a, b], [c], x) & delta(x) & unit\_\,step(x) + \\ +\Maple & hypergeom([a, b], [c], x) & Dirac(x) & Heaviside(x) \\ +\Mathematica & HypergeometricPFQ[\{a,b\},\{c\},x] & + \m{2}{@<< Calculus\`{}DiracDelta\`{}} \\ +\MuPAD & & dirac(x) & heaviside(x) \\ +\Octave & & & \\ +\Pari & & & \\ +\Reduce & hypergeometric(\{a, b\}, \{c\}, x) & & \\ +\Scilab & & & \\ +\Sumit & & & \\ +\Yacas & & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|l} +& \h{Define $|x|$ via a piecewise function} \\ +\hline +\Axiom & \\ +\Derive & a(x):= -x*CHI(-inf, x, 0) + x*CHI(0, x, inf) \\ +\DoCon & \\ +\GAP & \\ +\Gmp & \\ +\Macsyma & a(x):= -x*unit\_\,step(-x) + x*unit\_\,step(x)\$ \\ +\Magnus & \\ +\Maxima & a(x):= -x*unit\_\,step(-x) + x*unit\_\,step(x)\$ \\ +\Maple & a:= x -> piecewise(x < 0, -x, x): \\ +\Mathematica & @<< Calculus\`{}DiracDelta\`{} \\ + & a[x\_]:= -x*UnitStep[-x] + x*UnitStep[x] \\ +\MuPAD & a:= proc(x) begin -x*heaviside(-x) + x*heaviside(x) \\ + & \q\q end\_\,proc: \\ +\Octave & \\ +\Pari & \\ +\Reduce & \\ +\Scilab & \\ +\Sumit & \\ +\Yacas & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& \h{Assume $x$ is real} & \h{Remove that assumption} \\ +\hline +\Axiom & & \\ +\Derive & x :epsilon Real & x:= \\ +\DoCon & & \\ +\GAP & & \\ +\Gmp & & \\ +\Macsyma & declare(x, real)\$ & remove(x, real)\$ \\ +\Magnus & & \\ +\Maxima & declare(x, real)\$ & remove(x, real)\$ \\ +\Maple & assume(x, real); & x:= 'x': \\ +\Mathematica & x/: Im[x] = 0; & Clear[x] \\ +\MuPAD & assume(x, Type::RealNum): & unassume(x, Type::RealNum): \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & & \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& \h{Assume $0 < x \le 1$} & \h{Remove that assumption} \\ +\hline +\Axiom & & \\ +\Derive & x :epsilon (0, 1] & x:= \\ +\DoCon & & \\ +\GAP & & \\ +\Gmp & & \\ +\Macsyma & assume(x > 0, x <= 1)\$ & forget(x > 0, x <= 1)\$ \\ +\Magnus & & \\ +\Maxima & assume(x > 0, x <= 1)\$ & forget(x > 0, x <= 1)\$ \\ +\Maple & assume(x > 0); & x:= 'x': \\ + & additionally(x <= 1); & \\ +\Mathematica & Assumptions -> 0 < x <= 1\,\fnm & \\ +\MuPAD & assume(x > 0): assume(x <= 1): & unassume(x): \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & & \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\footnotetext{This is an option for {\tt Integrate}.} + +\begin{tabular}{l|l} +& \h{Basic simplification of an expression $e$} \\ +\hline +\Axiom & simplify(e) \OR\ normalize(e) \OR\ complexNormalize(e) \\ +\Derive & e \\ +\DoCon & \\ +\GAP & e \\ +\Gmp & \\ +\Macsyma & ratsimp(e) \OR\ radcan(e) \\ +\Magnus & \\ +\Maxima & ratsimp(e) \OR\ radcan(e) \\ +\Maple & simplify(e) \\ +\Mathematica & Simplify[e] \OR\ FullSimplify[e] \\ +\MuPAD & simplify(e) \OR\ normal(e) \\ +\Octave & \\ +\Pari & \\ +\Reduce & e \\ +\Scilab & \\ +\Sumit & \\ +\Yacas & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& \h{Use an unknown function} & \h{Numerically evaluate an expr.} \\ +\hline +\Axiom & f:= operator('f); \q f(x) & exp(1) :: Complex Float \\ +\Derive & f(x):= & Precision:= Approximate \\ + & f(x) & APPROX(EXP(1)) \\ + & & Precision:= Exact \\ +\DoCon & & \\ +\GAP & & EvalF(123/456)\\ +\Gmp & & \\ +\Macsyma & f(x) & sfloat(exp(1)); \\ +\Magnus & & \\ +\Maxima & f(x) & sfloat(exp(1)); \\ +\Maple & f(x) & evalf(exp(1)); \\ +\Mathematica & f[x] & N[Exp[1]] \\ +\MuPAD & f(x) & float(exp(1)); \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & operator f; \q f(x) & on rounded; \q exp(1); \\ + & & off rounded; \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& $ n \bmod m$ & \h{Solve $e \equiv 0 \bmod m$ for $x$} \\ +\hline +\Axiom & rem(n, m) & solve(e = 0 :: PrimeField(m), x) \\ +\Derive & MOD(n, m) & SOLVE\_\,MOD(e = 0, x, m) \\ +\DoCon & & \\ +\GAP & n mod m & \h{solve using finite fields}\\ +\Gmp & & \\ +\Macsyma & mod(n, m) & modulus: m\$ \q solve(e = 0, x) \\ +\Magnus & & \\ +\Maxima & mod(n, m) & modulus: m\$ \q solve(e = 0, x) \\ +\Maple & n mod m & msolve(e = 0, m) \\ +\Mathematica & Mod[n, m] & Solve[\{e == 0, Modulus == m\}, x] \\ +\MuPAD & n mod m & solve(poly(e = 0, [x], IntMod(m)), x) \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & on modular; & load\_\,package(modsr)\$ \q on modular; \\ + & setmod m\$ \q n & setmod m\$ \q m\_solve(e = 0, x) \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& \h{Put over common denominator} & \h{Expand into separate fractions} \\ +\hline +\Axiom & a/b + c/d & (a*d + b*c)/(b*d) :: \_ \\ + & & \q MPOLY([a], FRAC POLY INT) \\ +\Derive & FACTOR(a/b + c/d, Trivial) & EXPAND((a*d + b*c)/(b*d)) \\ +\DoCon & & \\ +\GAP & a/b+c/d &\\ +\Gmp & & \\ +\Macsyma & xthru(a/b + c/d) & expand((a*d + b*c)/(b*d)) \\ +\Magnus & & \\ +\Maxima & xthru(a/b + c/d) & expand((a*d + b*c)/(b*d)) \\ +\Maple & normal(a/b + c/d) & expand((a*d + b*c)/(b*d)) \\ +\Mathematica & Together[a/b + c/d] & Apart[(a*d + b*c)/(b*d)] \\ +\MuPAD & normal(a/b + c/d) & expand((a*d + b*c)/(b*d)) \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & a/b + c/d & on div; (a*d + b*c)/(b*d) \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|l} +& \h{Manipulate the root of a polynomial} \\ +\hline +\Axiom & a:= rootOf(x**2 - 2); \q a**2 \\ +\Derive & \\ +\DoCon & \\ +\GAP & x:=X(Rationals,"x");\\ +&\q a:=RootOfDefiningPolynomial(AlgebraicExtension(Rationals,x\^{}2-2)); +a\^{}2\\ +\Gmp & \\ +\Macsyma & algebraic:true\$ \q tellrat(a\^{}2 - 2)\$ \q rat(a\^{}2); \\ +\Magnus & \\ +\Maxima & algebraic:true\$ \q tellrat(a\^{}2 - 2)\$ \q rat(a\^{}2); \\ +\Maple & a:= RootOf(x\^{}2 - 2): \q simplify(a\^{}2); \\ +\Mathematica & a = Root[\#\^{}2 - 2 \&, 2] \q a\^{}2 \\ +\MuPAD & \\ +\Octave & \\ +\Pari & \\ +\Reduce & load\_\,package(arnum)\$ \q defpoly(a\^{}2 - 2); \q a\^{}2; \\ +\Scilab & \\ +\Sumit & \\ +\Yacas & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& \h{Noncommutative multiplication} & \h{Solve a pair of equations} \\ +\hline +\Axiom & & solve([eqn1, eqn2], [x, y]) \\ +\Derive & x :epsilon Nonscalar & SOLVE([eqn1, eqn2], [x, y]) \\ + & y :epsilon Nonscalar & \\ + & x . y & \\ +\DoCon & & \\ +\GAP &*&\\ +\Gmp & & \\ +\Macsyma & x . y & solve([eqn1, eqn2], [x, y]) \\ +\Magnus & & \\ +\Maxima & x . y & solve([eqn1, eqn2], [x, y]) \\ +\Maple & x \&* y & solve(\{eqn1, eqn2\}, \{x, y\}) \\ +\Mathematica & x ** y & Solve[\{eqn1, eqn2\}, \{x, y\}] \\ +\MuPAD & & solve(\{eqn1, eqn2\}, \{x, y\}) \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & operator x, y; & solve(\{eqn1, eqn2\}, \{x, y\}) \\ + & noncom x, y; & \\ + & x() * y() & \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& \m{2}{\rm Decrease/increase angles in trigonometric functions} \\ +\hline +\Axiom & \m{2}{simplify(normalize(sin(2*x)))} \\ +\Derive & Trigonometry:= Expand & Trigonometry:= Collect \\ + & sin(2*x) & 2*sin(x)*cos(x) \\ +\DoCon & & \\ +\GAP & & \\ +\Gmp & & \\ +\Macsyma & trigexpand(sin(2*x)) & trigreduce(2*sin(x)*cos(x)) \\ +\Magnus & & \\ +\Maxima & trigexpand(sin(2*x)) & trigreduce(2*sin(x)*cos(x)) \\ +\Maple & expand(sin(2*x)) & combine(2*sin(x)*cos(x)) \\ +\Mathematica & TrigExpand[Sin[2*x]] & TrigReduce[2*Sin[x]*Cos[x]] \\ +\MuPAD & expand(sin(2*x)) & combine(2*sin(x)*cos(x), sincos) + \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & load\_\,package(assist)\$ \\ + & trigexpand(sin(2*x)) & trigreduce(2*sin(x)*cos(x)) \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|l} +& \h{Gr\"obner basis} \\ +\hline +\Axiom & groebner([p1, p2, ...]) \\ +\Derive & \\ +\DoCon & \\ +\GAP & \\ +\Gmp & \\ +\Macsyma & grobner([p1, p2, ...]) \\ +\Magnus & \\ +\Maxima & grobner([p1, p2, ...]) \\ +\Maple & Groebner[gbasis]([p1, p2, ...], plex(x1, x2, ...)) \\ +\Mathematica & GroebnerBasis[\{p1, p2, ...\}, \{x1, x2, ...\}] \\ +\MuPAD & groebner::gbasis([p1, p2, ...]) \\ +\Octave & \\ +\Pari & \\ +\Reduce & load\_\,package(groebner)\$ \q groebner(\{p1, p2, ...\}) \\ +\Scilab & \\ +\Sumit & \\ +\Yacas & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|l} +& \h{Factorization of $e$ over $i = \sqrt{-1}$} \\ +\hline +\Axiom & factor(e, [rootOf(i**2 + 1)]) \\ +\Derive & FACTOR(e, Complex) \\ +\DoCon & \\ +\GAP & Factors(GaussianIntegers,e)\\ +\Gmp & \\ +\Macsyma & gfactor(e); \OR\ factor(e, i\^{}2 + 1); \\ +\Magnus & \\ +\Maxima & gfactor(e); \OR\ factor(e, i\^{}2 + 1); \\ +\Maple & factor(e, I); \\ +\Mathematica & Factor[e, Extension -> I] \\ +\MuPAD & QI:= Dom::AlgebraicExtension(Dom::Rational, i\^{}2 + 1); \\ + & QI::name:= "QI": \q Factor(poly(e, QI)); \\ +\Octave & \\ +\Pari & \\ +\Reduce & on complex, factor; \q e; \q off complex, factor; \\ +\Scilab & \\ +\Sumit & \\ +\Yacas & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& \h{Real part} & \h{Convert a complex expr.\ to rectangular form} \\ +\hline +\Axiom & real(f(z)) & complexForm(f(z)) \\ +\Derive & RE(f(z)) & f(z) \\ +\DoCon & & \\ +\GAP & (f(z)+GaloisCyc(f(z),-1))/2&\\ +\Gmp & & \\ +\Macsyma & realpart(f(z)) & rectform(f(z)) \\ +\Magnus & & \\ +\Maxima & realpart(f(z)) & rectform(f(z)) \\ +\Maple & Re(f(z)) & evalc(f(z)) \\ +\Mathematica & Re[f[z]] & ComplexExpand[f[z]] \\ +\MuPAD & Re(f(z)) & rectform(f(z)) \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & repart(f(z)) & repart(f(z)) + i*impart(f(z)) \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|lll} +& \h{Matrix addition} & \h{Matrix multiplication} & \h{Matrix transpose} \\ +\hline +\Axiom & A + B & A * B & transpose(A) \\ +\Derive & A + B & A . B & A\`{} \\ +\DoCon & & & \\ +\GAP & A + B & A * B & TransposedMat(A)\\ +\Gmp & & & \\ +\Macsyma & A + B & A . B & transpose(A) \\ +\Magnus & & & \\ +\Maxima & A + B & A . B & transpose(A) \\ +\Maple & evalm(A + B) & evalm(A \&* B) & linalg[transpose](A) \\ +\Mathematica & A + B & A . B & Transpose[A] \\ +\MuPAD & A + B & A * B & transpose(A) \\ +\Octave & & & \\ +\Pari & & & \\ +\Reduce & A + B & A * B & tp(A) \\ +\Scilab & & & \\ +\Sumit & & & \\ +\Yacas & & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|l} +& \h{Solve the matrix equation $A x = b$} \\ +\hline +\Axiom & solve(A, transpose(b)) . 1 . particular :: Matrix \_\_\_ \\ +\Derive & \\ +\DoCon & \\ +\GAP & SolutionMat(TransposedMat(A),b)\\ +\Gmp & \\ +\Macsyma & xx: genvector('x, mat\_nrows(b))\$ \\ + & x: part(matlinsolve(A . xx = b, xx), 1, 2) \\ +\Magnus & \\ +\Maxima & xx: genvector('x, mat\_nrows(b))\$ \\ + & x: part(matlinsolve(A . xx = b, xx), 1, 2) \\ +\Maple & x:= linalg[linsolve](A, b) \\ +\Mathematica & x = LinearSolve[A, b] \\ +\MuPAD & \\ +\Octave & \\ +\Pari & \\ +\Reduce & \\ +\Scilab & \\ +\Sumit & \\ +\Yacas & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& \h{Sum: $\sum_{i = 1}^n f(i)$} & \h{Product: $\prod_{i = 1}^n f(i)$} \\ +\hline +\Axiom & sum(f(i), i = 1..n) & product(f(i), i = 1..n) \\ +\Derive & SUM(f(i), i, 1, n) & PRODUCT(f(i), i, 1, n) \\ +\DoCon & & \\ +\GAP & Sum([1..n],f) & Product([1..n],f)\\ +\Gmp & & \\ +\Macsyma & closedform( & closedform( \\ + & \q sum(f(i), i, 1, n)) & \q product(f(i), i, 1, n)) \\ +\Magnus & & \\ +\Maxima & closedform( & closedform( \\ + & \q sum(f(i), i, 1, n)) & \q product(f(i), i, 1, n)) \\ +\Maple & sum(f(i), i = 1..n) & product(f(i), i = 1..n) \\ +\Mathematica & Sum[f[i], \{i, 1, n\}] & Product[f[i], \{i, 1, n\}] \\ +\MuPAD & sum(f(i), i = 1..n) & product(f(i), i = 1..n) \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & sum(f(i), i, 1, n) & prod(f(i), i, 1, n) \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& \h{Limit: $\lim_{x \rightarrow 0-} f(x)$} & \h{Taylor/Laurent/etc.\ series} \\ +\hline +\Axiom & limit(f(x), x = 0, "left") & series(f(x), x = 0, 3) \\ +\Derive & LIM(f(x), x, 0, -1) & TAYLOR(f(x), x, 0, 3) \\ +\DoCon & & \\ +\GAP & & \\ +\Gmp & & \\ +\Macsyma & limit(f(x), x, 0, minus) & taylor(f(x), x, 0, 3) \\ +\Magnus & & \\ +\Maxima & limit(f(x), x, 0, minus) & taylor(f(x), x, 0, 3) \\ +\Maple & limit(f(x), x = 0, left) & series(f(x), x = 0, 4) \\ +\Mathematica & Limit[f[x], x->0, Direction->1] & Series[f[x],\{x, 0, 3\}] \\ +\MuPAD & limit(f(x), x = 0, Left) & series(f(x), x = 0, 4) \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & limit!-(f(x), x, 0) & taylor(f(x), x, 0, 3) \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& \h{Differentiate: $\frac{d^3 f(x, y)}{dx \, dy^2}$} & + \h{Integrate: $\int_0^1 f(x) \, dx$} \\ +\hline +\Axiom & D(f(x, y), [x, y], [1, 2]) & integrate(f(x), x = 0..1) \\ +\Derive & DIF(DIF(f(x, y), x), y, 2) & INT(f(x), x, 0, 1) \\ +\DoCon & & \\ +\GAP & & \\ +\Gmp & & \\ +\Macsyma & diff(f(x, y), x, 1, y, 2) & integrate(f(x), x, 0, 1) \\ +\Magnus & & \\ +\Maxima & diff(f(x, y), x, 1, y, 2) & integrate(f(x), x, 0, 1) \\ +\Maple & diff(f(x, y), x, y\$2) & int(f(x), x = 0..1) \\ +\Mathematica & D[f[x, y], x, \{y, 2\}] & Integrate[f[x], \{x, 0, 1\}] \\ +\MuPAD & diff(f(x, y), x, y\$2) & int(f(x), x = 0..1) \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & df(f(x, y), x, y, 2) & int(f(x), x, 0, 1) \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|ll} +& \h{Laplace transform} & \h{Inverse Laplace transform} \\ +\hline +\Axiom & laplace(e, t, s) & inverseLaplace(e, s, t) \\ +\Derive & LAPLACE(e, t, s) & \\ +\DoCon & & \\ +\GAP & & \\ +\Gmp & & \\ +\Macsyma & laplace(e, t, s) & ilt(e, s, t) \\ +\Magnus & & \\ +\Maxima & laplace(e, t, s) & ilt(e, s, t) \\ +\Maple & inttrans[laplace](e,t,s) & inttrans[invlaplace](e,s,t) \\ +\Mathematica & \m{2}{\q @<< Calculus\`{}LaplaceTransform\`{}} \\ + & LaplaceTransform[e, t, s] & {\st InverseLaplaceTransform[e,s,t]} + \\ +\MuPAD & transform::laplace(e,t,s) & transform::ilaplace(e, s, t) \\ +\Octave & & \\ +\Pari & & \\ +\Reduce & \m{2}{\q load\_\,package(laplace)\$ \q load\_\,package(defint)\$} + \\ + & laplace(e, t, s) & invlap(e, t, s) \\ +\Scilab & & \\ +\Sumit & & \\ +\Yacas & & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|l} +& \h{Solve an ODE (with the initial condition $y'(0) = 1$)} \\ +\hline +\Axiom & solve(eqn, y, x) \\ +\Derive & APPLY\_\,IC(RHS(ODE(eqn, x, y, y\_)), [x, 0], [y, 1]) \\ +\DoCon & \\ +\GAP & \\ +\Gmp & \\ +\Macsyma & ode\_ibc(ode(eqn, y(x), x), x = 0, diff(y(x), x) = 1) \\ +\Magnus & \\ +\Maxima & ode\_ibc(ode(eqn, y(x), x), x = 0, diff(y(x), x) = 1) \\ +\Maple & dsolve(\{eqn, D(y)(0) = 1\}, y(x)) \\ +\Mathematica & DSolve[\{eqn, y'[0] == 1\}, y[x], x] \\ +\MuPAD & solve(ode(\{eqn, D(y)(0) = 1\}, y(x))) \\ +\Octave & \\ +\Pari & \\ +\Reduce & odesolve(eqn, y(x), x) \\ +\Scilab & \\ +\Sumit & \\ +\Yacas & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|l} +& \h{Define the differential operator $L = D_x + I$ and apply it to $\sin x$} \\ +\hline +\Axiom & DD : LODO(Expression Integer, e +-> D(e, x)) := D(); \\ + & L:= DD + 1; \q L(sin(x)) \\ +\Derive & \\ +\DoCon & \\ +\GAP & \\ +\Gmp & \\ +\Macsyma & load(opalg)\$ \q L: (diffop(x) - 1)\$ \q L(sin(x)); \\ +\Magnus & \\ +\Maxima & load(opalg)\$ \q L: (diffop(x) - 1)\$ \q L(sin(x)); \\ +\Maple & id:= x -> x: \q L:= (D + id): \q L(sin)(x); \\ +\Mathematica & L = D[\#, x]\& + Identity; \q Through[L[Sin[x]]] \\ +\MuPAD & L:= (D + id): \q L(sin)(x); \\ +\Octave & \\ +\Pari & \\ +\Reduce & \\ +\Scilab & \\ +\Sumit & \\ +\Yacas & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|l} +& \h{2D plot of two separate curves overlayed} \\ +\hline +\Axiom & draw(x, x = 0..1); \q draw(acsch(x), x = 0..1); \\ +\Derive & [Plot Overlay] \\ +\DoCon & \\ +\GAP & \\ +\Gmp & \\ +\Macsyma & plot(x, x, 0, 1)\$ \q plot(acsch(x), x, 0, 1)\$ \\ +\Magnus & \\ +\Maxima & plot(x, x, 0, 1)\$ \q plot(acsch(x), x, 0, 1)\$ \\ +\Maple & plot(\{x, arccsch(x)\}, x = 0..1): \\ +\Mathematica & Plot[\{x, ArcCsch[x]\}, \{x, 0, 1\}]; \\ +\MuPAD & plotfunc(x, acsch(x), x = 0..1): \\ +\Octave & \\ +\Pari & \\ +\Reduce & load\_\,package(gnuplot)\$ \q plot(y = x, x = (0 .. 1))\$ \\ + & plot(y = acsch(x), x = (0 .. 1))\$ \\ +\Scilab & \\ +\Sumit & \\ +\Yacas & \\ +\end{tabular} \\[10pt] + +\begin{tabular}{l|l} +& \h{Simple 3D plotting} \\ +\hline +\Axiom & draw(abs(x*y), x = 0..1, y = 0..1); \\ +\Derive & [Plot Overlay] \\ +\DoCon & \\ +\GAP & \\ +\Gmp & \\ +\Macsyma & plot3d(abs(x*y), x, 0, 1, y, 0, 1)\$ \\ +\Magnus & \\ +\Maxima & plot3d(abs(x*y), x, 0, 1, y, 0, 1)\$ \\ +\Maple & plot3d(abs(x*y), x = 0..1, y = 0..1): \\ +\Mathematica & Plot3D[Abs[x*y], \{x, 0, 1\}, \{y, 0, 1\}]; \\ +\MuPAD & plotfunc(abs(x*y), x = 0..1, y = 0..1): \\ +\Octave & \\ +\Pari & \\ +\Reduce & load\_\,package(gnuplot)\$ \\ + & plot(z = abs(x*y), x = (0 .. 1), y = (0 .. 1))\$ \\ +\Scilab & \\ +\Sumit & \\ +\Yacas & \\ +\end{tabular} \\[10pt] + +%\begin{tabular}{l|l} +%& \h{} \\ +%\hline +%\Axiom & \\ +%\Derive & \\ +%\DoCon & \\ +%\GAP & \\ +%\Gmp & \\ +%\Macsyma & \\ +%\Magnus & \\ +%\Maxima & \\ +%\Maple & \\ +%\Mathematica & \\ +%\MuPAD & \\ +%\Octave & \\ +%\Pari & \\ +%\Reduce & \\ +%\Scilab & \\ +%\Sumit & \\ +%\Yacas & \\ +%\end{tabular} \\[10pt] + +\end{tt} +\endgroup +\end{document} diff --git a/src/axiom-website/screenshot2.jpg b/src/axiom-website/screenshot2.jpg new file mode 100644 index 0000000..459d9c3 Binary files /dev/null and b/src/axiom-website/screenshot2.jpg differ diff --git a/src/axiom-website/screenshot3.png b/src/axiom-website/screenshot3.png new file mode 100644 index 0000000..7f43b07 Binary files /dev/null and b/src/axiom-website/screenshot3.png differ diff --git a/src/axiom-website/screenshots.html b/src/axiom-website/screenshots.html new file mode 100644 index 0000000..eef5c8b --- /dev/null +++ b/src/axiom-website/screenshots.html @@ -0,0 +1,301 @@ + + + + Axiom Computer Algebra System + + + + +
+ [ + + Home + + ] +   + [ + + Screenshots + + ] +   + [ + + FAQ + + ] +   + [ + + Download + + ] +   + [ + + Documentation + + ] +   + [ + + Current State + + ] +   + [ + + Community + + ] +   + [ + + Developers + + ] +   + [ + + Patches + + ] +
+ +
+
+ +
+ +
    + +
  • +

    + + Some computations. An important thing: everything is + mathematically typed in Axiom. + +
    + 

    +(1) -> 1+1
+
+   (1)  2
+                                                        Type: PositiveInteger
+(2) -> integrate(1/x^(1/3),x)
+
+         3+-+2
+        3\|x
+   (2)  ------
+           2
+                                          Type: Union(Expression Integer,...)
+
    +

    +
    • + +
  • +

    + + Some matrix computations under TeXmacs. Please notice the hierarchical + editing capabilities of TeXmacs. + +
    +
    +
    +
    +

    +
    • + +
  • +

    + + Some more complicated computations: + +
    +
    + 

    +)cl all
+ 
+   All user variables and function definitions have been cleared.
+
+Word := OrderedFreeMonoid(Symbol)
+ 
+
+   (1)  OrderedFreeMonoid Symbol
+                                                                 Type: Domain
+poly:= XPR(Integer,Word)
+ 
+
+   (2)  XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
+                                                                 Type: Domain
+p:poly := 2 * x - 3 * y + 1
+ 
+
+   (3)  1 + 2x - 3y
+                      Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
+q:poly := 2 * x + 1
+ 
+
+   (4)  1 + 2x
+                      Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
+p + q
+ 
+
+   (5)  2 + 4x - 3y
+                      Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
+p * q
+ 
+
+                        2
+   (6)  1 + 4x - 3y + 4x  - 6y x
+                      Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
+(p +q)^2 -p^2 -q^2 - 2*p*q
+ 
+
+   (7)  - 6x y + 6y x
+                      Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
+M := SquareMatrix(2,Fraction Integer)
+ 
+
+   (8)  SquareMatrix(2,Fraction Integer)
+                                                                 Type: Domain
+poly1:= XPR(M,Word)
+ 
+
+   (9)
+   XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol)
+                                                                 Type: Domain
+m1:M := matrix [[i*j**2 for i in 1..2] for j in 1..2]
+ 
+
+         +1  2+
+   (10)  |    |
+         +4  8+
+                                       Type: SquareMatrix(2,Fraction Integer)
+m2:M := m1 - 5/4
+ 
+
+         +  1    +
+         |- -  2 |
+         |  4    |
+   (11)  |       |
+         |     27|
+         | 4   --|
+         +      4+
+                                       Type: SquareMatrix(2,Fraction Integer)
+m3: M := m2**2
+ 
+
+         +129     +
+         |---  13 |
+         | 16     |
+   (12)  |        |
+         |     857|
+         |26   ---|
+         +      16+
+                                       Type: SquareMatrix(2,Fraction Integer)
+pm:poly1   := m1*x + m2*y + m3*z - 2/3
+ 
+
+         +  2     +             +  1    +    +129     +
+         |- -   0 |             |- -  2 |    |---  13 |
+         |  3     |   +1  2+    |  4    |    | 16     |
+   (13)  |        | + |    |x + |       |y + |        |z
+         |       2|   +4  8+    |     27|    |     857|
+         | 0   - -|             | 4   --|    |26   ---|
+         +       3+             +      4+    +      16+
+Type: XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol)
+qm:poly1 := pm - m1*x
+ 
+
+         +  2     +   +  1    +    +129     +
+         |- -   0 |   |- -  2 |    |---  13 |
+         |  3     |   |  4    |    | 16     |
+   (14)  |        | + |       |y + |        |z
+         |       2|   |     27|    |     857|
+         | 0   - -|   | 4   --|    |26   ---|
+         +       3+   +      4+    +      16+
+Type: XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol)
+qm**3
+ 
+
+   (15)
+     +   8      +   +  1  8+    +43   52 +    +  129       +
+     |- --   0  |   |- -  -|    |--   -- |    |- ---  - 26 |
+     |  27      |   |  3  3|    | 4    3 |    |   8        | 2
+     |          | + |      |y + |        |z + |            |y
+     |         8|   |16    |    |104  857|    |         857|
+     | 0    - --|   |--   9|    |---  ---|    |- 52   - ---|
+     +        27+   + 3    +    + 3    12+    +          8 +
+   + 
+     +  3199     831 +      +  3199     831 +      +  103169     6409 +
+     |- ----   - --- |      |- ----   - --- |      |- ------   - ---- |
+     |   32       4  |      |   32       4  |      |    128        4  | 2
+     |               |y z + |               |z y + |                  |z
+     |  831     26467|      |  831     26467|      |   6409     820977|
+     |- ---   - -----|      |- ---   - -----|      | - ----   - ------|
+     +   2        32 +      +   2        32 +      +     2        128 +
+   + 
+     +3199   831 +     +103169   6409 +      +103169   6409 +
+     |----   --- |     |------   ---- |      |------   ---- |
+     | 64     8  | 3   |  256      8  | 2    |  256      8  |
+     |           |y  + |              |y z + |              |y z y
+     |831   26467|     | 6409   820977|      | 6409   820977|
+     |---   -----|     | ----   ------|      | ----   ------|
+     + 4      64 +     +   4      256 +      +   4      256 +
+   + 
+     +3178239   795341 +       +103169   6409 +       +3178239   795341 +
+     |-------   ------ |       |------   ---- |       |-------   ------ |
+     |  1024      128  |   2   |  256      8  |   2   |  1024      128  |
+     |                 |y z  + |              |z y  + |                 |z y z
+     |795341   25447787|       | 6409   820977|       |795341   25447787|
+     |------   --------|       | ----   ------|       |------   --------|
+     +  64       1024  +       +   4      256 +       +  64       1024  +
+   + 
+     +3178239   795341 +      +98625409  12326223 +
+     |-------   ------ |      |--------  -------- |
+     |  1024      128  | 2    |  4096       256   | 3
+     |                 |z y + |                   |z
+     |795341   25447787|      |12326223  788893897|
+     |------   --------|      |--------  ---------|
+     +  64       1024  +      +   128       4096  +
+Type: XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol)
+
    +

    +
    • + +
+ +
+
+ +Axiom is now running under Windows. This is a screenshot of Axiom +running on Windows in a TeXmacs window: + +
+
+ +
+ + +