diff --git a/changelog b/changelog index bb20757..26d39fa 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,9 @@ +20080413 tpd src/input/Makefile add integration regression testing +20080413 tpd src/input/schaum30.input integrals of tanh(ax) +20080413 tpd src/input/schaum29.input integrals of sinh(ax) and cosh(ax) +20080413 tpd src/input/schaum28.input integrals of cosh(ax) +20080413 tpd src/input/schaum27.input integrals of sinh(ax) +20080413 tpd src/input/schaum26.input integrals of ln x 20080409 tpd src/input/Makefile add integration regression testing 20080409 tpd src/input/schaum25.input integrals of e^(a*x) 20080409 tpd readme add Max Tegmark diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index 9134ab7..57ff1e9 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -361,7 +361,8 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress alist.regress \ schaum13.regress schaum14.regress schaum15.regress schaum16.regress \ schaum17.regress schaum18.regress schaum19.regress schaum20.regress \ schaum21.regress schaum22.regress schaum23.regress schaum24.regress \ - schaum25.regress \ + schaum25.regress schaum26.regress schaum27.regress schaum28.regress \ + schaum29.regress schaum30.regress \ scherk.regress scope.regress seccsc.regress \ segbind.regress seg.regress \ series2.regress series.regress sersolve.regress set.regress \ @@ -642,6 +643,8 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input ${OUT}/algfacob.input \ ${OUT}/schaum17.input ${OUT}/schaum18.input ${OUT}/schaum19.input \ ${OUT}/schaum20.input ${OUT}/schaum21.input ${OUT}/schaum22.input \ ${OUT}/schaum23.input ${OUT}/schaum24.input ${OUT}/schaum25.input \ + ${OUT}/schaum26.input ${OUT}/schaum27.input ${OUT}/schaum28.input \ + ${OUT}/schaum29.input ${OUT}/schaum30.input \ ${OUT}/saddle.input \ ${OUT}/scherk.input ${OUT}/scope.input ${OUT}/seccsc.input \ ${OUT}/segbind.input ${OUT}/seg.input ${OUT}/series2.input \ @@ -952,7 +955,9 @@ DOCFILES= \ ${DOC}/schaum19.input.dvi ${DOC}/schaum20.input.dvi \ ${DOC}/schaum21.input.dvi ${DOC}/schaum22.input.dvi \ ${DOC}/schaum23.input.dvi ${DOC}/schaum24.input.dvi \ - ${DOC}/schaum25.input.dvi \ + ${DOC}/schaum25.input.dvi ${DOC}/schaum26.input.dvi \ + ${DOC}/schaum27.input.dvi ${DOC}/schaum28.input.dvi \ + ${DOC}/schaum29.input.dvi ${DOC}/schaum30.input.dvi \ ${DOC}/s01eaf.input.dvi ${DOC}/s13aaf.input.dvi \ ${DOC}/s13acf.input.dvi ${DOC}/s13adf.input.dvi \ ${DOC}/s14aaf.input.dvi ${DOC}/s14abf.input.dvi \ diff --git a/src/input/schaum26.input.pamphlet b/src/input/schaum26.input.pamphlet new file mode 100644 index 0000000..7bf7244 --- /dev/null +++ b/src/input/schaum26.input.pamphlet @@ -0,0 +1,324 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum26.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.525~~~~~$\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 of 16 +aa:=integrate(log(x),x) +--R +--R +--R (1) x log(x) - x +--R Type: Union(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 2 of 16 +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 +@ + +\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 3 of 16 +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 +@ + +\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 4 of 16 +aa:=integrate(log(x)/x,x) +--R +--R +--R 2 +--R log(x) +--R (1) ------- +--R 2 +--R Type: Union(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 5 of 16 +aa:=integrate(log(x)/x^2,x) +--R +--R +--R - log(x) - 1 +--R (1) ------------ +--R x +--R Type: Union(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 6 of 16 +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 +@ + +\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 7 of 16 +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 +@ + +\section{\cite{1}:14.532~~~~~$\displaystyle +\int{\frac{dx}{x\ln{x}}}$} +$$\int{\frac{1}{x\ln{x}}}= +\ln(\ln{x}) +$$ +<<*>>= +)clear all + +--S 8 of 16 +aa:=integrate(1/(x*log(x)),x) +--R +--R +--R (1) log(log(x)) +--R Type: Union(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 9 of 16 +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 10 of 16 +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 11 of 16 +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 12 of 16 +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 13 of 16 +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 +@ + +\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 14 of 16 +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 +@ + +\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 15a of 16 +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 15b of 16 +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/input/schaum27.input.pamphlet b/src/input/schaum27.input.pamphlet new file mode 100644 index 0000000..bb9f4d2 --- /dev/null +++ b/src/input/schaum27.input.pamphlet @@ -0,0 +1,619 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum27.input} +\author{Timothy 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 of 22 +aa:=integrate(sinh(a*x),x) +--R +--R cosh(a x) +--R (1) --------- +--R a +--R Type: Union(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 2 of 22 +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 +@ + +\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 3 of 22 +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 +@ + +\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 4 of 22 +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 5 of 22 +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 6 of 22 +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 +@ + +\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 7 of 22 +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 8 of 22 +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 +@ + +\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 9 of 22 +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 +@ + +\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 10 of 22 +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 +@ + +\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 11 of 22 +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 +@ + +\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 12 of 22 +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 +@ + +\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 13 of 22 +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 +@ + +\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 14 of 22 +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 +@ + +\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 15 of 22 +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 +@ + +\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 16 of 22 +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 +@ + +\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 17 of 22 +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 +@ + +\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 18 of 22 +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 19 of 22 +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 20 of 22 +aa:=integrate(sinh(a*x)/a^n,x) +--R +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1 +--R (1) ------------------------------------------------- +--R n +--R (2a sinh(a x) + 2a cosh(a x))a +--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 21 of 22 +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 22 of 22 +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/input/schaum28.input.pamphlet b/src/input/schaum28.input.pamphlet new file mode 100644 index 0000000..b3fc3c6 --- /dev/null +++ b/src/input/schaum28.input.pamphlet @@ -0,0 +1,846 @@ +\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 of 28 +aa:=integrate(cosh(a*x),x) +--R +--R +--R sinh(a x) +--R (1) --------- +--R a +--R Type: Union(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 2 of 28 +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 +@ + +\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 3 of 28 +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 +@ + +\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 4 of 28 +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 5 of 28 +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 6 of 28 +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 +@ + +\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 7 of 28 +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~~~~~$\displaystyle +\int{\cosh^2{ax}}~dx$} +$$\int{\cosh^2{ax}}= +\frac{x}{2}+\frac{\sinh{ax}\cosh{ax}}{2} +$$ +<<*>>= +)clear all + +--S 8 of 28 +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 +@ + +\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 9 of 28 +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 +@ + +\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 10 of 28 +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 +@ + +\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 11 of 28 +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 +@ + +\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 12 of 28 +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 +@ + +\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 13 of 28 +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 +@ + +\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 14 of 28 +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 +@ + +\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 15 of 28 +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 +@ + +\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 16 of 28 +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 +@ + +\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 17 of 28 +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 +@ + +\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 18 of 28 +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 +@ + +\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 19 of 28 +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 +@ + +\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 20 of 28 +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 +@ + +\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 21 of 28 +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 +@ + +\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 22 of 28 +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 +@ + +\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 23 of 28 +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 +@ + +\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 24 of 28 +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 25 of 28 +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 26 of 28 +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 27 of 28 +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 28 of 28 +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/input/schaum29.input.pamphlet b/src/input/schaum29.input.pamphlet new file mode 100644 index 0000000..37e2800 --- /dev/null +++ b/src/input/schaum29.input.pamphlet @@ -0,0 +1,365 @@ +\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 of 14 +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 +@ + +\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 2 of 14 +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 +@ + +\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 3 of 14 +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 +@ + +\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 4 of 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 +@ + +\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 5 of 14 +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 +@ + +\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 6 of 14 +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 +@ + +\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 7 of 14 +aa:=integrate(1/(sinh(a*x)^2*cos(a*x)),x) +--R +--R +--R x +--R ++ 1 +--I (1) | -------------------- d%R +--R ++ 2 +--I cos(%R a)sinh(%R a) +--R Type: Union(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 8 of 14 +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 +@ + +\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 9 of 14 +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 +@ + +\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 10 of 14 +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 +@ + +\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 11 of 14 +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 +@ + +\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 12 of 14 +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 +@ + +\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 13 of 14 +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 +@ + +\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 14 of 14 +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 + +)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/input/schaum30.input.pamphlet b/src/input/schaum30.input.pamphlet new file mode 100644 index 0000000..0e1aaf4 --- /dev/null +++ b/src/input/schaum30.input.pamphlet @@ -0,0 +1,286 @@ +\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 of 11 +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 +@ + +\section{\cite{1}:14.605~~~~~$\displaystyle +\int{\tanh^2{ax}}~dx$} +$$\int{\tanh^2{ax}}= +x-\frac{\tanh{ax}}{a} +$$ +<<*>>= +)clear all + +--S 2 of 11 +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 +@ + +\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 3 of 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 +@ + +\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 4 of 11 +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 +@ + +\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 5 of 11 +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 +@ + +\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 6 of 11 +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 +@ + +\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 7 of 11 +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 8 of 11 +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 +@ + +\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 9 of 11 +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 10 of 11 +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 +@ + +\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 11 of 11 +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 +\end{thebibliography} +\end{document}