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[Axiom-developer] 20080402.01.tpd.patch (CATS integration regression tes
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20080402.01.tpd.patch (CATS integration regression testing) |
|
Date: |
Thu, 3 Apr 2008 01:21:38 -0600 |
More integrals
=========================================================================
diff --git a/changelog b/changelog
index 6f6fcb0..102ba8b 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,7 @@
+20080402 tpd src/input/Makefile add integration regression testing
+20080402 tpd src/input/schaum17.input integrals of sin(ax)
+20080402 tpd src/input/schaum16.input integrals of x^n \pm a^n
+20080402 tpd src/input/schaum15.input integrals of x^4 \pm a^4
20080401 tpd src/input/Makefile add integration regression testing
20080401 tpd src/input/schaum14.input integrals of x^3+a^3
20080401 tpd src/input/schaum13.input integrals of sqrt(ax^2+bx+c)
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet
index 2f109e9..6de47f3 100644
--- a/src/input/Makefile.pamphlet
+++ b/src/input/Makefile.pamphlet
@@ -358,7 +358,8 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress
alist.regress \
schaum1.regress schaum2.regress schaum3.regress schaum4.regress \
schaum5.regress schaum6.regress schaum7.regress schaum8.regress \
schaum9.regress schaum10.regress schaum11.regress schaum12.regress \
- schaum13.regress schaum14.regress \
+ schaum13.regress schaum14.regress schaum15.regress schaum16.regress \
+ schaum17.regress \
scherk.regress scope.regress seccsc.regress \
segbind.regress seg.regress \
series2.regress series.regress sersolve.regress set.regress \
@@ -635,7 +636,8 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input
${OUT}/algfacob.input \
${OUT}/schaum5.input ${OUT}/schaum6.input ${OUT}/schaum7.input \
${OUT}/schaum8.input ${OUT}/schaum9.input ${OUT}/schaum10.input \
${OUT}/schaum11.input ${OUT}/schaum12.input ${OUT}/schaum13.input \
- ${OUT}/schaum14.input \
+ ${OUT}/schaum14.input ${OUT}/schaum15.input ${OUT}/schaum16.input \
+ ${OUT}/schaum17.input \
${OUT}/saddle.input \
${OUT}/scherk.input ${OUT}/scope.input ${OUT}/seccsc.input \
${OUT}/segbind.input ${OUT}/seg.input ${OUT}/series2.input \
@@ -941,6 +943,8 @@ DOCFILES= \
${DOC}/schaum9.input.dvi ${DOC}/schaum10.input.dvi \
${DOC}/schaum11.input.dvi ${DOC}/schaum12.input.dvi \
${DOC}/schaum13.input.dvi ${DOC}/schaum14.input.dvi \
+ ${DOC}/schaum15.input.dvi ${DOC}/schaum16.input.dvi \
+ ${DOC}/schaum17.input.dvi \
${DOC}/s01eaf.input.dvi ${DOC}/s13aaf.input.dvi \
${DOC}/s13acf.input.dvi ${DOC}/s13adf.input.dvi \
${DOC}/s14aaf.input.dvi ${DOC}/s14abf.input.dvi \
diff --git a/src/input/schaum15.input.pamphlet
b/src/input/schaum15.input.pamphlet
new file mode 100644
index 0000000..51a6094
--- /dev/null
+++ b/src/input/schaum15.input.pamphlet
@@ -0,0 +1,409 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/input schaum15.input}
+\author{Timothy Daly}
+\maketitle
+\eject
+\tableofcontents
+\eject
+\section{\cite{1}:14.311~~~~~$\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 of 14
+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
+@
+
+\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 2 of 14
+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
+@
+
+\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 3 of 14
+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
+@
+
+\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 4 of 14
+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
+@
+
+\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 5 of 14
+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
+@
+
+\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 6 of 14
+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
+@
+
+\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 7 of 14
+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
+@
+
+\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 8 of 14
+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
+@
+
+\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 9 of 14
+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
+@
+
+\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 10 of 14
+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
+@
+
+\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 11 of 14
+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
+@
+
+\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 12 of 14
+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
+@
+
+\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 13 of 14
+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
+@
+
+\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 14 of 14
+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
+
+)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
+\end{thebibliography}
+\end{document}
diff --git a/src/input/schaum16.input.pamphlet
b/src/input/schaum16.input.pamphlet
new file mode 100644
index 0000000..2246f9a
--- /dev/null
+++ b/src/input/schaum16.input.pamphlet
@@ -0,0 +1,394 @@
+\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 of 14
+aa:=integrate(1/x*(x^n+a^n),x)
+--R
+--R
+--R n log(x) n
+--R %e + n log(x)a
+--R (1) -----------------------
+--R n
+--R Type: Union(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 2 of 14
+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
+@
+
+\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 3 of 14
+aa:=integrate(x^m/(x^n+a^n)^r,x)
+--R
+--R
+--R x m
+--R ++ %J
+--R (1) | ----------- d%J
+--R ++ n n r
+--R (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 4 of 14
+aa:=integrate(1/(x^m*(x^n+a^n)^r),x)
+--R
+--R
+--R x
+--R ++ 1
+--R (1) | -------------- d%J
+--R ++ m n n r
+--R %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 5 of 14
+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
+@
+
+\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 6 of 14
+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
+@
+
+\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 7 of 14
+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
+@
+
+\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 8 of 14
+aa:=integrate(x^m/(x^n-a^n)^r,x)
+--R
+--R
+--R x m
+--R ++ %J
+--R (1) | ------------- d%J
+--R ++ n n r
+--R (- 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 9 of 14
+aa:=integrate(1/(x^m*(x^n-a^n)^r),x)
+--R
+--R
+--R x
+--R ++ 1
+--R (1) | ---------------- d%J
+--R ++ m n n r
+--R %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 10 of 14
+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
+@
+
+\section{\cite{1}:14.335~~~~~$\displaystyle
+\int{\frac{x^{p-1}~dx}{x^{2m}+a^{2m}}}$ provided $0<p\le 2m$}
+$$\begin{array}{rl}
+\displaystyle\int{\frac{x^{p-1}}{x^{2m}+a^{2m}}}=
+&\displaystyle\frac{1}{ma^{2m-p}}\sum_{k=1}^m{\sin\frac{(2k-1)p\pi}{2m}
+\tan^{-1}\left(\frac{x+a\cos\left((2k-1)\pi/2m\right)}
+{a\sin\left((2k-1)\pi/2m\right)}\right)}\\
+&\\
+&\displaystyle-\frac{1}{2ma^{2m-p}}\sum_{k=1}^m{\cos\frac{(2k-1)p\pi}{2m}
+\ln\left(x^2+2ax\cos\frac{(2k-1)\pi}{2m}+a^2\right)}
+\end{array}
+$$
+<<*>>=
+)clear all
+
+--S 11 of 14
+aa:=integrate(x^(p-1)/(x^(2*m)+a^(2*m)),x)
+--R
+--R
+--R x p - 1
+--R ++ %J
+--R (1) | ---------- d%J
+--R ++ 2m 2m
+--R 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 $0<p\le 2m$}
+$$\begin{array}{rl}
+\displaystyle
+\int{\frac{x^{p-1}}{x^{2m}-a^{2m}}}=
+&\displaystyle\frac{1}{2ma^{2m-p}}\sum_{k=1}^{m-1}\cos\frac{kp\pi}{m}
+\ln\left(x^2-2ax\cos\frac{k\pi}{m}+a^2\right)\\
+&\\
+&\displaystyle-\frac{1}{ma^{2m-p}}\sum_{k=1}^{m-1}\sin\frac{kp\pi}{m}
+\tan^{-1}\left(\frac{x-a\cos(k\pi/m)}{a\sin(k\pi/m)}\right)\\
+&\\
+&\displaystyle+\frac{1}{2ma^{2m-p}}\left(\ln(x-a)+(-1)^p\ln(x+a)\right)
+\end{array}
+$$
+<<*>>=
+)clear all
+
+--S 12 of 14
+aa:=integrate(x^(p-1)/(x^(2*m)-a^(2*m)),x)
+--R
+--R
+--R x p - 1
+--R ++ %J
+--R (1) | - ---------- d%J
+--R ++ 2m 2m
+--R 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 $0<p\le 2m+1$}
+$$\begin{array}{rl}
+\displaystyle\int{\frac{x^{p-1}}{x^{2m+1}+a^{2m+1}}}=&\hbox{\hskip 6.5cm}
+\end{array}
+$$
+$$\begin{array}{rl}
+\hbox{\hskip 1cm}&\displaystyle
+\frac{2(-1)^{p-1}}{(2m+1)a^{2m-p+1}}\sum_{k=1}^m{\sin\frac{2kp\pi}{2m+1}
+\tan^{-1}\left(\frac{x+a\cos\left(2k\pi/(2m+1)\right)}
+{a\sin\left(2k\pi/(2m+1)\right)}\right)}\\
+&\displaystyle
+-\frac{(-1)^{p-1}}{(2m+1)a^{2m-p+1}}\sum_{k=1}^m{\cos\frac{2kp\pi}{2m+1}
+\ln\left(x^2+2ax\cos\frac{2k\pi}{2m+1}+a^2\right)}\\
+&\\
+&\displaystyle+\frac{(-1)^{p-1}\ln(x+a)}{(2m+1)a^{2m-p+1}}
+\end{array}
+$$
+<<*>>=
+)clear all
+
+--S 13 of 14
+aa:=integrate(x^(p-1)/(x^(2*m+1)+a^(2*m+1)),x)
+--R
+--R
+--R x p - 1
+--R ++ %J
+--R (1) | ------------------ d%J
+--R ++ 2m + 1 2m + 1
+--R 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 $0<p\le 2m+1$}
+$$\begin{array}{rl}
+\displaystyle\int{\frac{x^{p-1}}{x^{2m+1}-a^{2m+1}}}=&\hbox{\hskip 6cm}
+\end{array}
+$$
+$$\begin{array}{rl}
+\hbox{\hskip 1cm}&\displaystyle
+\frac{2}{(2m+1)a^{2m-p+1}}\sum_{k=1}^m{\sin\frac{2kp\pi}{2m+1}
+\tan^{-1}\left(\frac{x-a\cos\left(2kp\pi/(2m+1)\right)}
+{a\sin\left(2k\pi/(2m+1)\right)}\right)}\\
+&\\
+&\hbox{\hskip 1cm}\displaystyle
++\frac{1}{(2m+1)a^{2m-p+1}}\sum_{k=1}^m{\cos\frac{2kp\pi}{2m+1}
+\ln\left(x^2-2ax\cos\frac{2k\pi}{2m+1}+a^2\right)}\\
+&\\
+&\hbox{\hskip 1cm}\displaystyle
++\frac{\ln(x-a)}{(2m+1)a^{2m-p+1}}
+\end{array}
+$$
+<<*>>=
+)clear all
+
+--S 14 of 14
+aa:=integrate(x^(p-1)/(x^(2*m+1)-a^(2*m+1)),x)
+--R
+--R
+--R x p - 1
+--R ++ %J
+--R (1) | - ------------------ d%J
+--R ++ 2m + 1 2m + 1
+--R 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/input/schaum17.input.pamphlet
b/src/input/schaum17.input.pamphlet
new file mode 100644
index 0000000..7ab11d7
--- /dev/null
+++ b/src/input/schaum17.input.pamphlet
@@ -0,0 +1,779 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/input schaum17.input}
+\author{Timothy Daly}
+\maketitle
+\eject
+\tableofcontents
+\eject
+\section{\cite{1}:14.339~~~~~$\displaystyle
+\int{\sin ax ~dx}$}
+$$\int{\sin ax}=
+-\frac{\cos{ax}}{a}
+$$
+<<*>>=
+)spool schaum17.output
+)set message test on
+)set message auto off
+)clear all
+
+--S 1 of 30
+aa:=integrate(sin(a*x),x)
+--R
+--R
+--R cos(a x)
+--R (1) - --------
+--R a
+--R Type: Union(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 2 of 30
+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
+@
+
+\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 3 of 30
+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
+@
+
+\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 4 of 30
+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
+@
+
+\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 5 of 30
+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 6 of 30
+aa:=integrate(sin(a*x)/x^2,x)
+--R
+--R
+--R x
+--R ++ sin(%I a)
+--R (1) | --------- d%I
+--R ++ 2
+--R %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 7 of 30
+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
+@
+
+\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 8 of 30
+aa:=integrate(x/sin(a*x),x)
+--R
+--R
+--R x
+--R ++ %I
+--R (1) | --------- d%I
+--R ++ 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 9 of 30
+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
+@
+
+\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 10 of 30
+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
+@
+
+\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 11 of 30
+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
+@
+
+\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 12 of 30
+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
+@
+
+\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 13 of 30
+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
+@
+
+\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 14 of 30
+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
+@
+
+\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 15 of 30
+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
+@
+
+\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 16 of 30
+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
+@
+
+\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 17 of 30
+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
+@
+
+\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 18 of 30
+aa:=integrate(1/(1+sin(ax)),x)
+--R
+--R
+--R x
+--R (1) -----------
+--R sin(ax) + 1
+--R Type: Union(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 19 of 30
+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
+@
+
+\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 20 of 30
+aa:=integrate(1/(1-sin(a*x))^2,x)
+--R
+--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
+@
+
+\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 21 of 30
+aa:=integrate(1/(1+sin(a*x))^2,x)
+--R
+--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
+@
+
+\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 22 of 30
+aa:=integrate(1/(p+q*sin(a*x)),x)
+--R
+--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
+@
+
+\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 23 of 30
+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
+@
+
+\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 24 of 30
+aa:=integrate(1/(p^2+a^2*sin(a*x)),x)
+--R
+--R
+--R (1)
+--R [
+--R log
+--R +---------+
+--R 2 2 4 4 4 | 4 4
+--R (a p sin(a x) + (- p + a )cos(a x) + a )\|- p + a
+--R +
+--R 6 4 2 2 4 6 2 4 6
+--R (p - a p )sin(a x) + (a p - a )cos(a x) + a p - a
+--R /
+--R 2 2
+--R a sin(a x) + p
+--R /
+--R +---------+
+--R | 4 4
+--R a\|- p + a
+--R ,
+--R +-------+
+--R 2 2 2 | 4 4
+--R (p sin(a x) + a cos(a x) + a )\|p - a
+--R 2atan(----------------------------------------)
+--R 4 4 4 4
+--R (p - a )cos(a x) + p - a
+--R -----------------------------------------------]
+--R +-------+
+--R | 4 4
+--R a\|p - a
+--R Type: Union(List Expression
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 25 of 30
+aa:=integrate(1/(p^2-q^2*sin(a*x)^2),x)
+--R
+--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
+@
+
+\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 26 of 30
+aa:=integrate(x^m*sin(a*x),x)
+--R
+--R
+--R x
+--R ++ m
+--R (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 26 of 30
+aa:=integrate(sin(a*x)/x^n,x)
+--R
+--R
+--R x
+--R ++ sin(%I a)
+--R (1) | --------- d%I
+--R ++ n
+--R %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 28 of 30
+aa:=integrate(sin(a*x)^n,x)
+--R
+--R
+--R x
+--R ++ n
+--R (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 29 of 30
+aa:=integrate(1/(sin(a*x))^n,x)
+--R
+--R
+--R x
+--R ++ 1
+--R (1) | ---------- d%I
+--R ++ n
+--R 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 30 of 30
+aa:=integrate(x/sin(a*x)^n,x)
+--R
+--R
+--R x
+--R ++ %I
+--R (1) | ---------- d%I
+--R ++ n
+--R 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}
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