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Examples from Timofeev Chapter 3 and Chapter 7

fricas
INT ==> integrate
Type: Void
fricas
pi := %pi

\label{eq1}\pi(1)
Type: Pi
fricas
SIN ==> sin
Type: Void
fricas
COS ==> cos
Type: Void
fricas
TAN ==> tan
Type: Void
fricas
COT ==> cot
Type: Void
fricas
SEC ==> sec
Type: Void
fricas
CSC ==> csc
Type: Void
fricas
LN ==> log
Type: Void
fricas
SQRT ==> sqrt
Type: Void
fricas
ATAN ==> atan
Type: Void
fricas
ACOS ==> acos
Type: Void
fricas
ASIN ==> asin
Type: Void
fricas
ATANH ==> atanh
Type: Void
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)boot $IOindex := 1
(EVAL-WHEN (EVAL LOAD) (SETQ |$IOindex| 1)) Value = 1
-- " Timofeev (1948) Ch. 3, examples 1 - 3 (p. 101) ... "
INT(1/((x-2)^3*(x+1)^2),x)

\label{eq2}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left(-{2 \ {{x}^{3}}}+{6 \ {{x}^{2}}}- 8 \right)}\ {\log \left({x + 1}\right)}}+ 
\
\
\displaystyle
{{\left({2 \ {{x}^{3}}}-{6 \ {{x}^{2}}}+ 8 \right)}\ {\log \left({x - 2}\right)}}+{6 \ {{x}^{2}}}-{{15}\  x}- 3 
(2)
Type: Union(Expression(Integer),...)
fricas
INT(1/((x+2)^3*(x+3)^4),x)

\label{eq3}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
-{{60}\ {{x}^{5}}}-{{780}\ {{x}^{4}}}-{{4020}\ {{x}^{3}}}- 
\
\
\displaystyle
{{10260}\ {{x}^{2}}}-{{12960}\  x}-{6480}
(3)
Type: Union(Expression(Integer),...)
fricas
INT(x^5/(3+x)^2,x)

\label{eq4}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({{1620}\  x}+{4860}\right)}\ {\log \left({x + 3}\right)}}+{{x}^{5}}-{5 \ {{x}^{4}}}+{{30}\ {{x}^{3}}}- 
\
\
\displaystyle
{{270}\ {{x}^{2}}}-{{1296}\  x}+{972}
(4)
Type: Union(Expression(Integer),...)
fricas
-- " Timofeev (1948) Ch. 3, examples 4 - 9 (p. 105) ... "
[]

\label{eq5}\left[ \right](5)
Type: List(None)
fricas
[]

\label{eq6}\left[ \right](6)
Type: List(None)
fricas
INT(x/(3+6*x+2*x^2),x)

\label{eq7}{\left(
\begin{array}{@{}l}
\displaystyle
{{\sqrt{3}}\ {\log \left({{{{\left({2 \  x}+ 3 \right)}\ {\sqrt{3}}}+{2 \ {{x}^{2}}}+{6 \  x}+ 6}\over{{2 \ {{x}^{2}}}+{6 \  x}+ 3}}\right)}}+ 
\
\
\displaystyle
{\log \left({{2 \ {{x}^{2}}}+{6 \  x}+ 3}\right)}
(7)
Type: Union(Expression(Integer),...)
fricas
INT((2*x-3)/(3+6*x+2*x^2)^3,x)

\label{eq8}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{8 \ {{x}^{4}}}+{{48}\ {{x}^{3}}}+ 
\
\
\displaystyle
{{96}\ {{x}^{2}}}+{{72}\  x}+{18}
(8)
Type: Union(Expression(Integer),...)
fricas
INT((x-1)/(x^2+5*x+4)^2,x)

\label{eq9}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left(-{7 \ {{x}^{2}}}-{{35}\  x}-{28}\right)}\ {\log \left({x + 4}\right)}}+ 
\
\
\displaystyle
{{\left({7 \ {{x}^{2}}}+{{35}\  x}+{28}\right)}\ {\log \left({x + 1}\right)}}+{{21}\  x}+{39}
(9)
Type: Union(Expression(Integer),...)
fricas
INT(1/(x^2+3*x+2)^5,x)

\label{eq10}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
-{{840}\ {{x}^{8}}}-{{10080}\ {{x}^{7}}}-{{52080}\ {{x}^{6}}}- 
\
\
\displaystyle
{{151200}\ {{x}^{5}}}-{{269640}\ {{x}^{4}}}- 
\
\
\displaystyle
{{302400}\ {{x}^{3}}}-{{208320}\ {{x}^{2}}}-{{80640}\  x}- 
\
\
\displaystyle
{13440}
(10)
Type: Union(Expression(Integer),...)
fricas
INT(1/(x^3*(7-6*x+2*x^2)^2),x)

\label{eq11}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left(-{{800}\ {{x}^{4}}}+{{2400}\ {{x}^{3}}}-{{2800}\ {{x}^{2}}}\right)}\ {\sqrt{5}}\ {\log \left({{2 \ {{x}^{2}}}-{6 \  x}+ 7}\right)}}+ 
\
\
\displaystyle
{{\left({{1600}\ {{x}^{4}}}-{{4800}\ {{x}^{3}}}+{{5600}\ {{x}^{2}}}\right)}\ {\sqrt{5}}\ {\log \left({x}\right)}}+ 
\
\
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{936}\ {{x}^{4}}}-{{2808}\ {{x}^{3}}}+ 
\
\
\displaystyle
{{3276}\ {{x}^{2}}}
(11)
Type: Union(Expression(Integer),...)
fricas
INT(x^9/(x^2+3*x+2)^5,x)

\label{eq12}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{17664}\ {{x}^{8}}}+{{211968}\ {{x}^{7}}}+ 
\
\
\displaystyle
{{1095168}\ {{x}^{6}}}+{{3179520}\ {{x}^{5}}}+ 
\
\
\displaystyle
{{5670144}\ {{x}^{4}}}+{{6359040}\ {{x}^{3}}}+ 
\
\
\displaystyle
{{4380672}\ {{x}^{2}}}+{{1695744}\  x}+{282624}
(12)
Type: Union(Expression(Integer),...)
fricas
INT((1+2*x)^2/(3+5*x+2*x^2)^5,x)

\label{eq13}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
-{{476160}\ {{x}^{8}}}-{{4761600}\ {{x}^{7}}}- 
\
\
\displaystyle
{{20712960}\ {{x}^{6}}}-{{51187200}\ {{x}^{5}}}- 
\
\
\displaystyle
{{78596160}\ {{x}^{4}}}-{{76780800}\ {{x}^{3}}}- 
\
\
\displaystyle
{{46604160}\ {{x}^{2}}}-{{16070400}\  x}- 
\
\
\displaystyle
{2410560}
(13)
Type: Union(Expression(Integer),...)
fricas
-- " Timofeev (1948) Ch. 3, examples 13 - 14 (p. 113) ... "
INT((a-b*x^2)^3/x^7,x)

\label{eq14}{-{{12}\ {{b}^{3}}\ {{x}^{6}}\ {\log \left({x}\right)}}-{{18}\  a \ {{b}^{2}}\ {{x}^{4}}}+{9 \ {{a}^{2}}\  b \ {{x}^{2}}}-{2 \ {{a}^{3}}}}\over{{12}\ {{x}^{6}}}(14)
Type: Union(Expression(Integer),...)
fricas
INT(x^13/(a^4+x^4)^5,x)

\label{eq15}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{15}\ {{x}^{16}}}+{{60}\ {{a}^{4}}\ {{x}^{12}}}+{{90}\ {{a}^{8}}\ {{x}^{8}}}+ 
\
\
\displaystyle
{{60}\ {{a}^{12}}\ {{x}^{4}}}+{{15}\ {{a}^{16}}}
(15)
Type: Union(Expression(Integer),...)
fricas
-- " ... end of Timofeev Ch. 3 "
fricas
)boot $IOindex := 1
(EVAL-WHEN (EVAL LOAD) (SETQ |$IOindex| 1)) Value = 1
-- " Timofeev (1948) Ch. 7, examples 1 - 4 (p. 334) ... "
INT(x^2*COS(x)^5,x)

\label{eq16}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{\left({{675}\ {{x}^{2}}}-{54}\right)}\ {{\cos \left({x}\right)}^{4}}}+ 
\
\
\displaystyle
{{\left({{900}\ {{x}^{2}}}-{272}\right)}\ {{\cos \left({x}\right)}^{2}}}+{{1800}\ {{x}^{2}}}- 
\
\
\displaystyle
{4144}
(16)
Type: Union(Expression(Integer),...)
fricas
INT(x^3*SIN(x)^3,x)

\label{eq17}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({{\left(-{9 \ {{x}^{2}}}+ 2 \right)}\ {{\cos \left({x}\right)}^{2}}}+{{63}\ {{x}^{2}}}-{122}\right)}\ {\sin \left({x}\right)}}+ 
\
\
\displaystyle
{{\left({9 \ {{x}^{3}}}-{6 \  x}\right)}\ {{\cos \left({x}\right)}^{3}}}+{{\left(-{{27}\ {{x}^{3}}}+{{126}\  x}\right)}\ {\cos \left({x}\right)}}
(17)
Type: Union(Expression(Integer),...)
fricas
INT(x^2*SIN(x)^6,x)

\label{eq18}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{\left(-{{576}\ {{x}^{2}}}+{32}\right)}\ {{\cos \left({x}\right)}^{5}}}+ 
\
\
\displaystyle
{{\left({{1872}\ {{x}^{2}}}-{194}\right)}\ {{\cos \left({x}\right)}^{3}}}+ 
\
\
\displaystyle
{{\left(-{{2376}\ {{x}^{2}}}+{897}\right)}\ {\cos \left({x}\right)}}
(18)
Type: Union(Expression(Integer),...)
fricas
INT(x^2*SIN(x)^2*COS(x),x)

\label{eq19}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({{\left(-{9 \ {{x}^{2}}}+ 2 \right)}\ {{\cos \left({x}\right)}^{2}}}+{9 \ {{x}^{2}}}-{14}\right)}\ {\sin \left({x}\right)}}- 
\
\
\displaystyle
{6 \  x \ {{\cos \left({x}\right)}^{3}}}+{{18}\  x \ {\cos \left({x}\right)}}
(19)
Type: Union(Expression(Integer),...)
fricas
-- " Timofeev (1948) Ch. 7, examples 5 - 9 (p. 342-343) ... "
INT(x*COS(x)^4/SIN(x)^2,x)

\label{eq20}{\left(
\begin{array}{@{}l}
\displaystyle
{4 \ {\sin \left({x}\right)}\ {\log \left({{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}}\right)}}-{4 \ {\sin \left({x}\right)}\ {\log \left({2 \over{{\cos \left({x}\right)}+ 1}}\right)}}+ 
\
\
\displaystyle
{{\left(-{{\cos \left({x}\right)}^{2}}-{3 \ {{x}^{2}}}+ 1 \right)}\ {\sin \left({x}\right)}}+{2 \  x \ {{\cos \left({x}\right)}^{3}}}- \
\
\displaystyle
{6 \  x \ {\cos \left({x}\right)}}
(20)
Type: Union(Expression(Integer),...)
fricas
INT(x*SIN(x)^3/COS(x)^4,x)

\label{eq21}{\left(
\begin{array}{@{}l}
\displaystyle
{5 \ {{\cos \left({x}\right)}^{3}}\ {\log \left({{{\sin \left({x}\right)}+{\cos \left({x}\right)}+ 1}\over{{\cos \left({x}\right)}+ 1}}\right)}}- 
\
\
\displaystyle
{5 \ {{\cos \left({x}\right)}^{3}}\ {\log \left({{{\sin \left({x}\right)}-{\cos \left({x}\right)}- 1}\over{{\cos \left({x}\right)}+ 1}}\right)}}-{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}- 
\
\
\displaystyle
{6 \  x \ {{\cos \left({x}\right)}^{2}}}+{2 \  x}
(21)
Type: Union(Expression(Integer),...)
fricas
INT(x*SIN(x)/COS(x)^3,x)

\label{eq22}{-{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ x}\over{2 \ {{\cos \left({x}\right)}^{2}}}(22)
Type: Union(Expression(Integer),...)
fricas
INT(x*SIN(x)^3/COS(x),x)

\label{eq23}\int^{
\displaystyle
x}{{{{\left(-{\%F \ {{\cos \left({\%F}\right)}^{2}}}+ \%F \right)}\ {\sin \left({\%F}\right)}}\over{\cos \left({\%F}\right)}}\ {d \%F}}(23)
Type: Union(Expression(Integer),...)
fricas
INT(x*SIN(x)^3/COS(x)^3,x)

\label{eq24}\int^{
\displaystyle
x}{{{{\left(-{\%F \ {{\cos \left({\%F}\right)}^{2}}}+ \%F \right)}\ {\sin \left({\%F}\right)}}\over{{\cos \left({\%F}\right)}^{3}}}\ {d \%F}}(24)
Type: Union(Expression(Integer),...)
fricas
-- " Timofeev (1948) Ch. 7, examples 10 - 11 (p. 344) ... "
INT((2*x+SIN(2*x))/(x*SIN(x)+COS(x))^2,x)

\label{eq25}{2 \  x \ {\sin \left({x}\right)}}\over{{x \ {\sin \left({x}\right)}}+{\cos \left({x}\right)}}(25)
Type: Union(Expression(Integer),...)
fricas
INT((x/(x*COS(x)-SIN(x)))^2,x)

\label{eq26}{-{x \ {\sin \left({x}\right)}}-{\cos \left({x}\right)}}\over{{\sin \left({x}\right)}-{x \ {\cos \left({x}\right)}}}(26)
Type: Union(Expression(Integer),...)




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