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Examples from Timofeev (1948) Ch. 1, divided into sections kindly provided by clicliclic vel Martin on sci.math.symbolic

fricas
INT ==> integrate
Type: Void
fricas
pi ==> %pi
Type: Void
fricas
SIN ==> sin
Type: Void
fricas
COS ==> cos
Type: Void
fricas
TAN ==> tan
Type: Void
fricas
COT ==> cot
Type: Void
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SEC ==> sec
Type: Void
fricas
CSC ==> csc
Type: Void
fricas
LN ==> log
Type: Void
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SQRT ==> sqrt
Type: Void
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ATAN ==> atan
Type: Void
fricas
ACOS ==> acos
Type: Void
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ASIN ==> asin
Type: Void
fricas
-- Timofeev (1948) Ch. 1, examples 1 - 21
INT(1/(a^2-b^2*x^2),x)

\label{eq1}{{\log \left({{b \  x}+ a}\right)}-{\log \left({{b \  x}- a}\right)}}\over{2 \  a \  b}(1)
Type: Union(Expression(Integer),...)
fricas
INT(1/(a^2+b^2*x^2),x)

\label{eq2}{\arctan \left({{b \  x}\over a}\right)}\over{a \  b}(2)
Type: Union(Expression(Integer),...)
fricas
INT(SEC(2*a*x),x)

\label{eq3}{\left(
\begin{array}{@{}l}
\displaystyle
{\log \left({{{\sin \left({2 \  a \  x}\right)}+{\cos \left({2 \  a \  x}\right)}+ 1}\over{{\cos \left({2 \  a \  x}\right)}+ 1}}\right)}- 
\
\
\displaystyle
{\log \left({{{\sin \left({2 \  a \  x}\right)}-{\cos \left({2 \  a \  x}\right)}- 1}\over{{\cos \left({2 \  a \  x}\right)}+ 1}}\right)}
(3)
Type: Union(Expression(Integer),...)
fricas
INT(1/(4*SIN(x/3)),x)

\label{eq4}{3 \ {\log \left({{\sin \left({x \over 3}\right)}\over{{\cos \left({x \over 3}\right)}+ 1}}\right)}}\over 4(4)
Type: Union(Expression(Integer),...)
fricas
INT(1/COS(3/4*pi-2*x),x)

\label{eq5}{\left(
\begin{array}{@{}l}
\displaystyle
{\log \left({{{\sin \left({{{8 \  x}-{3 \  \pi}}\over 4}\right)}+{\cos \left({{{8 \  x}-{3 \  \pi}}\over 4}\right)}+ 1}\over{{\cos \left({{{8 \  x}-{3 \  \pi}}\over 4}\right)}+ 1}}\right)}- 
\
\
\displaystyle
{\log \left({{{\sin \left({{{8 \  x}-{3 \  \pi}}\over 4}\right)}-{\cos \left({{{8 \  x}-{3 \  \pi}}\over 4}\right)}- 1}\over{{\cos \left({{{8 \  x}-{3 \  \pi}}\over 4}\right)}+ 1}}\right)}
(5)
Type: Union(Expression(Integer),...)
fricas
INT(SEC(x)*TAN(x),x)

\label{eq6}{{\cos \left({x}\right)}+ 1}\over{\cos \left({x}\right)}(6)
Type: Union(Expression(Integer),...)
fricas
INT(CSC(x)*COT(x),x)

\label{eq7}-{1 \over{\sin \left({x}\right)}}(7)
Type: Union(Expression(Integer),...)
fricas
INT(TAN(x)/SIN(2*x),x)

\label{eq8}{\tan \left({x}\right)}\over 2(8)
Type: Union(Expression(Integer),...)
fricas
INT(1/(1+COS(x)),x)

\label{eq9}{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}(9)
Type: Union(Expression(Integer),...)
fricas
INT(1/(1-COS(x)),x)

\label{eq10}{-{\cos \left({x}\right)}- 1}\over{\sin \left({x}\right)}(10)
Type: Union(Expression(Integer),...)
fricas
INT(SIN(x)/(a-b*COS(x)),x)

\label{eq11}{-{\log \left({2 \over{{\cos \left({x}\right)}+ 1}}\right)}+{\log \left({{-{2 \  b \ {\cos \left({x}\right)}}+{2 \  a}}\over{{\cos \left({x}\right)}+ 1}}\right)}}\over b(11)
Type: Union(Expression(Integer),...)
fricas
INT(COS(x)/(a^2+b^2*SIN(x)^2),x)

\label{eq12}{\left(
\begin{array}{@{}l}
\displaystyle
{\arctan \left({{{\left({2 \ {{b}^{2}}\ {\cos \left({x}\right)}}+{2 \ {{b}^{2}}}+{{a}^{2}}\right)}\ {\sin \left({x}\right)}}\over{{a \  b \ {{\cos \left({x}\right)}^{2}}}+{2 \  a \  b \ {\cos \left({x}\right)}}+{a \  b}}}\right)}- 
\
\
\displaystyle
{\arctan \left({{a \ {\sin \left({x}\right)}}\over{{2 \  b \ {\cos \left({x}\right)}}+{2 \  b}}}\right)}
(12)
Type: Union(Expression(Integer),...)
fricas
INT(COS(x)/(a^2-b^2*SIN(x)^2),x)

\label{eq13}{{\log \left({{{2 \  b \ {\sin \left({x}\right)}}+{2 \  a}}\over{{\cos \left({x}\right)}+ 1}}\right)}-{\log \left({{-{2 \  b \ {\sin \left({x}\right)}}+{2 \  a}}\over{{\cos \left({x}\right)}+ 1}}\right)}}\over{2 \  a \  b}(13)
Type: Union(Expression(Integer),...)
fricas
INT(SIN(2*x)/(b^2*SIN(x)^2+a^2),x)

\label{eq14}{-{2 \ {\log \left({2 \over{{\cos \left({x}\right)}+ 1}}\right)}}+{\log \left({{-{4 \ {{b}^{2}}\ {{\cos \left({x}\right)}^{2}}}+{4 \ {{b}^{2}}}+{4 \ {{a}^{2}}}}\over{{{\cos \left({x}\right)}^{2}}+{2 \ {\cos \left({x}\right)}}+ 1}}\right)}}\over{{b}^{2}}(14)
Type: Union(Expression(Integer),...)
fricas
INT(SIN(2*x)/(b^2*SIN(x)^2-a^2),x)

\label{eq15}{{\log \left({{{4 \ {{b}^{2}}\ {{\cos \left({x}\right)}^{2}}}-{4 \ {{b}^{2}}}+{4 \ {{a}^{2}}}}\over{{{\cos \left({x}\right)}^{2}}+{2 \ {\cos \left({x}\right)}}+ 1}}\right)}-{2 \ {\log \left({2 \over{{\cos \left({x}\right)}+ 1}}\right)}}}\over{{b}^{2}}(15)
Type: Union(Expression(Integer),...)
fricas
INT(SIN(2*x)/(b^2*COS(x)^2+a^2),x)

\label{eq16}{-{\log \left({{{4 \ {{b}^{2}}\ {{\cos \left({x}\right)}^{2}}}+{4 \ {{a}^{2}}}}\over{{{\cos \left({x}\right)}^{2}}+{2 \ {\cos \left({x}\right)}}+ 1}}\right)}+{2 \ {\log \left({2 \over{{\cos \left({x}\right)}+ 1}}\right)}}}\over{{b}^{2}}(16)
Type: Union(Expression(Integer),...)
fricas
INT(SIN(2*x)/(b^2*COS(x)^2-a^2),x)

\label{eq17}{-{\log \left({{{4 \ {{b}^{2}}\ {{\cos \left({x}\right)}^{2}}}-{4 \ {{a}^{2}}}}\over{{{\cos \left({x}\right)}^{2}}+{2 \ {\cos \left({x}\right)}}+ 1}}\right)}+{2 \ {\log \left({2 \over{{\cos \left({x}\right)}+ 1}}\right)}}}\over{{b}^{2}}(17)
Type: Union(Expression(Integer),...)
fricas
INT(1/(4-COS(x)^2),x)

\label{eq18}{\left(
\begin{array}{@{}l}
\displaystyle
{\arctan \left({{{\sqrt{3}}\ {\sin \left({x}\right)}}\over{{4 \ {\cos \left({x}\right)}}+ 4}}\right)}+ 
\
\
\displaystyle
{\arctan \left({{{\left({5 \ {\cos \left({x}\right)}}+ 8 \right)}\ {\sin \left({x}\right)}}\over{{2 \ {\sqrt{3}}\ {{\cos \left({x}\right)}^{2}}}+{4 \ {\sqrt{3}}\ {\cos \left({x}\right)}}+{2 \ {\sqrt{3}}}}}\right)}
(18)
Type: Union(Expression(Integer),...)
fricas
INT(%e^x/(%e^(2*x)-1),x)

\label{eq19}{-{\log \left({{{e}^{x}}+ 1}\right)}+{\log \left({{{e}^{x}}- 1}\right)}}\over 2(19)
Type: Union(Expression(Integer),...)
fricas
INT(1/(x*LN(x)),x)

\label{eq20}\log \left({\log \left({x}\right)}\right)(20)
Type: Union(Expression(Integer),...)
fricas
INT(1/(x*(1+LN(x)^2)),x)

\label{eq21}\arctan \left({\log \left({x}\right)}\right)(21)
Type: Union(Expression(Integer),...)
fricas
INT(1/(x*(1-LN(x))),x)

\label{eq22}-{\log \left({{\log \left({x}\right)}- 1}\right)}(22)
Type: Union(Expression(Integer),...)
fricas
INT(1/(x*(1+LN(x/a))),x)

\label{eq23}\log \left({{\log \left({x \over a}\right)}+ 1}\right)(23)
Type: Union(Expression(Integer),...)
fricas
-- Timofeev (1948) Ch. 1, examples 22 - 40 (p. 27-28)
INT(((1-SQRT(x)+x)/x)^2,x)

\label{eq24}{{6 \  x \ {\log \left({\sqrt{x}}\right)}}+{{\left(-{4 \  x}+ 4 \right)}\ {\sqrt{x}}}+{{x}^{2}}- 1}\over x(24)
Type: Union(Expression(Integer),...)
fricas
INT((2-x^(2/3))*(x+SQRT(x))/x^(3/2),x)

\label{eq25}{{{168}\ {\log \left({\root{6}\of{x}}\right)}}-{{21}\ {{\root{6}\of{x}}^{4}}}+{{56}\ {{\root{6}\of{x}}^{3}}}-{{12}\  x \ {\root{6}\of{x}}}}\over{1
4}(25)
Type: Union(Expression(Integer),...)
fricas
INT((2*x-1)/(2*x+3),x)

\label{eq26}-{2 \ {\log \left({{2 \  x}+ 3}\right)}}+ x(26)
Type: Union(Expression(Integer),...)
fricas
INT((2*x-5)/(3*x^2-2),x)

\label{eq27}{{{15}\ {\log \left({{{{\left({3 \ {{x}^{2}}}+ 2 \right)}\ {\sqrt{6}}}+{{12}\  x}}\over{{3 \ {{x}^{2}}}- 2}}\right)}}+{2 \ {\sqrt{6}}\ {\log \left({{3 \ {{x}^{2}}}- 2}\right)}}}\over{6 \ {\sqrt{6}}}(27)
Type: Union(Expression(Integer),...)
fricas
INT((2*x-5)/(3*x^2+2),x)

\label{eq28}{{{\sqrt{6}}\ {\log \left({{3 \ {{x}^{2}}}+ 2}\right)}}-{{15}\ {\arctan \left({{x \ {\sqrt{6}}}\over 2}\right)}}}\over{3 \ {\sqrt{6}}}(28)
Type: Union(Expression(Integer),...)
fricas
INT(SIN(x)*SIN(x/4),x)

\label{eq29}{{\left(-{{96}\ {{\cos \left({x \over 4}\right)}^{4}}}+{{112}\ {{\cos \left({x \over 4}\right)}^{2}}}-{16}\right)}\ {\sin \left({x \over 4}\right)}}\over{15}(29)
Type: Union(Expression(Integer),...)
fricas
INT(COS(3*x)*COS(4*x),x)

\label{eq30}{{\left({{32}\ {{\cos \left({x}\right)}^{6}}}-{{40}\ {{\cos \left({x}\right)}^{4}}}+{{12}\ {{\cos \left({x}\right)}^{2}}}+ 3 \right)}\ {\sin \left({x}\right)}}\over 7(30)
Type: Union(Expression(Integer),...)
fricas
INT(TAN(x)*TAN(x-a),x)

\label{eq31}{{\log \left({{{\tan \left({a}\right)}\ {\tan \left({x}\right)}}+ 1}\right)}-{x \ {\tan \left({a}\right)}}}\over{\tan \left({a}\right)}(31)
Type: Union(Expression(Integer),...)
fricas
INT(SIN(x)^2,x)

\label{eq32}{-{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ x}\over 2(32)
Type: Union(Expression(Integer),...)
fricas
INT(COS(x)^2,x)

\label{eq33}{{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ x}\over 2(33)
Type: Union(Expression(Integer),...)
fricas
INT(SIN(x)*COS(x)^3,x)

\label{eq34}-{{{\cos \left({x}\right)}^{4}}\over 4}(34)
Type: Union(Expression(Integer),...)
fricas
INT(COS(x)^3/SIN(x)^4,x)

\label{eq35}{{3 \ {{\cos \left({x}\right)}^{2}}}- 2}\over{{\left({3 \ {{\cos \left({x}\right)}^{2}}}- 3 \right)}\ {\sin \left({x}\right)}}(35)
Type: Union(Expression(Integer),...)
fricas
INT(1/(SIN(x)^2*COS(x)^2),x)

\label{eq36}{-{2 \ {{\cos \left({x}\right)}^{2}}}+ 1}\over{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}(36)
Type: Union(Expression(Integer),...)
fricas
INT(COT(3/4*x)^2,x)

\label{eq37}{-{3 \  x \ {\sin \left({{3 \  x}\over 2}\right)}}-{4 \ {\cos \left({{3 \  x}\over 2}\right)}}- 4}\over{3 \ {\sin \left({{3 \  x}\over 2}\right)}}(37)
Type: Union(Expression(Integer),...)
fricas
INT((1+TAN(2*x))^2,x)

\label{eq38}{{\log \left({{{\tan \left({2 \  x}\right)}^{2}}+ 1}\right)}+{\tan \left({2 \  x}\right)}}\over 2(38)
Type: Union(Expression(Integer),...)
fricas
INT((TAN(x)-COT(x))^2,x)

\label{eq39}{{{\tan \left({x}\right)}^{2}}-{4 \  x \ {\tan \left({x}\right)}}- 1}\over{\tan \left({x}\right)}(39)
Type: Union(Expression(Integer),...)
fricas
INT((TAN(x)-SEC(x))^2,x)

\label{eq40}{-{x \ {\sin \left({x}\right)}}+{{\left(- x - 4 \right)}\ {\cos \left({x}\right)}}- x - 4}\over{{\sin \left({x}\right)}+{\cos \left({x}\right)}+ 1}(40)
Type: Union(Expression(Integer),...)
fricas
INT(SIN(x)/(1+SIN(x)),x)

\label{eq41}{{x \ {\sin \left({x}\right)}}+{{\left(x + 2 \right)}\ {\cos \left({x}\right)}}+ x + 2}\over{{\sin \left({x}\right)}+{\cos \left({x}\right)}+ 1}(41)
Type: Union(Expression(Integer),...)
fricas
INT(COS(x)/(1-COS(x)),x)

\label{eq42}{-{x \ {\sin \left({x}\right)}}-{\cos \left({x}\right)}- 1}\over{\sin \left({x}\right)}(42)
Type: Union(Expression(Integer),...)
fricas
INT((%e^(x/2)-1)^3*%e^(-x/2),x)

\label{eq43}{{{{e}^{x \over 2}}^{3}}-{6 \ {{{e}^{x \over 2}}^{2}}}+{3 \  x \ {{e}^{x \over 2}}}+ 2}\over{{e}^{x \over 2}}(43)
Type: Union(Expression(Integer),...)
fricas
-- Timofeev (1948) Ch. 1, examples 41 - 65 (p. 35-37)
INT(1/(x^2-6*x+5),x)

\label{eq44}{-{\log \left({x - 1}\right)}+{\log \left({x - 5}\right)}}\over 4(44)
Type: Union(Expression(Integer),...)
fricas
INT(x^2/(13-6*x^3+x^6),x)

\label{eq45}{\arctan \left({{{{x}^{3}}- 3}\over 2}\right)}\over 6(45)
Type: Union(Expression(Integer),...)
fricas
INT((x+2)/(x^2-4*x-1),x)

\label{eq46}{\left(
\begin{array}{@{}l}
\displaystyle
{4 \ {\log \left({{{{\left({{x}^{2}}-{4 \  x}+ 9 \right)}\ {\sqrt{5}}}-{{10}\  x}+{20}}\over{{{x}^{2}}-{4 \  x}- 1}}\right)}}+ 
\
\
\displaystyle
{{\sqrt{5}}\ {\log \left({{{x}^{2}}-{4 \  x}- 1}\right)}}
(46)
Type: Union(Expression(Integer),...)
fricas
INT(1/(1+(x+1)^(1/3)),x)

\label{eq47}{{6 \ {\log \left({{\root{3}\of{x + 1}}+ 1}\right)}}+{3 \ {{\root{3}\of{x + 1}}^{2}}}-{6 \ {\root{3}\of{x + 1}}}}\over 2(47)
Type: Union(Expression(Integer),...)
fricas
INT(1/((a*x+b)*SQRT(x)),x)

\label{eq48}\begin{array}{@{}l}
\displaystyle
\left[{{\log \left({{{2 \  a \  b \ {\sqrt{x}}}+{{\left({a \  x}- b \right)}\ {\sqrt{-{a \  b}}}}}\over{{a \  x}+ b}}\right)}\over{\sqrt{-{a \  b}}}}, \:{{2 \ {\arctan \left({{{\sqrt{a \  b}}\ {\sqrt{x}}}\over b}\right)}}\over{\sqrt{a \  b}}}\right] 
(48)
Type: Union(List(Expression(Integer)),...)
fricas
INT(x^3*SQRT(1+x^2),x)

\label{eq49}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left(-{{48}\ {{x}^{9}}}-{{76}\ {{x}^{7}}}-{3 \ {{x}^{5}}}+{{35}\ {{x}^{3}}}+{{10}\  x}\right)}\ {\sqrt{{{x}^{2}}+ 1}}}+ 
\
\
\displaystyle
{{48}\ {{x}^{10}}}+{{100}\ {{x}^{8}}}+{{35}\ {{x}^{6}}}-{{40}\ {{x}^{4}}}-{{25}\ {{x}^{2}}}- 2 
(49)
Type: Union(Expression(Integer),...)
fricas
INT(x/SQRT(a^4-x^4),x)

\label{eq50}-{\arctan \left({{{\sqrt{-{{x}^{4}}+{{a}^{4}}}}-{{a}^{2}}}\over{{x}^{2}}}\right)}(50)
Type: Union(Expression(Integer),...)
fricas
INT(1/(x*SQRT(x^2-a^2)),x)

\label{eq51}{2 \ {\arctan \left({{{\sqrt{{{x}^{2}}-{{a}^{2}}}}- x}\over a}\right)}}\over a(51)
Type: Union(Expression(Integer),...)
fricas
INT(1/(x*SQRT(a^2-x^2)),x)

\label{eq52}{\log \left({{{\sqrt{-{{x}^{2}}+{{a}^{2}}}}- a}\over x}\right)}\over a(52)
Type: Union(Expression(Integer),...)
fricas
INT(1/(x*SQRT(x^2+a^2)),x)

\label{eq53}{-{\log \left({{\sqrt{{{x}^{2}}+{{a}^{2}}}}- x + a}\right)}+{\log \left({{\sqrt{{{x}^{2}}+{{a}^{2}}}}- x - a}\right)}}\over a(53)
Type: Union(Expression(Integer),...)
fricas
INT(1/SQRT(2+x-x^2),x)

\label{eq54}-{\arctan \left({{2 \ {\sqrt{-{{x}^{2}}+ x + 2}}}\over{{2 \  x}- 1}}\right)}(54)
Type: Union(Expression(Integer),...)
fricas
INT(1/SQRT(3*x^2-4*x+5),x)

\label{eq55}{\log{\left({{{\left(-{{18}\  x}+{12}\right)}\ {\sqrt{{3 \ {{x}^{2}}}-{4 \  x}+ 5}}}+{{\left(-{{18}\ {{x}^{2}}}+{{24}\  x}-{1
9}\right)}\ {\sqrt{3}}}}\right)}}\over{2 \ {\sqrt{3}}}(55)
Type: Union(Expression(Integer),...)
fricas
INT(1/SQRT(x-x^2),x)

\label{eq56}-{2 \ {\arctan \left({{\sqrt{-{{x}^{2}}+ x}}\over x}\right)}}(56)
Type: Union(Expression(Integer),...)
fricas
INT((2*x+1)/SQRT(2+x-x^2),x)

\label{eq57}-{2 \ {\arctan \left({{2 \ {\sqrt{-{{x}^{2}}+ x + 2}}}\over{{2 \  x}- 1}}\right)}}-{2 \ {\sqrt{-{{x}^{2}}+ x + 2}}}(57)
Type: Union(Expression(Integer),...)
fricas
INT(1/(x*SQRT(2+x-x^2)),x)

\label{eq58}{\left({\sqrt{2}}\ {\log{\left({{{{\left(-{4 \  x}-{16}\right)}\ {\sqrt{2}}\ {\sqrt{-{{x}^{2}}+ x + 2}}}-{7 \ {{x}^{2}}}+{{16}\  x}+{32}}\over{{x}^{2}}}\right)}}\right)}\over 4(58)
Type: Union(Expression(Integer),...)
fricas
INT(1/((x-2)*SQRT(2+x-x^2)),x)

\label{eq59}{2 \ {\sqrt{-{{x}^{2}}+ x + 2}}}\over{{3 \  x}- 6}(59)
Type: Union(Expression(Integer),...)
fricas
INT((2+3*SIN(x))/(SIN(x)*(1-COS(x))),x)

\label{eq60}{{{\left({2 \ {\cos \left({x}\right)}}- 2 \right)}\ {\log \left({{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}}\right)}}+{6 \ {\sin \left({x}\right)}}+{\cos \left({x}\right)}+ 1}\over{{2 \ {\cos \left({x}\right)}}- 2}(60)
Type: Union(Expression(Integer),...)
fricas
INT(1/(2+3*COS(x)^2),x)

\label{eq61}{\left(
\begin{array}{@{}l}
\displaystyle
{\arctan \left({{{\sqrt{10}}\ {\sin \left({x}\right)}}\over{{4 \ {\cos \left({x}\right)}}+ 4}}\right)}- 
\
\
\displaystyle
{\arctan \left({{{\left({\cos \left({x}\right)}- 4 \right)}\ {\sin \left({x}\right)}}\over{{{\sqrt{10}}\ {{\cos \left({x}\right)}^{2}}}+{2 \ {\sqrt{10}}\ {\cos \left({x}\right)}}+{\sqrt{10}}}}\right)}
(61)
Type: Union(Expression(Integer),...)
fricas
INT((1-TAN(x))/SIN(2*x),x)

\label{eq62}{{\log \left({\tan \left({x}\right)}\right)}-{\tan \left({x}\right)}}\over 2(62)
Type: Union(Expression(Integer),...)
fricas
INT((1+TAN(x)^2)/(1-TAN(x)^2),x)

\label{eq63}{{\log \left({{\tan \left({x}\right)}+ 1}\right)}-{\log \left({{\tan \left({x}\right)}- 1}\right)}}\over 2(63)
Type: Union(Expression(Integer),...)
fricas
INT((a^2-4*COS(x)^2)^(3/4)*SIN(2*x),x)

\label{eq64}{{\left(-{4 \ {{\cos \left({x}\right)}^{2}}}+{{a}^{2}}\right)}\ {{\root{4}\of{-{4 \ {{\cos \left({x}\right)}^{2}}}+{{a}^{2}}}}^{3}}}\over 7(64)
Type: Union(Expression(Integer),...)
fricas
INT(SIN(2*x)/(a^2-4*SIN(x)^2)^(1/3),x)

\label{eq65}-{{3 \ {{\root{3}\of{{4 \ {{\cos \left({x}\right)}^{2}}}+{{a}^{2}}- 4}}^{2}}}\over 8}(65)
Type: Union(Expression(Integer),...)
fricas
INT(1/SQRT(a^(2*x)-1),x)

\label{eq66}{\arctan \left({\sqrt{{{e}^{2 \  x \ {\log \left({a}\right)}}}- 1}}\right)}\over{\log \left({a}\right)}(66)
Type: Union(Expression(Integer),...)
fricas
INT(%e^(x/2)/SQRT(%e^x-1),x)

\label{eq67}-{2 \ {\log \left({{\sqrt{{{{e}^{x \over 2}}^{2}}- 1}}-{{e}^{x \over 2}}}\right)}}(67)
Type: Union(Expression(Integer),...)
fricas
INT(ATAN(x)^n/(1+x^2),x)

\label{eq68}{{\arctan \left({x}\right)}\ {{e}^{n \ {\log \left({\arctan \left({x}\right)}\right)}}}}\over{n + 1}(68)
Type: Union(Expression(Integer),...)
fricas
INT(ASIN(x/a)^(3/2)/SQRT(a^2-x^2),x)

\label{eq69}{2 \ {{\arctan \left({x \over{\sqrt{-{{x}^{2}}+{{a}^{2}}}}}\right)}^{2}}\ {\sqrt{2 \ {\arctan \left({x \over{\sqrt{-{{x}^{2}}+{{a}^{2}}}}}\right)}}}}\over{5 \ {\sqrt{2}}}(69)
Type: Union(Expression(Integer),...)
fricas
INT(1/(ACOS(x)^3*SQRT(1-x^2)),x)

\label{eq70}1 \over{2 \ {{\arccos \left({x}\right)}^{2}}}(70)
Type: Union(Expression(Integer),...)
fricas
-- Timofeev (1948) Ch. 1, examples 66 - 81 (p. 41-42)
INT(LN(x)^2*x,x)

\label{eq71}{{2 \ {{x}^{2}}\ {{\log \left({x}\right)}^{2}}}-{2 \ {{x}^{2}}\ {\log \left({x}\right)}}+{{x}^{2}}}\over 4(71)
Type: Union(Expression(Integer),...)
fricas
INT(LN(x)/x^5,x)

\label{eq72}{-{4 \ {\log \left({x}\right)}}- 1}\over{{16}\ {{x}^{4}}}(72)
Type: Union(Expression(Integer),...)
fricas
INT(x^2*LN((x-1)/x),x)

\label{eq73}{-{2 \ {\log \left({x - 1}\right)}}+{2 \ {{x}^{3}}\ {\log \left({{x - 1}\over x}\right)}}-{{x}^{2}}-{2 \  x}}\over 6(73)
Type: Union(Expression(Integer),...)
fricas
INT(COS(x)^5,x)

\label{eq74}{{\left({3 \ {{\cos \left({x}\right)}^{4}}}+{4 \ {{\cos \left({x}\right)}^{2}}}+ 8 \right)}\ {\sin \left({x}\right)}}\over{15}(74)
Type: Union(Expression(Integer),...)
fricas
INT(SIN(x)^2*COS(x)^4,x)

\label{eq75}{{{\left(-{8 \ {{\cos \left({x}\right)}^{5}}}+{2 \ {{\cos \left({x}\right)}^{3}}}+{3 \ {\cos \left({x}\right)}}\right)}\ {\sin \left({x}\right)}}+{3 \  x}}\over{48}(75)
Type: Union(Expression(Integer),...)
fricas
INT(1/SIN(x)^5,x)

\label{eq76}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({3 \ {{\cos \left({x}\right)}^{4}}}-{6 \ {{\cos \left({x}\right)}^{2}}}+ 3 \right)}\ {\log \left({{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}}\right)}}+ 
\
\
\displaystyle
{3 \ {{\cos \left({x}\right)}^{3}}}-{5 \ {\cos \left({x}\right)}}
(76)
Type: Union(Expression(Integer),...)
fricas
INT(SIN(x)/%e^x,x)

\label{eq77}{-{\sin \left({x}\right)}-{\cos \left({x}\right)}}\over{2 \ {{e}^{x}}}(77)
Type: Union(Expression(Integer),...)
fricas
INT(%e^(2*x)*SIN(3*x),x)

\label{eq78}{{2 \ {{e}^{2 \  x}}\ {\sin \left({3 \  x}\right)}}-{3 \ {\cos \left({3 \  x}\right)}\ {{e}^{2 \  x}}}}\over{13}(78)
Type: Union(Expression(Integer),...)
fricas
INT(a^x*COS(x),x)

\label{eq79}{{\left({\sin \left({x}\right)}+{{\cos \left({x}\right)}\ {\log \left({a}\right)}}\right)}\ {{e}^{x \ {\log \left({a}\right)}}}}\over{{{\log \left({a}\right)}^{2}}+ 1}(79)
Type: Union(Expression(Integer),...)
fricas
INT(COS(LN(x)),x)

\label{eq80}{{x \ {\sin \left({\log \left({x}\right)}\right)}}+{x \ {\cos \left({\log \left({x}\right)}\right)}}}\over 2(80)
Type: Union(Expression(Integer),...)
fricas
INT(SEC(x)^2*LN(COS(x)),x)

\label{eq81}{{{\sin \left({x}\right)}\ {\log \left({\cos \left({x}\right)}\right)}}+{\sin \left({x}\right)}-{x \ {\cos \left({x}\right)}}}\over{\cos \left({x}\right)}(81)
Type: Union(Expression(Integer),...)
fricas
INT(x*TAN(x)^2,x)

\label{eq82}{-{\log \left({{{\tan \left({x}\right)}^{2}}+ 1}\right)}+{2 \  x \ {\tan \left({x}\right)}}-{{x}^{2}}}\over 2(82)
Type: Union(Expression(Integer),...)
fricas
INT(ASIN(x)/x^2,x)

\label{eq83}{\left(
\begin{array}{@{}l}
\displaystyle
-{x \ {\log \left({{\sqrt{-{{x}^{2}}+ 1}}+ 1}\right)}}+{x \ {\log \left({{\sqrt{-{{x}^{2}}+ 1}}- 1}\right)}}+ 
\
\
\displaystyle
{x \ {\arctan \left({{\sqrt{-{{x}^{2}}+ 1}}\over x}\right)}}+{{\left(x - 2 \right)}\ {\arcsin \left({x}\right)}}
(83)
Type: Union(Expression(Integer),...)
fricas
INT(ASIN(x)^2,x)

\label{eq84}{2 \ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{x \ {{\arcsin \left({x}\right)}^{2}}}-{2 \  x}(84)
Type: Union(Expression(Integer),...)
fricas
INT(x^2*ATAN(x)/(1+x^2),x)

\label{eq85}{-{\log \left({{{x}^{2}}+ 1}\right)}-{{\arctan \left({x}\right)}^{2}}+{2 \  x \ {\arctan \left({x}\right)}}}\over 2(85)
Type: Union(Expression(Integer),...)
fricas
INT(ACOS(SQRT(x/(x+1))),x)

\label{eq86}{{{\left(x + 1 \right)}\ {\sqrt{1 \over{x + 1}}}\ {\arccos \left({\sqrt{x \over{x + 1}}}\right)}}+{\sqrt{x \over{x + 1}}}}\over{\sqrt{1 \over{x + 1}}}(86)
Type: Union(Expression(Integer),...)

See also FriCASTimofeev37, FriCASTimofeev5, FriCASTimofeev9




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