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FriCAS can now handle large class of integrals expressible in terms of exponential integral, error functions, incomplete Gamma function with rational first argument, logarithmic integral and polylogarithms. Like

fricas
integrate(1/log(x), x)

\label{eq1}li \left({x}\right)(1)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(log(x) + 1), x)

\label{eq2}{li \left({x \  e}\right)}\over e(2)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(log(x)^2-1), x)

\label{eq3}{-{li \left({x \  e}\right)}+{{{e}^{2}}\ {li \left({x \over e}\right)}}}\over{2 \  e}(3)
Type: Union(Expression(Integer),...)
fricas
integrate(exp(x + a)/x, x)

\label{eq4}{Ei \left({x}\right)}\ {{e}^{a}}(4)
Type: Union(Expression(Integer),...)
fricas
integrate(exp(x + a)/x^2, x)

\label{eq5}{-{{e}^{x + a}}+{x \ {Ei \left({x}\right)}\ {{e}^{a}}}}\over x(5)
Type: Union(Expression(Integer),...)
fricas
integrate(exp(x)/(x^2 - 1), x)

\label{eq6}{-{Ei \left({x + 1}\right)}+{{{e}^{2}}\ {Ei \left({x - 1}\right)}}}\over{2 \  e}(6)
Type: Union(Expression(Integer),...)
fricas
integrate(x/(exp(x) - 1), x)

\label{eq7}{{2 \  x \ {\log \left({-{{e}^{x}}+ 1}\right)}}+{2 \ {dilog \left({-{{e}^{x}}+ 1}\right)}}-{{x}^{2}}}\over 2(7)
Type: Union(Expression(Integer),...)
fricas
integrate(x^3/(exp(x) - 1), x)

\label{eq8}{\left(
\begin{array}{@{}l}
\displaystyle
{{24}\ {polylog \left({4, \:{{e}^{x}}}\right)}}-{{24}\  x \ {polylog \left({3, \:{{e}^{x}}}\right)}}+ 
\
\
\displaystyle
{4 \ {{x}^{3}}\ {\log \left({-{{e}^{x}}+ 1}\right)}}+{{12}\ {{x}^{2}}\ {dilog \left({-{{e}^{x}}+ 1}\right)}}-{{x}^{4}}
(8)
Type: Union(Expression(Integer),...)
fricas
integrate(2*x*exp(x)/(exp(x)^2 - 1), x)

\label{eq9}\begin{array}{@{}l}
\displaystyle
-{x \ {\log \left({{{e}^{x}}+ 1}\right)}}+{x \ {\log \left({-{{e}^{x}}+ 1}\right)}}-{dilog \left({{{e}^{x}}+ 1}\right)}+ 
\
\
\displaystyle
{dilog \left({-{{e}^{x}}+ 1}\right)}
(9)
Type: Union(Expression(Integer),...)
fricas
integrate(x/sinh(x), x)

\label{eq10}\begin{array}{@{}l}
\displaystyle
-{x \ {\log \left({{\sinh \left({x}\right)}+{\cosh \left({x}\right)}+ 1}\right)}}+ 
\
\
\displaystyle
{x \ {\log \left({-{\sinh \left({x}\right)}-{\cosh \left({x}\right)}+ 1}\right)}}- 
\
\
\displaystyle
{dilog \left({{\sinh \left({x}\right)}+{\cosh \left({x}\right)}+ 1}\right)}+ 
\
\
\displaystyle
{dilog \left({-{\sinh \left({x}\right)}-{\cosh \left({x}\right)}+ 1}\right)}
(10)
Type: Union(Expression(Integer),...)
fricas
integrate(log(sinh(x)), x)

\label{eq11}{\left(
\begin{array}{@{}l}
\displaystyle
-{2 \  x \ {\log \left({{\sinh \left({x}\right)}+{\cosh \left({x}\right)}+ 1}\right)}}+{2 \  x \ {\log \left({\sinh \left({x}\right)}\right)}}- 
\
\
\displaystyle
{2 \  x \ {\log \left({-{\sinh \left({x}\right)}-{\cosh \left({x}\right)}+ 1}\right)}}- 
\
\
\displaystyle
{2 \ {dilog \left({{\sinh \left({x}\right)}+{\cosh \left({x}\right)}+ 1}\right)}}- 
\
\
\displaystyle
{2 \ {dilog \left({-{\sinh \left({x}\right)}-{\cosh \left({x}\right)}+ 1}\right)}}+{{x}^{2}}
(11)
Type: Union(Expression(Integer),...)
fricas
integrate(exp((-x^2-2*x-1)/x^2)/x^2, x)

\label{eq12}-{{{\erf \left({{x + 1}\over x}\right)}\ {\sqrt{\pi}}}\over 2}(12)
Type: Union(Expression(Integer),...)
fricas
integrate(x^3*exp(-x^3), x)

\label{eq13}{-{\Gamma \left({{1 \over 3}, \:{{x}^{3}}}\right)}-{3 \  x \ {{e}^{-{{x}^{3}}}}}}\over 9(13)
Type: Union(Expression(Integer),...)
fricas
integrate(x^2*exp(-(x+1)^3), x)

\label{eq14}{\left(
\begin{array}{@{}l}
\displaystyle
{2 \ {\Gamma \left({{2 \over 3}, \:{{{x}^{3}}+{3 \ {{x}^{2}}}+{3 \  x}+ 1}}\right)}}- 
\
\
\displaystyle
{\Gamma \left({{1 \over 3}, \:{{{x}^{3}}+{3 \ {{x}^{2}}}+{3 \  x}+ 1}}\right)}- 
\
\
\displaystyle
{{e}^{-{{x}^{3}}-{3 \ {{x}^{2}}}-{3 \  x}- 1}}
(14)
Type: Union(Expression(Integer),...)

FriCAS can introduce new algebraic constants when needed:

fricas
integrate(1/(log(x)^2-3), x)

\label{eq15}{-{li \left({x \ {{e}^{\sqrt{3}}}}\right)}+{{{{e}^{\sqrt{3}}}^{2}}\ {li \left({x \over{{e}^{\sqrt{3}}}}\right)}}}\over{2 \ {\sqrt{3}}\ {{e}^{\sqrt{3}}}}(15)
Type: Union(Expression(Integer),...)
fricas
integrate(exp(x)/(x^2 - 5), x)

\label{eq16}{{{Ei \left({-{\sqrt{5}}+ x}\right)}\ {{{e}^{\sqrt{5}}}^{2}}}-{Ei \left({{\sqrt{5}}+ x}\right)}}\over{2 \ {\sqrt{5}}\ {{e}^{\sqrt{5}}}}(16)
Type: Union(Expression(Integer),...)

The method is robust, FriCAS can handle both

fricas
integrate(((x+1)*exp(x))/log(x*exp(x)), x)

\label{eq17}li \left({x \ {{e}^{x}}}\right)(17)
Type: Union(Expression(Integer),...)
fricas
integrate(((x+1)*exp(x))/(x + log(x)), x)

\label{eq18}li \left({x \ {{e}^{x}}}\right)(18)
Type: Union(Expression(Integer),...)

while Mathematca 8 can handle the first form, but not the second one (Maple 15 and Maxima 5.30.0 can not handle any).

Similarly FriCAS has no troubles with

fricas
integrate(((-4*x-8)*log(x)+(-2*x^2-4*x))/(3*x*exp(2*log(x)+x)^2-x), x)

\label{eq19}\begin{array}{@{}l}
\displaystyle
{{\left(-{2 \ {\log \left({x}\right)}}- x \right)}\ {\log \left({{{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}}+ 
\
\
\displaystyle
{{\left(-{2 \ {\log \left({x}\right)}}- x \right)}\ {\log \left({{-{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}}- 
\
\
\displaystyle
{dilog \left({{{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}- 
\
\
\displaystyle
{dilog \left({{-{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}+{4 \ {{\log \left({x}\right)}^{2}}}+{4 \  x \ {\log \left({x}\right)}}+ 
\
\
\displaystyle
{{x}^{2}}
(19)
Type: Union(Expression(Integer),...)
fricas
integrate(((-4*x-8)*log(x)+(-2*x^2-4*x))/(3*x^3*exp(log(x)+x)^2-x), x)

\label{eq20}\begin{array}{@{}l}
\displaystyle
{{\left(-{2 \ {\log \left({x}\right)}}- x \right)}\ {\log \left({{{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}}+ 
\
\
\displaystyle
{{\left(-{2 \ {\log \left({x}\right)}}- x \right)}\ {\log \left({{-{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}}- 
\
\
\displaystyle
{dilog \left({{{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}- 
\
\
\displaystyle
{dilog \left({{-{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}+{4 \ {{\log \left({x}\right)}^{2}}}+{4 \  x \ {\log \left({x}\right)}}+ 
\
\
\displaystyle
{{x}^{2}}
(20)
Type: Union(Expression(Integer),...)
fricas
integrate(((2*x^4-x^3+3*x^2+2*x+2)*exp(x/(x^2+2)))/(x^3+2*x), x)

\label{eq21}{{\left({{x}^{2}}+ 2 \right)}\ {{e}^{x \over{{{x}^{2}}+ 2}}}}+{Ei \left({x \over{{{x}^{2}}+ 2}}\right)}(21)
Type: Union(Expression(Integer),...)

none of Mathematca 8, Maple 15 and Maxima 5.30.0 can handle them.

FriCAS can also handle some integrals involving special functions of algebraic arguments:

fricas
integrate(((26*x+23)*x^(1/2)+4*x^2+50*x-6)*exp(2*x^(1/2)+x)/((16*x^2+36*x)*x^(1/2)+(2*x^3+42*x^2)), x)

\label{eq22}{{{e}^{{2 \ {\sqrt{x}}}+ x}}+{{\left({3 \ {\sqrt{x}}}+ x \right)}\ {Ei \left({{2 \ {\sqrt{x}}}+ x}\right)}}}\over{{3 \ {\sqrt{x}}}+ x}(22)
Type: Union(Expression(Integer),...)




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