2.1 \(\int \arcsin (x) \ln (x)\,dx\)

2.1.1 Mathematica

Clear[x] 
Integrate[ArcSin[x] Log[x], x]
 

\[ -2 \sqrt {1-x^2}+\left (\sqrt {1-x^2}-1\right ) \log (x)+\log \left (\sqrt {1-x^2}+1\right )+x (\log (x)-1) \sin ^{-1}(x) \]

2.1.2 Rubi

<< Rubi` 
Clear[x] 
Int[ArcSin[x] Log[x], x]
 

\[ -2 \sqrt {1-x^2}+\sqrt {1-x^2} \log (x)+\tanh ^{-1}\left (\sqrt {1-x^2}\right )-x \sin ^{-1}(x)+x \log (x) \sin ^{-1}(x) \]

2.1.3 Maple

restart; 
int(arcsin(x)*log(x),x);
 

\[ 2\,{\frac {1}{1+ \left ( \tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) \right ) ^{2}} \left ( \arcsin \left ( x \right ) \tan \left ( 1/ 2\,\arcsin \left ( x \right ) \right ) \ln \left ( 2\,{\frac {\tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) }{1+ \left ( \tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) \right ) ^{2}}} \right ) - \left ( \tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) \right ) ^{2}\ln \left ( 2\,{\frac {\tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) }{1+ \left ( \tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) \right ) ^{2}}} \right ) -\arcsin \left ( x \right ) \tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) -2 \right ) }-\ln \left ( 1+ \left ( \tan \left ( {\frac {\arcsin \left ( x \right ) }{2}} \right ) \right ) ^{2} \right ) \]

2.1.4 Fricas

setSimplifyDenomsFlag(true) 
ii:=integrate(asin(x)*log(x),x); 
latex(ii)
 

\[ {{\log \left ( {{{\sqrt {{-{{x} \sp {2}}+1}}}+1}} \right )} -{\log \left ( {{{\sqrt {{-{{x} \sp {2}}+1}}} -1}} \right )}+{{\left ( {2 \ {\log \left ( {x} \right )}} -4 \right )} \ {\sqrt {{-{{x} \sp {2}}+1}}}}+{2 \ x \ {\arcsin \left ( {x} \right )} \ {\log \left ( {x} \right )}} -{2 \ x \ {\arcsin \left ( {x} \right )}}} \over 2 \]

2.1.5 Maxima

ii : integrate(asin(x)*log(x),x); 
tex(ii);
 

\[ \log \left ({{2\,\sqrt {1-x^2}}\over {\left | x\right | }}+{{2}\over { \left | x\right | }}\right )+\arcsin x\,\left (x\,\log x-x\right )+\sqrt { 1-x^2}\,\log x-2\,\sqrt {1-x^2} \]

2.1.6 XCAS

ii := integrate(asin(x)*log(x),x); 
latex(ii);
 

\[ \sqrt {-x^{2}+1} \ln x+\frac {2 \ln \left (\sqrt {-x^{2}+1}+1\right )}{2 x^{2} \left (\frac 1{\sqrt {-x^{2}+1}+1}\right )^{2}+2}-\frac {2 \ln \left |x\right |}{2 x^{2} \left (\frac 1{\sqrt {-x^{2}+1}+1}\right )^{2}+2}+x \arcsin x\cdot \ln x-\frac {4 x \arcsin x}{\left (\sqrt {-x^{2}+1}+1\right ) \left (2 x^{2} \left (\frac 1{\sqrt {-x^{2}+1}+1}\right )^{2}+2\right )}+\frac {4 x^{2} \left (\frac 1{\sqrt {-x^{2}+1}+1}\right )^{2}}{2 x^{2} \left (\frac 1{\sqrt {-x^{2}+1}+1}\right )^{2}+2}+\frac {2 x^{2} \ln \left (\sqrt {-x^{2}+1}+1\right ) \left (\frac 1{\sqrt {-x^{2}+1}+1}\right )^{2}}{2 x^{2} \left (\frac 1{\sqrt {-x^{2}+1}+1}\right )^{2}+2}-\frac {2 x^{2} \ln \left |x\right |\cdot \left (\frac 1{\sqrt {-x^{2}+1}+1}\right )^{2}}{2 x^{2} \left (\frac 1{\sqrt {-x^{2}+1}+1}\right )^{2}+2}-\frac {4}{2 x^{2} \left (\frac 1{\sqrt {-x^{2}+1}+1}\right )^{2}+2} \]

2.1.7 Sympy

>python 
Python 3.7.3 (default, Mar 27 2019, 22:11:17) 
[GCC 7.3.0] :: Anaconda, Inc. on linux 
Type "help", "copyright", "credits" or "license" for more information. 
>>> from sympy import * 
>>> x = symbols('x') 
>>> ii = integrate(asin(x)*log(x),x) 
>>> latex(ii)
 

\[ x \log {\left (x \right )} \operatorname {asin}{\left (x \right )} - x \operatorname {asin}{\left (x \right )} + \sqrt {1 - x^{2}} \log {\left (x \right )} - \sqrt {1 - x^{2}} - \begin {cases} - \frac {x}{\sqrt {-1 + \frac {1}{x^{2}}}} - \operatorname {acosh}{\left (\frac {1}{x} \right )} + \frac {1}{x \sqrt {-1 + \frac {1}{x^{2}}}} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > 1 \\\frac {i x}{\sqrt {1 - \frac {1}{x^{2}}}} + i \operatorname {asin}{\left (\frac {1}{x} \right )} - \frac {i}{x \sqrt {1 - \frac {1}{x^{2}}}} & \text {otherwise} \end {cases} \]

2.1.8 MuPad

evalin(symengine,'int(asin(x)*log(x),x)')
 

\[ \text {did not solve} \]