\[ \int x \coth ^{-1}(a x) \, dx \]
Optimal antiderivative \[ \frac {x}{2 a}+\frac {x^{2} \mathrm {arccoth}\left (a x \right )}{2}-\frac {\arctanh \left (a x \right )}{2 a^{2}} \]
command
integrate(x*arccoth(a*x),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ a {\left (\frac {1}{a^{3} {\left (\frac {a x + 1}{a x - 1} - 1\right )}} + \frac {{\left (a x + 1\right )} \log \left (-\frac {\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} + 1}{\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} - 1}\right )}{{\left (a x - 1\right )} a^{3} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{2}}\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int x \operatorname {arcoth}\left (a x\right )\,{d x} \]________________________________________________________________________________________