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