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