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