\[ \int \frac {\coth ^{-1}(a x)}{x^4} \, dx \]
Optimal antiderivative \[ -\frac {a}{6 x^{2}}-\frac {\mathrm {arccoth}\left (a x \right )}{3 x^{3}}+\frac {a^{3} \ln \left (x \right )}{3}-\frac {a^{3} \ln \left (-a^{2} x^{2}+1\right )}{6} \]
command
integrate(arccoth(a*x)/x^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{3} \, {\left (a^{2} \log \left (\frac {{\left | a x + 1 \right |}}{{\left | a x - 1 \right |}}\right ) - a^{2} \log \left ({\left | \frac {a x + 1}{a x - 1} + 1 \right |}\right ) - \frac {2 \, {\left (a x + 1\right )} a^{2}}{{\left (a x - 1\right )} {\left (\frac {a x + 1}{a x - 1} + 1\right )}^{2}} - \frac {{\left (\frac {3 \, {\left (a x + 1\right )}^{2} a^{2}}{{\left (a x - 1\right )}^{2}} + a^{2}\right )} \log \left (-\frac {\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} + 1}{\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} - 1}\right )}{{\left (\frac {a x + 1}{a x - 1} + 1\right )}^{3}}\right )} a \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\operatorname {arcoth}\left (a x\right )}{x^{4}}\,{d x} \]________________________________________________________________________________________