\[ \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^2} \, dx \]
Optimal antiderivative \[ \frac {3 x^{2}}{2 b^{2}}+\frac {3 x \left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )}{b^{3}}-\frac {x^{3}}{b \,\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )}+\frac {3 \left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{2} \ln \left (\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )}{b^{4}} \]
command
integrate(x^3/arccoth(tanh(b*x+a))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {i \, \pi ^{3} + 6 \, \pi ^{2} a - 12 i \, \pi a^{2} - 8 \, a^{3}}{4 \, {\left (2 \, b^{5} x + i \, \pi b^{4} + 2 \, a b^{4}\right )}} + \frac {x^{2}}{2 \, b^{2}} + \frac {{\left (-i \, \pi - 2 \, a\right )} x}{b^{3}} - \frac {3 \, {\left (\pi ^{2} - 4 i \, \pi a - 4 \, a^{2}\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{4 \, b^{4}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {x^{3}}{\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}\,{d x} \]________________________________________________________________________________________