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