\[ \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx \]
Optimal antiderivative \[ \frac {3 x}{b^{3}}-\frac {x^{3}}{2 b \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{2}}-\frac {3 x^{2}}{2 b^{2} \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )}+\frac {3 \left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right ) \ln \left (\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )}{b^{4}} \]
command
integrate(x^3/arccoth(tanh(b*x+a))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {12 \, \pi ^{2} b x - 48 i \, \pi a b x - 48 \, a^{2} b x + 5 i \, \pi ^{3} + 30 \, \pi ^{2} a - 60 i \, \pi a^{2} - 40 \, a^{3}}{4 \, {\left (4 \, b^{6} x^{2} + 4 i \, \pi b^{5} x + 8 \, a b^{5} x - \pi ^{2} b^{4} + 4 i \, \pi a b^{4} + 4 \, a^{2} b^{4}\right )}} + \frac {x}{b^{3}} + \frac {3 \, {\left (-i \, \pi - 2 \, a\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{2 \, b^{4}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {x^{3}}{\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}\,{d x} \]________________________________________________________________________________________