\[ \int \frac {x^4}{\coth ^{-1}(\tanh (a+b x))^2} \, dx \]
Optimal antiderivative \[ \frac {4 x^{3}}{3 b^{2}}+\frac {2 x^{2} \left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )}{b^{3}}+\frac {4 x \left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{2}}{b^{4}}-\frac {x^{4}}{b \,\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )}+\frac {4 \left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{3} \ln \left (\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )}{b^{5}} \]
command
integrate(x^4/arccoth(tanh(b*x+a))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {x^{3}}{3 \, b^{2}} - \frac {\pi ^{4} - 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} + 32 i \, \pi a^{3} + 16 \, a^{4}}{8 \, {\left (2 \, b^{6} x + i \, \pi b^{5} + 2 \, a b^{5}\right )}} - \frac {{\left (i \, \pi + 2 \, a\right )} x^{2}}{2 \, b^{3}} - \frac {3 \, {\left (\pi ^{2} - 4 i \, \pi a - 4 \, a^{2}\right )} x}{4 \, b^{4}} + \frac {{\left (i \, \pi ^{3} + 6 \, \pi ^{2} a - 12 i \, \pi a^{2} - 8 \, a^{3}\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{2 \, b^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {x^{4}}{\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}\,{d x} \]________________________________________________________________________________________