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