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