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