\[ \int \frac {\tanh ^{-1}(a x)^p}{1-a^2 x^2} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (a x \right )^{1+p}}{a \left (1+p \right )} \]
command
integrate(arctanh(a*x)^p/(-a^2*x^2+1),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\left (\frac {1}{2} \, \log \left (-\frac {a x + 1}{a x - 1}\right )\right )^{p + 1}}{a {\left (p + 1\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int -\frac {\operatorname {artanh}\left (a x\right )^{p}}{a^{2} x^{2} - 1}\,{d x} \]________________________________________________________________________________________