\[ \int \frac {\sqrt {\tanh ^{-1}(a x)}}{1-a^2 x^2} \, dx \]
Optimal antiderivative \[ \frac {2 \arctanh \left (a x \right )^{\frac {3}{2}}}{3 a} \]
command
integrate(arctanh(a*x)^(1/2)/(-a^2*x^2+1),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\sqrt {2} \log \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {3}{2}}}{6 \, a} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int -\frac {\sqrt {\operatorname {artanh}\left (a x\right )}}{a^{2} x^{2} - 1}\,{d x} \]________________________________________________________________________________________