\[ \int \frac {\left (F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}\right )^n}{1-a^2 x^2} \, dx \]
Optimal antiderivative \[ -\frac {\left (F^{\frac {\sqrt {-a x +1}}{\sqrt {a x +1}}}\right )^{n} \expIntegral \! \left (\frac {n \ln \left (F \right ) \sqrt {-a x +1}}{\sqrt {a x +1}}\right ) F^{-\frac {n \sqrt {-a x +1}}{\sqrt {a x +1}}}}{a} \]
command
integrate((F^((-a*x+1)^(1/2)/(a*x+1)^(1/2)))^n/(-a^2*x^2+1),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {could not integrate} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ -\frac {{\rm Ei}\left (\frac {\sqrt {-a x + 1} n \log \left (F\right )}{\sqrt {a x + 1}}\right )}{a} \]