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