\[ \int \frac {\tanh ^{-1}(a x)^2}{c x-a c x^2} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (a x \right )^{2} \ln \left (2-\frac {2}{-a x +1}\right )}{c}+\frac {\arctanh \left (a x \right ) \polylog \left (2, -1+\frac {2}{-a x +1}\right )}{c}-\frac {\polylog \left (3, -1+\frac {2}{-a x +1}\right )}{2 c} \]
command
integrate(arctanh(a*x)^2/(-a*c*x^2+c*x),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\log \left (\frac {2 \, a x}{a x - 1}\right ) \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 2 \, {\rm Li}_2\left (-\frac {2 \, a x}{a x - 1} + 1\right ) \log \left (-\frac {a x + 1}{a x - 1}\right ) - 2 \, {\rm polylog}\left (3, -\frac {a x + 1}{a x - 1}\right )}{4 \, c} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\operatorname {artanh}\left (a x\right )^{2}}{a c x^{2} - c x}, x\right ) \]