\[ \int \frac {\tanh ^{-1}(a x)^4}{x (c-a c x)} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (a x \right )^{4} \ln \left (2-\frac {2}{-a x +1}\right )}{c}+\frac {2 \arctanh \left (a x \right )^{3} \polylog \left (2, -1+\frac {2}{-a x +1}\right )}{c}-\frac {3 \arctanh \left (a x \right )^{2} \polylog \left (3, -1+\frac {2}{-a x +1}\right )}{c}+\frac {3 \arctanh \left (a x \right ) \polylog \left (4, -1+\frac {2}{-a x +1}\right )}{c}-\frac {3 \polylog \left (5, -1+\frac {2}{-a x +1}\right )}{2 c} \]
command
integrate(arctanh(a*x)^4/x/(-a*c*x+c),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\log \left (\frac {2 \, a x}{a x - 1}\right ) \log \left (-\frac {a x + 1}{a x - 1}\right )^{4} + 4 \, {\rm Li}_2\left (-\frac {2 \, a x}{a x - 1} + 1\right ) \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} - 12 \, \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} {\rm polylog}\left (3, -\frac {a x + 1}{a x - 1}\right ) + 24 \, \log \left (-\frac {a x + 1}{a x - 1}\right ) {\rm polylog}\left (4, -\frac {a x + 1}{a x - 1}\right ) - 24 \, {\rm polylog}\left (5, -\frac {a x + 1}{a x - 1}\right )}{16 \, c} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\operatorname {artanh}\left (a x\right )^{4}}{a c x^{2} - c x}, x\right ) \]