\[ \int \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx \]
Optimal antiderivative \[ -2 a e x -2 b e x \,\mathrm {arccoth}\left (c x \right )+\frac {e \left (a +b \,\mathrm {arccoth}\left (c x \right )\right )^{2}}{b c}-\frac {b e \ln \left (-c^{2} x^{2}+1\right )}{c}+x \left (a +b \,\mathrm {arccoth}\left (c x \right )\right ) \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )+\frac {b \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )^{2}}{4 c e} \]
command
integrate((a+b*arccoth(c*x))*(d+e*log(-c^2*x^2+1)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{2} \, b e x \log \left (-c x + 1\right )^{2} - \frac {1}{2} \, {\left (-i \, \pi b e - b d - 2 \, a e + 2 \, b e\right )} x \log \left (c x + 1\right ) + \frac {1}{2} \, {\left (b e x + \frac {b e}{c}\right )} \log \left (c x + 1\right )^{2} - \frac {b e \log \left (c x - 1\right )^{2}}{2 \, c} - \frac {1}{2} \, {\left (-i \, \pi b d + 2 i \, \pi b e - 2 \, a d + 4 \, a e\right )} x - \frac {1}{2} \, {\left ({\left (-i \, \pi b e + b d - 2 \, a e - 2 \, b e\right )} x - \frac {2 \, b e \log \left (c x - 1\right )}{c}\right )} \log \left (-c x + 1\right ) + \frac {{\left (i \, \pi b e + b d + 2 \, a e - 2 \, b e\right )} \log \left (c x + 1\right )}{2 \, c} + \frac {{\left (-i \, \pi b e + b d - 2 \, a e - 2 \, b e\right )} \log \left (c x - 1\right )}{2 \, c} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int {\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}\,{d x} \]________________________________________________________________________________________