\[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx \]
Optimal antiderivative \[ \frac {b \left (d -e \right ) x}{2 c}-\frac {b e x}{c}+\frac {d \,x^{2} \left (a +b \,\mathrm {arccoth}\left (c x \right )\right )}{2}-\frac {e \,x^{2} \left (a +b \,\mathrm {arccoth}\left (c x \right )\right )}{2}-\frac {b \left (d -e \right ) \arctanh \left (c x \right )}{2 c^{2}}+\frac {b e \arctanh \left (c x \right )}{c^{2}}+\frac {b e x \ln \left (-c^{2} x^{2}+1\right )}{2 c}-\frac {e \left (-c^{2} x^{2}+1\right ) \left (a +b \,\mathrm {arccoth}\left (c x \right )\right ) \ln \left (-c^{2} x^{2}+1\right )}{2 c^{2}} \]
command
integrate(x*(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
\[ -\frac {1}{4} \, b e x^{2} \log \left (-c x + 1\right )^{2} - \frac {1}{4} \, {\left (-i \, \pi b d + i \, \pi b e - 2 \, a d + 2 \, a e\right )} x^{2} + \frac {1}{4} \, {\left (b e x^{2} - \frac {b e}{c^{2}}\right )} \log \left (c x + 1\right )^{2} - \frac {1}{4} \, {\left ({\left (-i \, \pi b e - b d - 2 \, a e + b e\right )} x^{2} - \frac {2 \, b e x}{c}\right )} \log \left (c x + 1\right ) - \frac {b e \log \left (c x - 1\right )^{2}}{4 \, c^{2}} - \frac {1}{4} \, {\left ({\left (-i \, \pi b e + b d - 2 \, a e - b e\right )} x^{2} - \frac {2 \, b e x}{c} - \frac {2 \, b e \log \left (c x - 1\right )}{c^{2}}\right )} \log \left (-c x + 1\right ) + \frac {{\left (b d - 3 \, b e\right )} x}{2 \, c} + \frac {{\left (-i \, \pi b e - b d - 2 \, a e + 3 \, b e\right )} \log \left (c x + 1\right )}{4 \, c^{2}} + \frac {{\left (-i \, \pi b e + b d - 2 \, a e - 3 \, b e\right )} \log \left (c x - 1\right )}{4 \, c^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int {\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )} x\,{d x} \]________________________________________________________________________________________