\[ \int x^2 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx \]
Optimal antiderivative \[ -\frac {2 a e x}{3 c^{2}}-\frac {5 b e \,x^{2}}{18 c}-\frac {2 a e \,x^{3}}{9}-\frac {2 b e x \,\mathrm {arccoth}\left (c x \right )}{3 c^{2}}-\frac {2 b e \,x^{3} \mathrm {arccoth}\left (c x \right )}{9}+\frac {b e \mathrm {arccoth}\left (c x \right )^{2}}{3 c^{3}}-\frac {\left (2 a +b \right ) e \ln \left (-c x +1\right )}{6 c^{3}}+\frac {\left (2 a -b \right ) e \ln \left (c x +1\right )}{6 c^{3}}-\frac {4 b e \ln \left (-c^{2} x^{2}+1\right )}{9 c^{3}}-\frac {b e \ln \left (-c^{2} x^{2}+1\right )^{2}}{12 c^{3}}+\frac {b \,x^{2} \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )}{6 c}+\frac {x^{3} \left (a +b \,\mathrm {arccoth}\left (c x \right )\right ) \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )}{3}+\frac {b \ln \left (-c^{2} x^{2}+1\right ) \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )}{6 c^{3}} \]
command
integrate(x^2*(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
\[ -\frac {1}{6} \, b e x^{3} \log \left (-c x + 1\right )^{2} - \frac {1}{18} \, {\left (-3 i \, \pi b d + 2 i \, \pi b e - 6 \, a d + 4 \, a e\right )} x^{3} + \frac {1}{6} \, {\left (b e x^{3} + \frac {b e}{c^{3}}\right )} \log \left (c x + 1\right )^{2} + \frac {{\left (3 \, b d - 5 \, b e\right )} x^{2}}{18 \, c} - \frac {1}{18} \, {\left ({\left (-3 i \, \pi b e - 3 \, b d - 6 \, a e + 2 \, b e\right )} x^{3} - \frac {3 \, b e x^{2}}{c} + \frac {6 \, b e x}{c^{2}}\right )} \log \left (c x + 1\right ) - \frac {1}{18} \, {\left ({\left (-3 i \, \pi b e + 3 \, b d - 6 \, a e - 2 \, b e\right )} x^{3} - \frac {3 \, b e x^{2}}{c} - \frac {6 \, b e x}{c^{2}} - \frac {6 \, b e \log \left (c x - 1\right )}{c^{3}}\right )} \log \left (-c x + 1\right ) - \frac {b e \log \left (c x - 1\right )^{2}}{6 \, c^{3}} - \frac {{\left (i \, \pi b e + 2 \, a e\right )} x}{3 \, c^{2}} + \frac {{\left (3 i \, \pi b e + 3 \, b d + 6 \, a e - 11 \, b e\right )} \log \left (c x + 1\right )}{18 \, c^{3}} + \frac {{\left (-3 i \, \pi b e + 3 \, b d - 6 \, a e - 11 \, b e\right )} \log \left (c x - 1\right )}{18 \, c^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int {\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )} x^{2}\,{d x} \]________________________________________________________________________________________