\[ \int x^4 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx \]
Optimal antiderivative \[ -\frac {2 a e x}{5 c^{4}}-\frac {77 b e \,x^{2}}{300 c^{3}}-\frac {2 a e \,x^{3}}{15 c^{2}}-\frac {9 b e \,x^{4}}{200 c}-\frac {2 a e \,x^{5}}{25}-\frac {2 b e x \,\mathrm {arccoth}\left (c x \right )}{5 c^{4}}-\frac {2 b e \,x^{3} \mathrm {arccoth}\left (c x \right )}{15 c^{2}}-\frac {2 b e \,x^{5} \mathrm {arccoth}\left (c x \right )}{25}+\frac {b e \mathrm {arccoth}\left (c x \right )^{2}}{5 c^{5}}-\frac {\left (4 a +3 b \right ) e \ln \left (-c x +1\right )}{20 c^{5}}+\frac {\left (4 a -3 b \right ) e \ln \left (c x +1\right )}{20 c^{5}}-\frac {23 b e \ln \left (-c^{2} x^{2}+1\right )}{75 c^{5}}-\frac {b e \ln \left (-c^{2} x^{2}+1\right )^{2}}{20 c^{5}}+\frac {b \,x^{2} \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )}{10 c^{3}}+\frac {b \,x^{4} \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )}{20 c}+\frac {x^{5} \left (a +b \,\mathrm {arccoth}\left (c x \right )\right ) \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )}{5}+\frac {b \ln \left (-c^{2} x^{2}+1\right ) \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )}{10 c^{5}} \]
command
integrate(x^4*(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
\[ -\frac {1}{10} \, b e x^{5} \log \left (-c x + 1\right )^{2} - \frac {1}{50} \, {\left (-5 i \, \pi b d + 2 i \, \pi b e - 10 \, a d + 4 \, a e\right )} x^{5} + \frac {{\left (10 \, b d - 9 \, b e\right )} x^{4}}{200 \, c} + \frac {1}{10} \, {\left (b e x^{5} + \frac {b e}{c^{5}}\right )} \log \left (c x + 1\right )^{2} - \frac {{\left (i \, \pi b e + 2 \, a e\right )} x^{3}}{15 \, c^{2}} - \frac {1}{300} \, {\left (6 \, {\left (-5 i \, \pi b e - 5 \, b d - 10 \, a e + 2 \, b e\right )} x^{5} - \frac {15 \, b e x^{4}}{c} + \frac {20 \, b e x^{3}}{c^{2}} - \frac {30 \, b e x^{2}}{c^{3}} + \frac {60 \, b e x}{c^{4}}\right )} \log \left (c x + 1\right ) - \frac {1}{300} \, {\left (6 \, {\left (-5 i \, \pi b e + 5 \, b d - 10 \, a e - 2 \, b e\right )} x^{5} - \frac {15 \, b e x^{4}}{c} - \frac {20 \, b e x^{3}}{c^{2}} - \frac {30 \, b e x^{2}}{c^{3}} - \frac {60 \, b e x}{c^{4}} - \frac {60 \, b e \log \left (c x - 1\right )}{c^{5}}\right )} \log \left (-c x + 1\right ) + \frac {{\left (30 \, b d - 77 \, b e\right )} x^{2}}{300 \, c^{3}} - \frac {b e \log \left (c x - 1\right )^{2}}{10 \, c^{5}} - \frac {{\left (i \, \pi b e + 2 \, a e\right )} x}{5 \, c^{4}} + \frac {{\left (30 i \, \pi b e + 30 \, b d + 60 \, a e - 137 \, b e\right )} \log \left (c x + 1\right )}{300 \, c^{5}} + \frac {{\left (-30 i \, \pi b e + 30 \, b d - 60 \, a e - 137 \, b e\right )} \log \left (c x - 1\right )}{300 \, c^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int {\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )} x^{4}\,{d x} \]________________________________________________________________________________________