\[ \int (d+e x)^3 \cosh ^{-1}(c x)^2 \, dx \]
Optimal antiderivative \[ 2 d^{3} x +\frac {4 d \,e^{2} x}{3 c^{2}}+\frac {3 d^{2} e \,x^{2}}{4}+\frac {3 e^{3} x^{2}}{32 c^{2}}+\frac {2 d \,e^{2} x^{3}}{9}+\frac {e^{3} x^{4}}{32}-\frac {d^{4} \mathrm {arccosh}\! \left (c x \right )^{2}}{4 e}-\frac {3 d^{2} e \mathrm {arccosh}\! \left (c x \right )^{2}}{4 c^{2}}-\frac {3 e^{3} \mathrm {arccosh}\! \left (c x \right )^{2}}{32 c^{4}}+\frac {\left (e x +d \right )^{4} \mathrm {arccosh}\! \left (c x \right )^{2}}{4 e}-\frac {2 d^{3} \mathrm {arccosh}\! \left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{c}-\frac {4 d \,e^{2} \mathrm {arccosh}\! \left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{3 c^{3}}-\frac {3 d^{2} e x \,\mathrm {arccosh}\! \left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{2 c}-\frac {3 e^{3} x \,\mathrm {arccosh}\! \left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{16 c^{3}}-\frac {2 d \,e^{2} x^{2} \mathrm {arccosh}\! \left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{3 c}-\frac {e^{3} x^{3} \mathrm {arccosh}\! \left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{8 c} \]
command
int((e*x+d)^3*arccosh(c*x)^2,x)
Maple 2022.1 output
\[\int \left (e x +d \right )^{3} \mathrm {arccosh}\left (c x \right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {72 \mathrm {arccosh}\left (c x \right )^{2} c^{4} x^{4} e^{3}+288 \mathrm {arccosh}\left (c x \right )^{2} c^{4} x^{3} d \,e^{2}+432 \mathrm {arccosh}\left (c x \right )^{2} c^{4} x^{2} d^{2} e +288 \mathrm {arccosh}\left (c x \right )^{2} c^{4} x \,d^{3}-36 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3} e^{3}-192 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{2} d \,e^{2}-432 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x \,d^{2} e -576 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} d^{3}-216 \mathrm {arccosh}\left (c x \right )^{2} c^{2} d^{2} e -54 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c x \,e^{3}-384 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c d \,e^{2}+9 c^{4} e^{3} x^{4}+64 c^{4} d \,e^{2} x^{3}+216 c^{4} d^{2} e \,x^{2}+576 x \,c^{4} d^{3}-27 \mathrm {arccosh}\left (c x \right )^{2} e^{3}+27 c^{2} x^{2} e^{3}+384 x \,c^{2} d \,e^{2}}{288 c^{4}} \]