\[ \int (d+e x)^2 \cosh ^{-1}(c x)^2 \, dx \]
Optimal antiderivative \[ 2 d^{2} x +\frac {4 e^{2} x}{9 c^{2}}+\frac {d e \,x^{2}}{2}+\frac {2 e^{2} x^{3}}{27}-\frac {d^{3} \mathrm {arccosh}\! \left (c x \right )^{2}}{3 e}-\frac {d e \mathrm {arccosh}\! \left (c x \right )^{2}}{2 c^{2}}+\frac {\left (e x +d \right )^{3} \mathrm {arccosh}\! \left (c x \right )^{2}}{3 e}-\frac {2 d^{2} \mathrm {arccosh}\! \left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{c}-\frac {4 e^{2} \mathrm {arccosh}\! \left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{9 c^{3}}-\frac {d e x \,\mathrm {arccosh}\! \left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{c}-\frac {2 e^{2} x^{2} \mathrm {arccosh}\! \left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{9 c} \]
command
int((e*x+d)^2*arccosh(c*x)^2,x)
Maple 2022.1 output
\[\int \left (e x +d \right )^{2} \mathrm {arccosh}\left (c x \right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {18 \mathrm {arccosh}\left (c x \right )^{2} c^{3} x^{3} e^{2}+54 \mathrm {arccosh}\left (c x \right )^{2} c^{3} x^{2} d e +54 \mathrm {arccosh}\left (c x \right )^{2} c^{3} x \,d^{2}-12 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2} e^{2}-54 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x d e -108 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} d^{2}-27 \mathrm {arccosh}\left (c x \right )^{2} c d e -24 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2}+4 e^{2} c^{3} x^{3}+27 c^{3} x^{2} d e +108 x \,c^{3} d^{2}+24 c x \,e^{2}}{54 c^{3}} \]