\[ \int \left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx \]
Optimal antiderivative \[ 2 b^{2} d x +\frac {4 b^{2} e x}{9 c^{2}}+\frac {2 b^{2} e \,x^{3}}{27}+d x \left (a +b \,\mathrm {arccosh}\! \left (c x \right )\right )^{2}+\frac {e \,x^{3} \left (a +b \,\mathrm {arccosh}\! \left (c x \right )\right )^{2}}{3}-\frac {2 b d \left (a +b \,\mathrm {arccosh}\! \left (c x \right )\right ) \sqrt {c x -1}\, \sqrt {c x +1}}{c}-\frac {4 b e \left (a +b \,\mathrm {arccosh}\! \left (c x \right )\right ) \sqrt {c x -1}\, \sqrt {c x +1}}{9 c^{3}}-\frac {2 b e \,x^{2} \left (a +b \,\mathrm {arccosh}\! \left (c x \right )\right ) \sqrt {c x -1}\, \sqrt {c x +1}}{9 c} \]
command
int((e*x^2+d)*(a+b*arccosh(c*x))^2,x)
Maple 2022.1 output
\[\int \left (e \,x^{2}+d \right ) \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {\frac {a^{2} \left (\frac {1}{3} c^{3} x^{3} e +c^{3} d x \right )}{c^{2}}+\frac {b^{2} \left (\frac {e \left (9 \mathrm {arccosh}\left (c x \right )^{2} c^{3} x^{3}-6 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c^{3} x^{3}+12 c x \right )}{27}+c^{2} d \left (\mathrm {arccosh}\left (c x \right )^{2} c x -2 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )\right )}{c^{2}}+\frac {2 a b \left (\frac {\mathrm {arccosh}\left (c x \right ) c^{3} x^{3} e}{3}+\mathrm {arccosh}\left (c x \right ) c^{3} d x -\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2} e +9 c^{2} d +2 e \right )}{9}\right )}{c^{2}}}{c} \]