\[ \int \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Optimal antiderivative \[ \frac {14 b^{2} d x}{9}+\frac {2 b^{2} c^{2} d \,x^{3}}{27}-\frac {2 b d \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{9 c}+\frac {2 d x \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{3}+\frac {d x \left (c^{2} x^{2}+1\right ) \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{3}-\frac {4 b d \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{3 c} \]
command
int((c^2*d*x^2+d)*(a+b*arcsinh(c*x))^2,x)
Maple 2022.1 output
\[\int \left (c^{2} d \,x^{2}+d \right ) \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {d \,a^{2} \left (\frac {1}{3} c^{3} x^{3}+c x \right )+d \,b^{2} \left (\frac {2 \arcsinh \left (c x \right )^{2} c x}{3}+\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{3}-\frac {4 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3}+\frac {40 c x}{27}-\frac {2 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{9}+\frac {2 c x \left (c^{2} x^{2}+1\right )}{27}\right )+2 d a b \left (\frac {\arcsinh \left (c x \right ) c^{3} x^{3}}{3}+\arcsinh \left (c x \right ) c x -\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {c^{2} x^{2}+1}}{9}\right )}{c} \]