\[ \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {\pi +c^2 \pi x^2}} \, dx \]
Optimal antiderivative \[ \frac {2 b x}{3 c^{3} \sqrt {\pi }}-\frac {b \,x^{3}}{9 c \sqrt {\pi }}-\frac {2 \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} \pi \,x^{2}+\pi }}{3 c^{4} \pi }+\frac {x^{2} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} \pi \,x^{2}+\pi }}{3 c^{2} \pi } \]
command
int(x^3*(a+b*arcsinh(c*x))/(Pi*c^2*x^2+Pi)^(1/2),x)
Maple 2022.1 output
\[\int \frac {x^{3} \left (a +b \arcsinh \left (c x \right )\right )}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\, dx\]
Maple 2021.1 output
\[ a \left (\frac {x^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}{3 \pi \,c^{2}}-\frac {2 \sqrt {\pi \,c^{2} x^{2}+\pi }}{3 \pi \,c^{4}}\right )+\frac {b \left (3 \arcsinh \left (c x \right ) c^{4} x^{4}-3 \arcsinh \left (c x \right ) c^{2} x^{2}-c^{3} x^{3} \sqrt {c^{2} x^{2}+1}-6 \arcsinh \left (c x \right )+6 c x \sqrt {c^{2} x^{2}+1}\right )}{9 c^{4} \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}} \]