\[ \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {\pi +c^2 \pi x^2}} \, dx \]
Optimal antiderivative \[ \frac {3 b \,x^{2}}{16 c^{3} \sqrt {\pi }}-\frac {b \,x^{4}}{16 c \sqrt {\pi }}+\frac {3 \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{16 b \,c^{5} \sqrt {\pi }}-\frac {3 x \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} \pi \,x^{2}+\pi }}{8 c^{4} \pi }+\frac {x^{3} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} \pi \,x^{2}+\pi }}{4 c^{2} \pi } \]
command
int(x^4*(a+b*arcsinh(c*x))/(Pi*c^2*x^2+Pi)^(1/2),x)
Maple 2022.1 output
\[\int \frac {x^{4} \left (a +b \arcsinh \left (c x \right )\right )}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\, dx\]
Maple 2021.1 output
\[ \frac {a \,x^{3} \sqrt {\pi \,c^{2} x^{2}+\pi }}{4 \pi \,c^{2}}-\frac {3 a x \sqrt {\pi \,c^{2} x^{2}+\pi }}{8 c^{4} \pi }+\frac {3 a \ln \left (\frac {\pi x \,c^{2}}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{8 c^{4} \sqrt {\pi \,c^{2}}}+\frac {b \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3}}{4 c^{2} \sqrt {\pi }}-\frac {b \,x^{4}}{16 c \sqrt {\pi }}-\frac {3 b \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x}{8 c^{4} \sqrt {\pi }}+\frac {3 b \,x^{2}}{16 c^{3} \sqrt {\pi }}+\frac {3 b \arcsinh \left (c x \right )^{2}}{16 c^{5} \sqrt {\pi }}+\frac {3 b}{16 c^{5} \sqrt {\pi }} \]