\[ \int \left (d+e x^2\right ) \sinh ^{-1}(a x) \log \left (c x^n\right ) \, dx \]
Optimal antiderivative \[ \frac {2 e n \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{27 a^{3}}-d n x \arcsinh \left (a x \right )-\frac {e n \,x^{3} \arcsinh \left (a x \right )}{9}-\frac {\left (3 a^{2} d -e \right ) n \arctanh \left (\sqrt {a^{2} x^{2}+1}\right )}{3 a^{3}}-\frac {e n \arctanh \left (\sqrt {a^{2} x^{2}+1}\right )}{9 a^{3}}-\frac {e \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} \ln \left (c \,x^{n}\right )}{9 a^{3}}+d x \arcsinh \left (a x \right ) \ln \left (c \,x^{n}\right )+\frac {e \,x^{3} \arcsinh \left (a x \right ) \ln \left (c \,x^{n}\right )}{3}+\frac {d n \sqrt {a^{2} x^{2}+1}}{a}+\frac {\left (3 a^{2} d -e \right ) n \sqrt {a^{2} x^{2}+1}}{3 a^{3}}-\frac {\left (3 a^{2} d -e \right ) \ln \left (c \,x^{n}\right ) \sqrt {a^{2} x^{2}+1}}{3 a^{3}} \]
command
int((e*x^2+d)*arcsinh(a*x)*ln(c*x^n),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(4077\) |
Maple 2021.1 output
\[ \int \left (e \,x^{2}+d \right ) \arcsinh \left (a x \right ) \ln \left (c \,x^{n}\right )\, dx \]________________________________________________________________________________________