\[ \int \left (d+e x^2\right ) \tan ^{-1}(a x) \log \left (c x^n\right ) \, dx \]
Optimal antiderivative \[ \frac {5 e n \,x^{2}}{36 a}-d n x \arctan \left (a x \right )-\frac {e n \,x^{3} \arctan \left (a x \right )}{9}-\frac {e \,x^{2} \ln \left (c \,x^{n}\right )}{6 a}+d x \arctan \left (a x \right ) \ln \left (c \,x^{n}\right )+\frac {e \,x^{3} \arctan \left (a x \right ) \ln \left (c \,x^{n}\right )}{3}+\frac {d n \ln \left (a^{2} x^{2}+1\right )}{2 a}-\frac {e n \ln \left (a^{2} x^{2}+1\right )}{18 a^{3}}-\frac {\left (3 a^{2} d -e \right ) \ln \left (c \,x^{n}\right ) \ln \left (a^{2} x^{2}+1\right )}{6 a^{3}}-\frac {\left (3 a^{2} d -e \right ) n \polylog \left (2, -a^{2} x^{2}\right )}{12 a^{3}} \]
command
int((e*x^2+d)*arctan(a*x)*ln(c*x^n),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(2700\) |
| default | \(\text {Expression too large to display}\) | \(78943\) |
Maple 2021.1 output
\[ \int \left (e \,x^{2}+d \right ) \arctan \left (a x \right ) \ln \left (c \,x^{n}\right )\, dx \]________________________________________________________________________________________