\[ \int \left (\frac {x^3}{\left (1+a^2 x^2\right ) \text {ArcTan}(a x)^3}-\frac {3 x^2}{2 a \text {ArcTan}(a x)^2}\right ) \, dx \]
Optimal antiderivative \[ -\frac {x^{3}}{2 a \arctan \left (a x \right )^{2}} \]
command
int(x^3/(a^2*x^2+1)/arctan(a*x)^3-3/2*x^2/a/arctan(a*x)^2,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\frac {2 x^{3}}{a \left (\ln \left (-i a x +1\right )-\ln \left (i a x +1\right )\right )^{2}}\) | \(30\) |
Maple 2021.1 output
\[ \int \frac {x^{3}}{\left (a^{2} x^{2}+1\right ) \arctan \left (a x \right )^{3}}-\frac {3 x^{2}}{2 a \arctan \left (a x \right )^{2}}\, dx \]________________________________________________________________________________________