\[ \int \frac {(a+b \text {ArcTan}(c x))^2}{x^2 \left (d+e x^2\right )} \, dx \]
Optimal antiderivative \[ -\frac {i c \left (a +b \arctan \left (c x \right )\right )^{2}}{d}-\frac {\left (a +b \arctan \left (c x \right )\right )^{2}}{d x}+\frac {2 b c \left (a +b \arctan \left (c x \right )\right ) \ln \left (2-\frac {2}{-i c x +1}\right )}{d}-\frac {i b^{2} c \polylog \left (2, -1+\frac {2}{-i c x +1}\right )}{d}+\frac {\left (a +b \arctan \left (c x \right )\right )^{2} \ln \left (\frac {2 c \left (\sqrt {-d}-x \sqrt {e}\right )}{\left (-i c x +1\right ) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right ) \sqrt {e}}{2 \left (-d \right )^{\frac {3}{2}}}-\frac {\left (a +b \arctan \left (c x \right )\right )^{2} \ln \left (\frac {2 c \left (\sqrt {-d}+x \sqrt {e}\right )}{\left (-i c x +1\right ) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right ) \sqrt {e}}{2 \left (-d \right )^{\frac {3}{2}}}-\frac {i b \left (a +b \arctan \left (c x \right )\right ) \polylog \left (2, 1-\frac {2 c \left (\sqrt {-d}-x \sqrt {e}\right )}{\left (-i c x +1\right ) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right ) \sqrt {e}}{2 \left (-d \right )^{\frac {3}{2}}}+\frac {i b \left (a +b \arctan \left (c x \right )\right ) \polylog \left (2, 1-\frac {2 c \left (\sqrt {-d}+x \sqrt {e}\right )}{\left (-i c x +1\right ) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right ) \sqrt {e}}{2 \left (-d \right )^{\frac {3}{2}}}+\frac {b^{2} \polylog \left (3, 1-\frac {2 c \left (\sqrt {-d}-x \sqrt {e}\right )}{\left (-i c x +1\right ) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right ) \sqrt {e}}{4 \left (-d \right )^{\frac {3}{2}}}-\frac {b^{2} \polylog \left (3, 1-\frac {2 c \left (\sqrt {-d}+x \sqrt {e}\right )}{\left (-i c x +1\right ) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right ) \sqrt {e}}{4 \left (-d \right )^{\frac {3}{2}}} \]
command
int((a+b*arctan(c*x))^2/x^2/(e*x^2+d),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(102882\) |
| default | \(\text {Expression too large to display}\) | \(102882\) |
Maple 2021.1 output
\[ \int \frac {\left (a +b \arctan \left (c x \right )\right )^{2}}{x^{2} \left (e \,x^{2}+d \right )}\, dx \]________________________________________________________________________________________