\[ \int \frac {(a+b \text {ArcTan}(c x))^3}{x (d+i c d x)} \, dx \]
Optimal antiderivative \[ \frac {\left (a +b \arctan \left (c x \right )\right )^{3} \ln \left (2-\frac {2}{i c x +1}\right )}{d}+\frac {3 i b \left (a +b \arctan \left (c x \right )\right )^{2} \polylog \left (2, -1+\frac {2}{i c x +1}\right )}{2 d}+\frac {3 b^{2} \left (a +b \arctan \left (c x \right )\right ) \polylog \left (3, -1+\frac {2}{i c x +1}\right )}{2 d}-\frac {3 i b^{3} \polylog \left (4, -1+\frac {2}{i c x +1}\right )}{4 d} \]
command
integrate((a+b*arctan(c*x))^3/x/(d+I*c*d*x),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-3 i \, b^{3} {\rm Li}_2\left (-\frac {2 \, c x}{c x - i} + 1\right ) \log \left (-\frac {c x + i}{c x - i}\right )^{2} - 12 \, a b^{2} {\rm Li}_2\left (-\frac {2 \, c x}{c x - i} + 1\right ) \log \left (-\frac {c x + i}{c x - i}\right ) - 12 i \, a^{2} b {\rm Li}_2\left (\frac {c x + i}{c x - i} + 1\right ) + 8 \, a^{3} \log \left (x\right ) - 8 \, a^{3} \log \left (\frac {c x - i}{c}\right ) - 6 i \, b^{3} {\rm polylog}\left (4, -\frac {c x + i}{c x - i}\right ) + {\left (-i \, b^{3} \log \left (-\frac {c x + i}{c x - i}\right )^{3} - 6 \, a b^{2} \log \left (-\frac {c x + i}{c x - i}\right )^{2}\right )} \log \left (\frac {2 \, c x}{c x - i}\right ) - 6 \, {\left (-i \, b^{3} \log \left (-\frac {c x + i}{c x - i}\right ) - 2 \, a b^{2}\right )} {\rm polylog}\left (3, -\frac {c x + i}{c x - i}\right )}{8 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {b^{3} \log \left (-\frac {c x + i}{c x - i}\right )^{3} - 6 i \, a b^{2} \log \left (-\frac {c x + i}{c x - i}\right )^{2} - 12 \, a^{2} b \log \left (-\frac {c x + i}{c x - i}\right ) + 8 i \, a^{3}}{8 \, c d x^{2} - 8 i \, d x}, x\right ) \]