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