\[ \int \frac {a+b \text {ArcTan}(c x)}{x^4 (d+i c d x)} \, dx \]
Optimal antiderivative \[ -\frac {b c}{6 d \,x^{2}}+\frac {i b \,c^{2}}{2 d x}+\frac {i b \,c^{3} \arctan \left (c x \right )}{2 d}+\frac {-a -b \arctan \left (c x \right )}{3 d \,x^{3}}+\frac {i c \left (a +b \arctan \left (c x \right )\right )}{2 d \,x^{2}}+\frac {c^{2} \left (a +b \arctan \left (c x \right )\right )}{d x}-\frac {4 b \,c^{3} \ln \left (x \right )}{3 d}+\frac {2 b \,c^{3} \ln \left (c^{2} x^{2}+1\right )}{3 d}+\frac {i c^{3} \left (a +b \arctan \left (c x \right )\right ) \ln \left (2-\frac {2}{i c x +1}\right )}{d}-\frac {b \,c^{3} \polylog \left (2, -1+\frac {2}{i c x +1}\right )}{2 d} \]
command
integrate((a+b*arctan(c*x))/x^4/(d+I*c*d*x),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {6 \, b c^{3} x^{3} {\rm Li}_2\left (\frac {c x + i}{c x - i} + 1\right ) - 4 \, {\left (-3 i \, a + 4 \, b\right )} c^{3} x^{3} \log \left (x\right ) + 5 \, b c^{3} x^{3} \log \left (\frac {c x + i}{c}\right ) + {\left (-12 i \, a + 11 \, b\right )} c^{3} x^{3} \log \left (\frac {c x - i}{c}\right ) + 6 \, {\left (2 \, a + i \, b\right )} c^{2} x^{2} - 2 \, {\left (-3 i \, a + b\right )} c x + {\left (6 i \, b c^{2} x^{2} - 3 \, b c x - 2 i \, b\right )} \log \left (-\frac {c x + i}{c x - i}\right ) - 4 \, a}{12 \, d x^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {b \log \left (-\frac {c x + i}{c x - i}\right ) - 2 i \, a}{2 \, c d x^{5} - 2 i \, d x^{4}}, x\right ) \]