\[ \int \frac {(d+i c d x)^4 (a+b \text {ArcTan}(c x))}{x^8} \, dx \]
Optimal antiderivative \[ -\frac {b c \,d^{4}}{42 x^{6}}-\frac {2 i b \,c^{2} d^{4}}{15 x^{5}}+\frac {47 b \,c^{3} d^{4}}{140 x^{4}}+\frac {5 i b \,c^{4} d^{4}}{9 x^{3}}-\frac {88 b \,c^{5} d^{4}}{105 x^{2}}-\frac {5 i b \,c^{6} d^{4}}{3 x}-\frac {d^{4} \left (a +b \arctan \left (c x \right )\right )}{7 x^{7}}-\frac {2 i c \,d^{4} \left (a +b \arctan \left (c x \right )\right )}{3 x^{6}}+\frac {6 c^{2} d^{4} \left (a +b \arctan \left (c x \right )\right )}{5 x^{5}}+\frac {i c^{3} d^{4} \left (a +b \arctan \left (c x \right )\right )}{x^{4}}-\frac {c^{4} d^{4} \left (a +b \arctan \left (c x \right )\right )}{3 x^{3}}-\frac {176 b \,c^{7} d^{4} \ln \left (x \right )}{105}+\frac {b \,c^{7} d^{4} \ln \left (-c x +i\right )}{210}+\frac {117 b \,c^{7} d^{4} \ln \left (c x +i\right )}{70} \]
command
integrate((d+I*c*d*x)**4*(a+b*atan(c*x))/x**8,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {176 b c^{7} d^{4} \log {\left (43427825 b^{2} c^{15} d^{8} x \right )}}{105} + \frac {b c^{7} d^{4} \log {\left (43427825 b^{2} c^{15} d^{8} x - 43427825 i b^{2} c^{14} d^{8} \right )}}{210} + \frac {117 b c^{7} d^{4} \log {\left (43427825 b^{2} c^{15} d^{8} x + 43427825 i b^{2} c^{14} d^{8} \right )}}{70} + \frac {\left (- 35 i b c^{4} d^{4} x^{4} - 105 b c^{3} d^{4} x^{3} + 126 i b c^{2} d^{4} x^{2} + 70 b c d^{4} x - 15 i b d^{4}\right ) \log {\left (- i c x + 1 \right )}}{210 x^{7}} + \frac {\left (35 i b c^{4} d^{4} x^{4} + 105 b c^{3} d^{4} x^{3} - 126 i b c^{2} d^{4} x^{2} - 70 b c d^{4} x + 15 i b d^{4}\right ) \log {\left (i c x + 1 \right )}}{210 x^{7}} + \frac {- 180 a d^{4} - 2100 i b c^{6} d^{4} x^{6} - 1056 b c^{5} d^{4} x^{5} + x^{4} \left (- 420 a c^{4} d^{4} + 700 i b c^{4} d^{4}\right ) + x^{3} \cdot \left (1260 i a c^{3} d^{4} + 423 b c^{3} d^{4}\right ) + x^{2} \cdot \left (1512 a c^{2} d^{4} - 168 i b c^{2} d^{4}\right ) + x \left (- 840 i a c d^{4} - 30 b c d^{4}\right )}{1260 x^{7}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________