\[ \int \frac {(a+b \text {ArcTan}(c x))^3}{(d+i c d x)^3} \, dx \]
Optimal antiderivative \[ \frac {3 b^{3}}{64 c \,d^{3} \left (-c x +i\right )^{2}}-\frac {21 i b^{3}}{64 c \,d^{3} \left (-c x +i\right )}+\frac {21 i b^{3} \arctan \left (c x \right )}{64 c \,d^{3}}+\frac {3 i b^{2} \left (a +b \arctan \left (c x \right )\right )}{16 c \,d^{3} \left (-c x +i\right )^{2}}+\frac {9 b^{2} \left (a +b \arctan \left (c x \right )\right )}{16 c \,d^{3} \left (-c x +i\right )}-\frac {9 b \left (a +b \arctan \left (c x \right )\right )^{2}}{32 c \,d^{3}}-\frac {3 b \left (a +b \arctan \left (c x \right )\right )^{2}}{8 c \,d^{3} \left (-c x +i\right )^{2}}+\frac {3 i b \left (a +b \arctan \left (c x \right )\right )^{2}}{8 c \,d^{3} \left (-c x +i\right )}-\frac {i \left (a +b \arctan \left (c x \right )\right )^{3}}{8 c \,d^{3}}+\frac {i \left (a +b \arctan \left (c x \right )\right )^{3}}{2 c \,d^{3} \left (i c x +1\right )^{2}} \]
command
integrate((a+b*atan(c*x))**3/(d+I*c*d*x)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {3 b \left (8 a^{2} - 12 i a b - 7 b^{2}\right ) \log {\left (- \frac {3 i b \left (8 a^{2} - 12 i a b - 7 b^{2}\right )}{c} + x \left (24 a^{2} b - 36 i a b^{2} - 21 b^{3}\right ) \right )}}{128 c d^{3}} + \frac {3 b \left (8 a^{2} - 12 i a b - 7 b^{2}\right ) \log {\left (\frac {3 i b \left (8 a^{2} - 12 i a b - 7 b^{2}\right )}{c} + x \left (24 a^{2} b - 36 i a b^{2} - 21 b^{3}\right ) \right )}}{128 c d^{3}} + \frac {\left (- b^{3} c^{2} x^{2} + 2 i b^{3} c x - 3 b^{3}\right ) \log {\left (- i c x + 1 \right )}^{3}}{64 c^{3} d^{3} x^{2} - 128 i c^{2} d^{3} x - 64 c d^{3}} + \frac {\left (b^{3} c^{2} x^{2} - 2 i b^{3} c x + 3 b^{3}\right ) \log {\left (i c x + 1 \right )}^{3}}{64 c^{3} d^{3} x^{2} - 128 i c^{2} d^{3} x - 64 c d^{3}} + \frac {\left (12 i a b^{2} c^{2} x^{2} + 24 a b^{2} c x + 36 i a b^{2} + 9 b^{3} c^{2} x^{2} - 6 i b^{3} c x + 15 b^{3}\right ) \log {\left (i c x + 1 \right )}^{2}}{128 c^{3} d^{3} x^{2} - 256 i c^{2} d^{3} x - 128 c d^{3}} + \frac {\left (12 i a b^{2} c^{2} x^{2} + 24 a b^{2} c x + 36 i a b^{2} + 6 b^{3} c^{2} x^{2} \log {\left (i c x + 1 \right )} + 9 b^{3} c^{2} x^{2} - 12 i b^{3} c x \log {\left (i c x + 1 \right )} - 6 i b^{3} c x + 18 b^{3} \log {\left (i c x + 1 \right )} + 15 b^{3}\right ) \log {\left (- i c x + 1 \right )}^{2}}{128 c^{3} d^{3} x^{2} - 256 i c^{2} d^{3} x - 128 c d^{3}} + \frac {- 32 i a^{3} - 48 a^{2} b + 48 i a b^{2} + 24 b^{3} + x \left (- 24 i a^{2} b c - 36 a b^{2} c + 21 i b^{3} c\right )}{64 c^{3} d^{3} x^{2} - 128 i c^{2} d^{3} x - 64 c d^{3}} + \frac {\left (48 a^{2} b - 12 i a b^{2} c^{2} x^{2} \log {\left (i c x + 1 \right )} - 24 a b^{2} c x \log {\left (i c x + 1 \right )} + 24 a b^{2} c x - 36 i a b^{2} \log {\left (i c x + 1 \right )} - 48 i a b^{2} - 3 b^{3} c^{2} x^{2} \log {\left (i c x + 1 \right )}^{2} - 9 b^{3} c^{2} x^{2} \log {\left (i c x + 1 \right )} + 6 i b^{3} c x \log {\left (i c x + 1 \right )}^{2} + 6 i b^{3} c x \log {\left (i c x + 1 \right )} - 18 i b^{3} c x - 9 b^{3} \log {\left (i c x + 1 \right )}^{2} - 15 b^{3} \log {\left (i c x + 1 \right )} - 24 b^{3}\right ) \log {\left (- i c x + 1 \right )}}{64 c^{3} d^{3} x^{2} - 128 i c^{2} d^{3} x - 64 c d^{3}} + \frac {\left (- 24 a^{2} b - 12 a b^{2} c x + 24 i a b^{2} + 9 i b^{3} c x + 12 b^{3}\right ) \log {\left (i c x + 1 \right )}}{32 c^{3} d^{3} x^{2} - 64 i c^{2} d^{3} x - 32 c d^{3}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________