\[ \int \frac {a+b \text {ArcTan}(c x)}{x^2 (d+i c d x)^3} \, dx \]
Optimal antiderivative \[ \frac {b c}{8 d^{3} \left (\mathrm {I}-c x \right )^{2}}-\frac {9 \,\mathrm {I} b c}{8 d^{3} \left (\mathrm {I}-c x \right )}+\frac {9 \,\mathrm {I} b c \arctan \! \left (c x \right )}{8 d^{3}}+\frac {-a -b \arctan \! \left (c x \right )}{d^{3} x}+\frac {\mathrm {I} c \left (a +b \arctan \! \left (c x \right )\right )}{2 d^{3} \left (\mathrm {I}-c x \right )^{2}}+\frac {2 c \left (a +b \arctan \! \left (c x \right )\right )}{d^{3} \left (\mathrm {I}-c x \right )}-\frac {3 \,\mathrm {I} a c \ln \! \left (x \right )}{d^{3}}+\frac {b c \ln \! \left (x \right )}{d^{3}}-\frac {3 \,\mathrm {I} c \left (a +b \arctan \! \left (c x \right )\right ) \ln \! \left (\frac {2}{1+\mathrm {I} c x}\right )}{d^{3}}-\frac {b c \ln \! \left (c^{2} x^{2}+1\right )}{2 d^{3}}+\frac {3 b c \polylog \! \left (2, \mathrm {-I} c x \right )}{2 d^{3}}-\frac {3 b c \polylog \! \left (2, \mathrm {I} c x \right )}{2 d^{3}}+\frac {3 b c \polylog \! \left (2, 1-\frac {2}{1+\mathrm {I} c x}\right )}{2 d^{3}} \]
command
integrate((a+b*arctan(c*x))/x^2/(d+I*c*d*x)^3,x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \text {Exception raised: RuntimeError} \]
Maxima 5.44 via sagemath 9.3 output
\[ -\frac {17 i \, b c^{3} x^{3} \arctan \left (1, c x\right ) + {\left (b {\left (34 \, \arctan \left (1, c x\right ) - 18 i\right )} + 48 \, a\right )} c^{2} x^{2} + {\left (b {\left (-17 i \, \arctan \left (1, c x\right ) - 20\right )} - 72 i \, a\right )} c x + {\left (12 \, b c^{3} x^{3} - 24 i \, b c^{2} x^{2} - 12 \, b c x\right )} \arctan \left (c x\right )^{2} + {\left (3 \, b c^{3} x^{3} - 6 i \, b c^{2} x^{2} - 3 \, b c x\right )} \log \left (c^{2} x^{2} + 1\right )^{2} + {\left (-12 i \, b c^{3} x^{3} - 24 \, b c^{2} x^{2} + 12 i \, b c x\right )} \arctan \left (c x\right ) \log \left (\frac {1}{4} \, c^{2} x^{2} + \frac {1}{4}\right ) + {\left (48 i \, b c^{3} x^{3} + 96 \, b c^{2} x^{2} - 48 i \, b c x\right )} \arctan \left (c x\right ) \log \left (c x\right ) + {\left ({\left (48 \, a - i \, b\right )} c^{3} x^{3} + {\left (-96 i \, a + 46 \, b\right )} c^{2} x^{2} - {\left (48 \, a + 71 i \, b\right )} c x - 16 \, b\right )} \arctan \left (c x\right ) + {\left (24 \, b c^{3} x^{3} - 48 i \, b c^{2} x^{2} - 24 \, b c x\right )} {\rm Li}_2\left (i \, c x + 1\right ) - {\left (24 \, b c^{3} x^{3} - 48 i \, b c^{2} x^{2} - 24 \, b c x\right )} {\rm Li}_2\left (\frac {1}{2} i \, c x + \frac {1}{2}\right ) - {\left (24 \, b c^{3} x^{3} - 48 i \, b c^{2} x^{2} - 24 \, b c x\right )} {\rm Li}_2\left (-i \, c x + 1\right ) - {\left (4 \, {\left ({\left (3 i \, \pi - 2\right )} b + 6 i \, a\right )} c^{3} x^{3} + {\left ({\left (24 \, \pi + 16 i\right )} b + 48 \, a\right )} c^{2} x^{2} + 4 \, {\left ({\left (-3 i \, \pi + 2\right )} b - 6 i \, a\right )} c x + {\left (6 \, b c^{3} x^{3} - 12 i \, b c^{2} x^{2} - 6 \, b c x\right )} \log \left (\frac {1}{4} \, c^{2} x^{2} + \frac {1}{4}\right )\right )} \log \left (c^{2} x^{2} + 1\right ) + {\left ({\left (48 i \, a - 16 \, b\right )} c^{3} x^{3} + 32 \, {\left (3 \, a + i \, b\right )} c^{2} x^{2} + {\left (-48 i \, a + 16 \, b\right )} c x\right )} \log \left (x\right ) - 16 \, a}{16 \, {\left (c^{2} d^{3} x^{3} - 2 i \, c d^{3} x^{2} - d^{3} x\right )}} \]