\[ \int \sec ^3(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx \]
Optimal antiderivative \[ \frac {64 i a^{3} \left (\sec ^{3}\left (d x +c \right )\right )}{105 d \sqrt {a +i a \tan \left (d x +c \right )}}+\frac {8 i a^{2} \left (\sec ^{3}\left (d x +c \right )\right ) \sqrt {a +i a \tan \left (d x +c \right )}}{21 d}+\frac {256 i a^{4} \left (\sec ^{3}\left (d x +c \right )\right )}{315 d \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {2 i a \left (\sec ^{3}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{9 d} \]
command
integrate(sec(d*x+c)^3*(a+I*a*tan(d*x+c))^(5/2),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {32 \, {\left (105 i \, \sqrt {2} a^{2} \cos \left (6 \, d x + 6 \, c\right ) + 126 i \, \sqrt {2} a^{2} \cos \left (4 \, d x + 4 \, c\right ) + 72 i \, \sqrt {2} a^{2} \cos \left (2 \, d x + 2 \, c\right ) - 105 \, \sqrt {2} a^{2} \sin \left (6 \, d x + 6 \, c\right ) - 126 \, \sqrt {2} a^{2} \sin \left (4 \, d x + 4 \, c\right ) - 72 \, \sqrt {2} a^{2} \sin \left (2 \, d x + 2 \, c\right ) + 16 i \, \sqrt {2} a^{2}\right )} \sqrt {a}}{315 \, {\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )}^{\frac {1}{4}} {\left ({\left (2 \, \cos \left (2 \, d x + 2 \, c\right )^{3} + {\left (2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 i \, \sin \left (2 \, d x + 2 \, c\right )^{3} + {\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \cos \left (4 \, d x + 4 \, c\right ) + 5 \, \cos \left (2 \, d x + 2 \, c\right )^{2} + {\left (i \, \cos \left (2 \, d x + 2 \, c\right )^{2} + i \, \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 i \, \cos \left (2 \, d x + 2 \, c\right ) + i\right )} \sin \left (4 \, d x + 4 \, c\right ) + 2 \, {\left (i \, \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 i \, \cos \left (2 \, d x + 2 \, c\right ) + i\right )} \sin \left (2 \, d x + 2 \, c\right ) + 4 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \cos \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) + {\left (2 i \, \cos \left (2 \, d x + 2 \, c\right )^{3} + {\left (2 i \, \cos \left (2 \, d x + 2 \, c\right ) + i\right )} \sin \left (2 \, d x + 2 \, c\right )^{2} - 2 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + {\left (i \, \cos \left (2 \, d x + 2 \, c\right )^{2} + i \, \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 i \, \cos \left (2 \, d x + 2 \, c\right ) + i\right )} \cos \left (4 \, d x + 4 \, c\right ) + 5 i \, \cos \left (2 \, d x + 2 \, c\right )^{2} - {\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sin \left (4 \, d x + 4 \, c\right ) - 2 \, {\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sin \left (2 \, d x + 2 \, c\right ) + 4 i \, \cos \left (2 \, d x + 2 \, c\right ) + i\right )} \sin \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right )\right )} d} \]
Maxima 5.44 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________