\[ \int \frac {(e \sec (c+d x))^{5/2}}{(a+i a \tan (c+d x))^4} \, dx \]
Optimal antiderivative \[ -\frac {2 e^{3} \sin \left (d x +c \right )}{77 a^{4} d \sqrt {e \sec \left (d x +c \right )}}-\frac {2 e^{2} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {e \sec \left (d x +c \right )}}{77 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} d}+\frac {4 i e^{2} \sqrt {e \sec \left (d x +c \right )}}{11 a d \left (a +i a \tan \left (d x +c \right )\right )^{3}}-\frac {4 i e^{4}}{77 d \left (e \sec \left (d x +c \right )\right )^{\frac {3}{2}} \left (a^{4}+i a^{4} \tan \left (d x +c \right )\right )} \]
command
integrate((e*sec(d*x+c))^(5/2)/(a+I*a*tan(d*x+c))^4,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (4 i \, \sqrt {2} e^{\left (6 i \, d x + 6 i \, c + \frac {5}{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \frac {\sqrt {2} {\left (7 i \, e^{\frac {5}{2}} + 4 i \, e^{\left (6 i \, d x + 6 i \, c + \frac {5}{2}\right )} + 17 i \, e^{\left (4 i \, d x + 4 i \, c + \frac {5}{2}\right )} + 20 i \, e^{\left (2 i \, d x + 2 i \, c + \frac {5}{2}\right )}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{154 \, a^{4} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {{\left (154 \, a^{4} d e^{\left (6 i \, d x + 6 i \, c\right )} {\rm integral}\left (\frac {i \, \sqrt {2} e^{2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{77 \, a^{4} d}, x\right ) + \sqrt {2} {\left (4 i \, e^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 17 i \, e^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 20 i \, e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i \, e^{2}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{154 \, a^{4} d} \]