\[ \int \frac {(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^4} \, dx \]
Optimal antiderivative \[ \frac {2 e^{3} \sin \left (d x +c \right )}{117 a^{4} d \left (e \sec \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {2 e^{2} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{39 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} d \sqrt {\cos \left (d x +c \right )}\, \sqrt {e \sec \left (d x +c \right )}}+\frac {4 i e^{2}}{13 a d \sqrt {e \sec \left (d x +c \right )}\, \left (a +i a \tan \left (d x +c \right )\right )^{3}}+\frac {4 i e^{4}}{117 d \left (e \sec \left (d x +c \right )\right )^{\frac {5}{2}} \left (a^{4}+i a^{4} \tan \left (d x +c \right )\right )} \]
command
integrate((e*sec(d*x+c))^(3/2)/(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
\[ \frac {{\left (24 i \, \sqrt {2} e^{\left (7 i \, d x + 7 i \, c + \frac {3}{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right ) + \frac {\sqrt {2} {\left (9 i \, e^{\frac {3}{2}} + 24 i \, e^{\left (8 i \, d x + 8 i \, c + \frac {3}{2}\right )} + 55 i \, e^{\left (6 i \, d x + 6 i \, c + \frac {3}{2}\right )} + 59 i \, e^{\left (4 i \, d x + 4 i \, c + \frac {3}{2}\right )} + 37 i \, e^{\left (2 i \, d x + 2 i \, c + \frac {3}{2}\right )}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-7 i \, d x - 7 i \, c\right )}}{468 \, a^{4} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {{\left (468 \, a^{4} d e^{\left (7 i \, d x + 7 i \, c\right )} {\rm integral}\left (-\frac {i \, \sqrt {2} e \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{39 \, a^{4} d}, x\right ) + \sqrt {2} {\left (24 i \, e e^{\left (8 i \, d x + 8 i \, c\right )} + 55 i \, e e^{\left (6 i \, d x + 6 i \, c\right )} + 59 i \, e e^{\left (4 i \, d x + 4 i \, c\right )} + 37 i \, e e^{\left (2 i \, d x + 2 i \, c\right )} + 9 i \, e\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}\right )} e^{\left (-7 i \, d x - 7 i \, c\right )}}{468 \, a^{4} d} \]