\[ \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^4} \, dx \]
Optimal antiderivative \[ \frac {2 e \sin \left (d x +c \right )}{33 a^{4} d \sqrt {e \sec \left (d x +c \right )}}+\frac {2 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {e \sec \left (d x +c \right )}}{33 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} d}+\frac {2 i \sqrt {e \sec \left (d x +c \right )}}{15 d \left (a +i a \tan \left (d x +c \right )\right )^{4}}+\frac {14 i \sqrt {e \sec \left (d x +c \right )}}{165 a d \left (a +i a \tan \left (d x +c \right )\right )^{3}}+\frac {4 i e^{2}}{33 d \left (e \sec \left (d x +c \right )\right )^{\frac {3}{2}} \left (a^{4}+i a^{4} \tan \left (d x +c \right )\right )} \]
command
integrate((e*sec(d*x+c))^(1/2)/(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
\[ \frac {{\left (-80 i \, \sqrt {2} e^{\left (8 i \, d x + 8 i \, c + \frac {1}{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \frac {\sqrt {2} {\left (11 i \, e^{\frac {1}{2}} + 85 i \, e^{\left (8 i \, d x + 8 i \, c + \frac {1}{2}\right )} + 166 i \, e^{\left (6 i \, d x + 6 i \, c + \frac {1}{2}\right )} + 128 i \, e^{\left (4 i \, d x + 4 i \, c + \frac {1}{2}\right )} + 58 i \, e^{\left (2 i \, d x + 2 i \, c + \frac {1}{2}\right )}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-8 i \, d x - 8 i \, c\right )}}{1320 \, a^{4} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {{\left (1320 \, a^{4} d e^{\left (8 i \, d x + 8 i \, c\right )} {\rm integral}\left (-\frac {i \, \sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{33 \, a^{4} d}, x\right ) + \sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (85 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 166 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 128 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 58 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 11 i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}\right )} e^{\left (-8 i \, d x - 8 i \, c\right )}}{1320 \, a^{4} d} \]