\[ \int \frac {1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))} \, dx \]
Optimal antiderivative \[ \frac {10 \sin \left (d x +c \right )}{21 a d e \sqrt {e \sec \left (d x +c \right )}}+\frac {10 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {e \sec \left (d x +c \right )}}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a d \,e^{2}}+\frac {2 i}{7 d \left (e \sec \left (d x +c \right )\right )^{\frac {3}{2}} \left (a +i a \tan \left (d x +c \right )\right )} \]
command
integrate(1/(e*sec(d*x+c))^(3/2)/(a+I*a*tan(d*x+c)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (-40 i \, \sqrt {2} e^{\left (4 i \, d x + 4 i \, c\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \frac {\sqrt {2} {\left (-7 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 9 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 19 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-4 i \, d x - 4 i \, c - \frac {3}{2}\right )}}{84 \, a d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {{\left (84 \, a d e^{2} e^{\left (4 i \, d x + 4 i \, c\right )} {\rm integral}\left (-\frac {5 i \, \sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{21 \, a d e^{2}}, x\right ) + \sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-7 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 9 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 19 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{84 \, a d e^{2}} \]