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