\[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+i a \tan (c+d x))^2} \, dx \]
Optimal antiderivative \[ \frac {10 \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{33 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} d \cos \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 \cos \left (d x +c \right ) \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{11 a^{2} d}+\frac {10 \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \tan \left (d x +c \right )}{33 a^{2} d}+\frac {4 i \left (\cos ^{2}\left (d x +c \right )\right ) \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{11 d \left (a^{2}+i a^{2} \tan \left (d x +c \right )\right )} \]
command
integrate((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
\[ \frac {{\left (\sqrt {\frac {1}{2}} {\left (3 i \, e^{\frac {3}{2}} - 11 i \, e^{\left (6 i \, d x + 6 i \, c + \frac {3}{2}\right )} + 41 i \, e^{\left (4 i \, d x + 4 i \, c + \frac {3}{2}\right )} + 15 i \, e^{\left (2 i \, d x + 2 i \, c + \frac {3}{2}\right )}\right )} \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} - 40 i \, \sqrt {2} e^{\left (5 i \, d x + 5 i \, c + \frac {3}{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{132 \, a^{2} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {{\left (132 \, a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} {\rm integral}\left (-\frac {10 i \, \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{33 \, {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}}, x\right ) + \sqrt {\frac {1}{2}} {\left (-11 i \, e e^{\left (6 i \, d x + 6 i \, c\right )} + 41 i \, e e^{\left (4 i \, d x + 4 i \, c\right )} + 15 i \, e e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i \, e\right )} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{132 \, a^{2} d} \]