\[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx \]
Optimal antiderivative \[ \frac {2 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {e \cos \left (d x +c \right )}}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} d \sqrt {\cos \left (d x +c \right )}}+\frac {2 i \sqrt {e \cos \left (d x +c \right )}}{9 d \left (a +i a \tan \left (d x +c \right )\right )^{2}}+\frac {2 i \sqrt {e \cos \left (d x +c \right )}}{9 d \left (a^{2}+i a^{2} \tan \left (d x +c \right )\right )} \]
command
integrate((e*cos(d*x+c))^(1/2)/(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
\[ \frac {{\left (\sqrt {\frac {1}{2}} {\left (i \, e^{\frac {1}{2}} + 15 i \, e^{\left (4 i \, d x + 4 i \, c + \frac {1}{2}\right )} + 4 i \, e^{\left (2 i \, d x + 2 i \, c + \frac {1}{2}\right )}\right )} \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} + 12 i \, \sqrt {2} e^{\left (4 i \, d x + 4 i \, c + \frac {1}{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{18 \, a^{2} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {\sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} {\left (-9 i \, e^{\left (5 i \, d x + 5 i \, c\right )} - 15 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 4 i \, e^{\left (3 i \, d x + 3 i \, c\right )} - 4 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i \, e^{\left (i \, d x + i \, c\right )} - i\right )} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} + 18 \, {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} {\rm integral}\left (\frac {\sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} {\left (-2 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 4 i \, e^{\left (i \, d x + i \, c\right )} - 2 i\right )} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{3 \, {\left (a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )} + 2 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - 2 \, a^{2} d e^{\left (i \, d x + i \, c\right )} + a^{2} d\right )}}, x\right )}{18 \, {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )}} \]