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