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