\[ \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2} \, dx \]
Optimal antiderivative \[ \frac {2 e \sin \left (d x +c \right )}{9 a^{2} d \left (e \sec \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {2 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} d \sqrt {\cos \left (d x +c \right )}\, \sqrt {e \sec \left (d x +c \right )}}+\frac {4 i e^{2}}{9 d \left (e \sec \left (d x +c \right )\right )^{\frac {5}{2}} \left (a^{2}+i a^{2} \tan \left (d x +c \right )\right )} \]
command
integrate(1/(e*sec(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
\[ \frac {{\left (24 i \, \sqrt {2} e^{\left (5 i \, d x + 5 i \, c\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right ) + \frac {\sqrt {2} {\left (15 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 19 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 5 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-5 i \, d x - 5 i \, c - \frac {1}{2}\right )}}{36 \, a^{2} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {\sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-9 i \, e^{\left (7 i \, d x + 7 i \, c\right )} - 15 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 5 i \, e^{\left (5 i \, d x + 5 i \, c\right )} - 19 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 5 i \, e^{\left (3 i \, d x + 3 i \, c\right )} - 5 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i \, e^{\left (i \, d x + i \, c\right )} - i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 36 \, {\left (a^{2} d e e^{\left (6 i \, d x + 6 i \, c\right )} - a^{2} d e e^{\left (5 i \, d x + 5 i \, c\right )}\right )} {\rm integral}\left (\frac {\sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 2 i \, e^{\left (i \, d x + i \, c\right )} - i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{3 \, {\left (a^{2} d e e^{\left (3 i \, d x + 3 i \, c\right )} - 2 \, a^{2} d e e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d e e^{\left (i \, d x + i \, c\right )}\right )}}, x\right )}{36 \, {\left (a^{2} d e e^{\left (6 i \, d x + 6 i \, c\right )} - a^{2} d e e^{\left (5 i \, d x + 5 i \, c\right )}\right )}} \]