\[ \int \frac {(e \sec (c+d x))^{11/2}}{a+i a \tan (c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {2 i e^{2} \left (e \sec \left (d x +c \right )\right )^{\frac {7}{2}}}{7 a d}+\frac {2 e^{3} \left (e \sec \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{5 a d}-\frac {6 e^{6} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a d \sqrt {\cos \left (d x +c \right )}\, \sqrt {e \sec \left (d x +c \right )}}+\frac {6 e^{5} \sin \left (d x +c \right ) \sqrt {e \sec \left (d x +c \right )}}{5 a d} \]
command
integrate((e*sec(d*x+c))^(11/2)/(a+I*a*tan(d*x+c)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (\frac {\sqrt {2} {\left (21 i \, e^{\left (7 i \, d x + 7 i \, c + \frac {11}{2}\right )} + 77 i \, e^{\left (5 i \, d x + 5 i \, c + \frac {11}{2}\right )} + 103 i \, e^{\left (3 i \, d x + 3 i \, c + \frac {11}{2}\right )} + 7 i \, e^{\left (i \, d x + i \, c + \frac {11}{2}\right )}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + 21 \, {\left (i \, \sqrt {2} e^{\frac {11}{2}} + i \, \sqrt {2} e^{\left (6 i \, d x + 6 i \, c + \frac {11}{2}\right )} + 3 i \, \sqrt {2} e^{\left (4 i \, d x + 4 i \, c + \frac {11}{2}\right )} + 3 i \, \sqrt {2} e^{\left (2 i \, d x + 2 i \, c + \frac {11}{2}\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{35 \, {\left (a d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {\sqrt {2} {\left (-42 i \, e^{5} e^{\left (7 i \, d x + 7 i \, c\right )} - 154 i \, e^{5} e^{\left (5 i \, d x + 5 i \, c\right )} - 206 i \, e^{5} e^{\left (3 i \, d x + 3 i \, c\right )} - 14 i \, e^{5} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 35 \, {\left (a d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )} {\rm integral}\left (\frac {3 i \, \sqrt {2} e^{5} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{5 \, a d}, x\right )}{35 \, {\left (a d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )}} \]