\[ \int \frac {(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2} \, dx \]
Optimal antiderivative \[ \frac {6 e^{5} \left (e \sec \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{7 a^{2} d}+\frac {18 e^{3} \left (e \sec \left (d x +c \right )\right )^{\frac {7}{2}} \sin \left (d x +c \right )}{35 a^{2} d}+\frac {6 e^{6} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {e \sec \left (d x +c \right )}}{7 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} d}-\frac {4 i e^{2} \left (e \sec \left (d x +c \right )\right )^{\frac {9}{2}}}{5 d \left (a^{2}+i a^{2} \tan \left (d x +c \right )\right )} \]
command
integrate((e*sec(d*x+c))^(13/2)/(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
\[ -\frac {2 \, {\left (\frac {\sqrt {2} {\left (-15 i \, e^{\frac {13}{2}} + 15 i \, e^{\left (6 i \, d x + 6 i \, c + \frac {13}{2}\right )} + 51 i \, e^{\left (4 i \, d x + 4 i \, c + \frac {13}{2}\right )} + 61 i \, e^{\left (2 i \, d x + 2 i \, c + \frac {13}{2}\right )}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + 15 \, {\left (i \, \sqrt {2} e^{\frac {13}{2}} + i \, \sqrt {2} e^{\left (6 i \, d x + 6 i \, c + \frac {13}{2}\right )} + 3 i \, \sqrt {2} e^{\left (4 i \, d x + 4 i \, c + \frac {13}{2}\right )} + 3 i \, \sqrt {2} e^{\left (2 i \, d x + 2 i \, c + \frac {13}{2}\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}}{35 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {\sqrt {2} {\left (-30 i \, e^{6} e^{\left (6 i \, d x + 6 i \, c\right )} - 102 i \, e^{6} e^{\left (4 i \, d x + 4 i \, c\right )} - 122 i \, e^{6} e^{\left (2 i \, d x + 2 i \, c\right )} + 30 i \, e^{6}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 35 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} {\rm integral}\left (-\frac {3 i \, \sqrt {2} e^{6} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{7 \, a^{2} d}, x\right )}{35 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}} \]