\[ \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3} \, dx \]
Optimal antiderivative \[ \frac {14 e \sin \left (d x +c \right )}{117 a^{3} d \left (e \sec \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {14 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{39 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} d \sqrt {\cos \left (d x +c \right )}\, \sqrt {e \sec \left (d x +c \right )}}+\frac {2 i}{13 d \sqrt {e \sec \left (d x +c \right )}\, \left (a +i a \tan \left (d x +c \right )\right )^{3}}+\frac {28 i e^{2}}{117 d \left (e \sec \left (d x +c \right )\right )^{\frac {5}{2}} \left (a^{3}+i a^{3} \tan \left (d x +c \right )\right )} \]
command
integrate(1/(e*sec(d*x+c))^(1/2)/(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (336 i \, \sqrt {2} e^{\left (7 i \, d x + 7 i \, c\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right ) + \frac {\sqrt {2} {\left (219 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 302 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 124 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 50 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 9 i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-7 i \, d x - 7 i \, c - \frac {1}{2}\right )}}{936 \, a^{3} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {\sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-117 i \, e^{\left (9 i \, d x + 9 i \, c\right )} - 219 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 34 i \, e^{\left (7 i \, d x + 7 i \, c\right )} - 302 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 124 i \, e^{\left (5 i \, d x + 5 i \, c\right )} - 124 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 50 i \, e^{\left (3 i \, d x + 3 i \, c\right )} - 50 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 9 i \, e^{\left (i \, d x + i \, c\right )} - 9 i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 936 \, {\left (a^{3} d e e^{\left (8 i \, d x + 8 i \, c\right )} - a^{3} d e e^{\left (7 i \, d x + 7 i \, c\right )}\right )} {\rm integral}\left (\frac {\sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-7 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 14 i \, e^{\left (i \, d x + i \, c\right )} - 7 i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{39 \, {\left (a^{3} d e e^{\left (3 i \, d x + 3 i \, c\right )} - 2 \, a^{3} d e e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d e e^{\left (i \, d x + i \, c\right )}\right )}}, x\right )}{936 \, {\left (a^{3} d e e^{\left (8 i \, d x + 8 i \, c\right )} - a^{3} d e e^{\left (7 i \, d x + 7 i \, c\right )}\right )}} \]