\[ \int \frac {(e \sec (c+d x))^{9/2}}{(a+i a \tan (c+d x))^3} \, dx \]
Optimal antiderivative \[ \frac {10 i e^{4} \sqrt {e \sec \left (d x +c \right )}}{3 a^{3} d}-\frac {10 e^{4} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {e \sec \left (d x +c \right )}}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} d}+\frac {4 i e^{2} \left (e \sec \left (d x +c \right )\right )^{\frac {5}{2}}}{3 a d \left (a +i a \tan \left (d x +c \right )\right )^{2}} \]
command
integrate((e*sec(d*x+c))^(9/2)/(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
\[ -\frac {2 \, {\left (-5 i \, \sqrt {2} e^{\left (2 i \, d x + 2 i \, c + \frac {9}{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \frac {\sqrt {2} {\left (-2 i \, e^{\frac {9}{2}} - 5 i \, e^{\left (2 i \, d x + 2 i \, c + \frac {9}{2}\right )}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{3 \, a^{3} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {{\left (3 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} {\rm integral}\left (\frac {5 i \, \sqrt {2} e^{4} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{3 \, a^{3} d}, x\right ) + \sqrt {2} {\left (10 i \, e^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 4 i \, e^{4}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{3 \, a^{3} d} \]