\[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{15/2}} \, dx \]
Optimal antiderivative \[ \frac {6 a^{3} \sin \left (d x +c \right )}{55 d \,e^{5} \left (e \sec \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {2 a^{3} \sin \left (d x +c \right )}{11 d \,e^{7} \sqrt {e \sec \left (d x +c \right )}}+\frac {2 a^{3} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {e \sec \left (d x +c \right )}}{11 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \,e^{8}}-\frac {2 i \left (a +i a \tan \left (d x +c \right )\right )^{3}}{15 d \left (e \sec \left (d x +c \right )\right )^{\frac {15}{2}}}-\frac {12 i \left (a^{3}+i a^{3} \tan \left (d x +c \right )\right )}{55 d \,e^{2} \left (e \sec \left (d x +c \right )\right )^{\frac {11}{2}}} \]
command
integrate((a+I*a*tan(d*x+c))^3/(e*sec(d*x+c))^(15/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (-480 i \, \sqrt {2} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \frac {\sqrt {2} {\left (-11 i \, a^{3} e^{\left (10 i \, d x + 10 i \, c\right )} - 73 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 218 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 446 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 235 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 55 i \, a^{3}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c - \frac {15}{2}\right )}}{2640 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {{\left (2640 \, d e^{8} e^{\left (2 i \, d x + 2 i \, c\right )} {\rm integral}\left (-\frac {i \, \sqrt {2} a^{3} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{11 \, d e^{8}}, x\right ) + \sqrt {2} {\left (-11 i \, a^{3} e^{\left (10 i \, d x + 10 i \, c\right )} - 73 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 218 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 446 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 235 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 55 i \, a^{3}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{2640 \, d e^{8}} \]