\[ \int \frac {(e \sec (c+d x))^{15/2}}{(a+i a \tan (c+d x))^4} \, dx \]
Optimal antiderivative \[ -\frac {154 e^{5} \left (e \sec \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{15 a^{4} d}+\frac {154 e^{8} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} d \sqrt {\cos \left (d x +c \right )}\, \sqrt {e \sec \left (d x +c \right )}}-\frac {154 e^{7} \sin \left (d x +c \right ) \sqrt {e \sec \left (d x +c \right )}}{5 a^{4} d}+\frac {4 i e^{2} \left (e \sec \left (d x +c \right )\right )^{\frac {11}{2}}}{a d \left (a +i a \tan \left (d x +c \right )\right )^{3}}+\frac {44 i e^{4} \left (e \sec \left (d x +c \right )\right )^{\frac {7}{2}}}{3 d \left (a^{4}+i a^{4} \tan \left (d x +c \right )\right )} \]
command
integrate((e*sec(d*x+c))^(15/2)/(a+I*a*tan(d*x+c))^4,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (\frac {\sqrt {2} {\left (-120 i \, e^{\frac {15}{2}} - 231 i \, e^{\left (6 i \, d x + 6 i \, c + \frac {15}{2}\right )} - 616 i \, e^{\left (4 i \, d x + 4 i \, c + \frac {15}{2}\right )} - 517 i \, e^{\left (2 i \, d x + 2 i \, c + \frac {15}{2}\right )}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + 231 \, {\left (-i \, \sqrt {2} e^{\left (5 i \, d x + 5 i \, c + \frac {15}{2}\right )} - 2 i \, \sqrt {2} e^{\left (3 i \, d x + 3 i \, c + \frac {15}{2}\right )} - i \, \sqrt {2} e^{\left (i \, d x + i \, c + \frac {15}{2}\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{15 \, {\left (a^{4} d e^{\left (5 i \, d x + 5 i \, c\right )} + 2 \, a^{4} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{4} d e^{\left (i \, d x + i \, c\right )}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {\sqrt {2} {\left (462 i \, e^{7} e^{\left (6 i \, d x + 6 i \, c\right )} + 1232 i \, e^{7} e^{\left (4 i \, d x + 4 i \, c\right )} + 1034 i \, e^{7} e^{\left (2 i \, d x + 2 i \, c\right )} + 240 i \, e^{7}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 15 \, {\left (a^{4} d e^{\left (5 i \, d x + 5 i \, c\right )} + 2 \, a^{4} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{4} d e^{\left (i \, d x + i \, c\right )}\right )} {\rm integral}\left (-\frac {77 i \, \sqrt {2} e^{7} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{5 \, a^{4} d}, x\right )}{15 \, {\left (a^{4} d e^{\left (5 i \, d x + 5 i \, c\right )} + 2 \, a^{4} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{4} d e^{\left (i \, d x + i \, c\right )}\right )}} \]