\[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^4 \, dx \]
Optimal antiderivative \[ \frac {22 i a^{4} \left (e \sec \left (d x +c \right )\right )^{\frac {3}{2}}}{9 d}-\frac {22 a^{4} e^{2} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {\cos \left (d x +c \right )}\, \sqrt {e \sec \left (d x +c \right )}}+\frac {22 a^{4} e \sin \left (d x +c \right ) \sqrt {e \sec \left (d x +c \right )}}{3 d}+\frac {2 i a \left (e \sec \left (d x +c \right )\right )^{\frac {3}{2}} \left (a +i a \tan \left (d x +c \right )\right )^{3}}{9 d}+\frac {10 i \left (e \sec \left (d x +c \right )\right )^{\frac {3}{2}} \left (a^{2}+i a^{2} \tan \left (d x +c \right )\right )^{2}}{21 d}+\frac {22 i \left (e \sec \left (d x +c \right )\right )^{\frac {3}{2}} \left (a^{4}+i a^{4} \tan \left (d x +c \right )\right )}{21 d} \]
command
integrate((e*sec(d*x+c))^(3/2)*(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
\[ -\frac {2 \, {\left (\frac {\sqrt {2} {\left (231 i \, a^{4} e^{\left (9 i \, d x + 9 i \, c + \frac {3}{2}\right )} + 406 i \, a^{4} e^{\left (7 i \, d x + 7 i \, c + \frac {3}{2}\right )} + 540 i \, a^{4} e^{\left (5 i \, d x + 5 i \, c + \frac {3}{2}\right )} + 330 i \, a^{4} e^{\left (3 i \, d x + 3 i \, c + \frac {3}{2}\right )} + 77 i \, a^{4} e^{\left (i \, d x + i \, c + \frac {3}{2}\right )}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + 231 \, {\left (i \, \sqrt {2} a^{4} e^{\frac {3}{2}} + i \, \sqrt {2} a^{4} e^{\left (8 i \, d x + 8 i \, c + \frac {3}{2}\right )} + 4 i \, \sqrt {2} a^{4} e^{\left (6 i \, d x + 6 i \, c + \frac {3}{2}\right )} + 6 i \, \sqrt {2} a^{4} e^{\left (4 i \, d x + 4 i \, c + \frac {3}{2}\right )} + 4 i \, \sqrt {2} a^{4} e^{\left (2 i \, d x + 2 i \, c + \frac {3}{2}\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{63 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {\sqrt {2} {\left (-462 i \, a^{4} e e^{\left (9 i \, d x + 9 i \, c\right )} - 812 i \, a^{4} e e^{\left (7 i \, d x + 7 i \, c\right )} - 1080 i \, a^{4} e e^{\left (5 i \, d x + 5 i \, c\right )} - 660 i \, a^{4} e e^{\left (3 i \, d x + 3 i \, c\right )} - 154 i \, a^{4} e e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 63 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} {\rm integral}\left (\frac {11 i \, \sqrt {2} a^{4} e \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{3 \, d}, x\right )}{63 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]