\[ \int (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))^3 \, dx \]
Optimal antiderivative \[ \frac {10 i a^{3} \left (e \sec \left (d x +c \right )\right )^{\frac {7}{2}}}{21 d}+\frac {2 a^{3} e \left (e \sec \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{3 d}-\frac {2 a^{3} e^{4} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {\cos \left (d x +c \right )}\, \sqrt {e \sec \left (d x +c \right )}}+\frac {2 a^{3} e^{3} \sin \left (d x +c \right ) \sqrt {e \sec \left (d x +c \right )}}{d}+\frac {2 i a \left (e \sec \left (d x +c \right )\right )^{\frac {7}{2}} \left (a +i a \tan \left (d x +c \right )\right )^{2}}{11 d}+\frac {10 i \left (e \sec \left (d x +c \right )\right )^{\frac {7}{2}} \left (a^{3}+i a^{3} \tan \left (d x +c \right )\right )}{33 d} \]
command
integrate((e*sec(d*x+c))^(7/2)*(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
\[ -\frac {2 \, {\left (\frac {\sqrt {2} {\left (231 i \, a^{3} e^{\left (11 i \, d x + 11 i \, c + \frac {7}{2}\right )} + 1309 i \, a^{3} e^{\left (9 i \, d x + 9 i \, c + \frac {7}{2}\right )} + 946 i \, a^{3} e^{\left (7 i \, d x + 7 i \, c + \frac {7}{2}\right )} + 870 i \, a^{3} e^{\left (5 i \, d x + 5 i \, c + \frac {7}{2}\right )} + 407 i \, a^{3} e^{\left (3 i \, d x + 3 i \, c + \frac {7}{2}\right )} + 77 i \, a^{3} e^{\left (i \, d x + i \, c + \frac {7}{2}\right )}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + 231 \, {\left (i \, \sqrt {2} a^{3} e^{\frac {7}{2}} + i \, \sqrt {2} a^{3} e^{\left (10 i \, d x + 10 i \, c + \frac {7}{2}\right )} + 5 i \, \sqrt {2} a^{3} e^{\left (8 i \, d x + 8 i \, c + \frac {7}{2}\right )} + 10 i \, \sqrt {2} a^{3} e^{\left (6 i \, d x + 6 i \, c + \frac {7}{2}\right )} + 10 i \, \sqrt {2} a^{3} e^{\left (4 i \, d x + 4 i \, c + \frac {7}{2}\right )} + 5 i \, \sqrt {2} a^{3} e^{\left (2 i \, d x + 2 i \, c + \frac {7}{2}\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{231 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {\sqrt {2} {\left (-462 i \, a^{3} e^{3} e^{\left (11 i \, d x + 11 i \, c\right )} - 2618 i \, a^{3} e^{3} e^{\left (9 i \, d x + 9 i \, c\right )} - 1892 i \, a^{3} e^{3} e^{\left (7 i \, d x + 7 i \, c\right )} - 1740 i \, a^{3} e^{3} e^{\left (5 i \, d x + 5 i \, c\right )} - 814 i \, a^{3} e^{3} e^{\left (3 i \, d x + 3 i \, c\right )} - 154 i \, a^{3} e^{3} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 231 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} {\rm integral}\left (\frac {i \, \sqrt {2} a^{3} e^{3} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{d}, x\right )}{231 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]