\[ \int (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^3 \, dx \]
Optimal antiderivative \[ \frac {26 i a^{3} \left (e \sec \left (d x +c \right )\right )^{\frac {5}{2}}}{35 d}+\frac {26 a^{3} e \left (e \sec \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{21 d}+\frac {26 a^{3} e^{2} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {e \sec \left (d x +c \right )}}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 i a \left (e \sec \left (d x +c \right )\right )^{\frac {5}{2}} \left (a +i a \tan \left (d x +c \right )\right )^{2}}{9 d}+\frac {26 i \left (e \sec \left (d x +c \right )\right )^{\frac {5}{2}} \left (a^{3}+i a^{3} \tan \left (d x +c \right )\right )}{63 d} \]
command
integrate((e*sec(d*x+c))^(5/2)*(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
\[ -\frac {2 \, {\left (\frac {\sqrt {2} {\left (-195 i \, a^{3} e^{\frac {5}{2}} + 195 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c + \frac {5}{2}\right )} - 1158 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c + \frac {5}{2}\right )} - 1456 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c + \frac {5}{2}\right )} - 858 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c + \frac {5}{2}\right )}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + 195 \, {\left (i \, \sqrt {2} a^{3} e^{\frac {5}{2}} + i \, \sqrt {2} a^{3} e^{\left (8 i \, d x + 8 i \, c + \frac {5}{2}\right )} + 4 i \, \sqrt {2} a^{3} e^{\left (6 i \, d x + 6 i \, c + \frac {5}{2}\right )} + 6 i \, \sqrt {2} a^{3} e^{\left (4 i \, d x + 4 i \, c + \frac {5}{2}\right )} + 4 i \, \sqrt {2} a^{3} e^{\left (2 i \, d x + 2 i \, c + \frac {5}{2}\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}}{315 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {\sqrt {2} {\left (-390 i \, a^{3} e^{2} e^{\left (8 i \, d x + 8 i \, c\right )} + 2316 i \, a^{3} e^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 2912 i \, a^{3} e^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 1716 i \, a^{3} e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 390 i \, a^{3} e^{2}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 315 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} {\rm integral}\left (-\frac {13 i \, \sqrt {2} a^{3} e^{2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{21 \, d}, x\right )}{315 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]