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