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