\[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {42 a^{4} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \,e^{2} \sqrt {\cos \left (d x +c \right )}\, \sqrt {e \sec \left (d x +c \right )}}+\frac {42 a^{4} \sin \left (d x +c \right ) \sqrt {e \sec \left (d x +c \right )}}{5 d \,e^{3}}-\frac {4 i a \left (a +i a \tan \left (d x +c \right )\right )^{3}}{5 d \left (e \sec \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {28 i \left (a^{4}+i a^{4} \tan \left (d x +c \right )\right )}{5 d \,e^{2} \sqrt {e \sec \left (d x +c \right )}} \]
command
integrate((a+I*a*tan(d*x+c))^4/(e*sec(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (21 i \, \sqrt {2} a^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right ) + \frac {\sqrt {2} {\left (2 i \, a^{4} e^{\left (3 i \, d x + 3 i \, c\right )} + 7 i \, a^{4} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-\frac {5}{2}\right )}}{5 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {\sqrt {2} {\left (-4 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 i \, a^{4} e^{\left (3 i \, d x + 3 i \, c\right )} + 28 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 14 i \, a^{4} e^{\left (i \, d x + i \, c\right )} + 42 i \, a^{4}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 5 \, {\left (d e^{3} e^{\left (i \, d x + i \, c\right )} - d e^{3}\right )} {\rm integral}\left (\frac {\sqrt {2} {\left (21 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 42 i \, a^{4} e^{\left (i \, d x + i \, c\right )} + 21 i \, a^{4}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{5 \, {\left (d e^{3} e^{\left (3 i \, d x + 3 i \, c\right )} - 2 \, d e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{3} e^{\left (i \, d x + i \, c\right )}\right )}}, x\right )}{5 \, {\left (d e^{3} e^{\left (i \, d x + i \, c\right )} - d e^{3}\right )}} \]