\[ \int \frac {(a+i a \tan (c+d x))^4}{\sqrt {e \sec (c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {154 i a^{4} \left (e \sec \left (d x +c \right )\right )^{\frac {3}{2}}}{15 d \,e^{2}}+\frac {154 a^{4} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {\cos \left (d x +c \right )}\, \sqrt {e \sec \left (d x +c \right )}}-\frac {154 a^{4} \sin \left (d x +c \right ) \sqrt {e \sec \left (d x +c \right )}}{5 d e}-\frac {4 i a \left (a +i a \tan \left (d x +c \right )\right )^{3}}{d \sqrt {e \sec \left (d x +c \right )}}-\frac {22 i \left (e \sec \left (d x +c \right )\right )^{\frac {3}{2}} \left (a^{4}+i a^{4} \tan \left (d x +c \right )\right )}{5 d \,e^{2}} \]
command
integrate((a+I*a*tan(d*x+c))^4/(e*sec(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (\frac {\sqrt {2} {\left (-111 i \, a^{4} e^{\left (5 i \, d x + 5 i \, c\right )} - 176 i \, a^{4} e^{\left (3 i \, d x + 3 i \, c\right )} - 77 i \, a^{4} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + 231 \, {\left (-i \, \sqrt {2} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 2 i \, \sqrt {2} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, \sqrt {2} a^{4}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{15 \, {\left (d e^{\frac {1}{2}} + d e^{\left (4 i \, d x + 4 i \, c + \frac {1}{2}\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c + \frac {1}{2}\right )}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {\sqrt {2} {\left (-240 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 222 i \, a^{4} e^{\left (5 i \, d x + 5 i \, c\right )} - 1034 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 352 i \, a^{4} e^{\left (3 i \, d x + 3 i \, c\right )} - 1232 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 154 i \, a^{4} e^{\left (i \, d x + i \, c\right )} - 462 i \, a^{4}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 15 \, {\left (d e e^{\left (5 i \, d x + 5 i \, c\right )} - d e e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e e^{\left (3 i \, d x + 3 i \, c\right )} - 2 \, d e e^{\left (2 i \, d x + 2 i \, c\right )} + d e e^{\left (i \, d x + i \, c\right )} - d e\right )} {\rm integral}\left (\frac {\sqrt {2} {\left (-77 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 154 i \, a^{4} e^{\left (i \, d x + i \, c\right )} - 77 i \, a^{4}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{5 \, {\left (d e e^{\left (3 i \, d x + 3 i \, c\right )} - 2 \, d e e^{\left (2 i \, d x + 2 i \, c\right )} + d e e^{\left (i \, d x + i \, c\right )}\right )}}, x\right )}{15 \, {\left (d e e^{\left (5 i \, d x + 5 i \, c\right )} - d e e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e e^{\left (3 i \, d x + 3 i \, c\right )} - 2 \, d e e^{\left (2 i \, d x + 2 i \, c\right )} + d e e^{\left (i \, d x + i \, c\right )} - d e\right )}} \]