\[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {10 i a^{4} \sqrt {e \sec \left (d x +c \right )}}{d \,e^{2}}-\frac {10 a^{4} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {e \sec \left (d x +c \right )}}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \,e^{2}}-\frac {4 i a \left (a +i a \tan \left (d x +c \right )\right )^{3}}{3 d \left (e \sec \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {2 i \sqrt {e \sec \left (d x +c \right )}\, \left (a^{4}+i a^{4} \tan \left (d x +c \right )\right )}{d \,e^{2}} \]
command
integrate((a+I*a*tan(d*x+c))^4/(e*sec(d*x+c))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (\frac {\sqrt {2} {\left (4 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 21 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 15 i \, a^{4}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + 15 \, {\left (-i \, \sqrt {2} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, \sqrt {2} a^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}}{3 \, {\left (d e^{\frac {3}{2}} + d e^{\left (2 i \, d x + 2 i \, c + \frac {3}{2}\right )}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {\sqrt {2} {\left (-8 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 42 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 30 i \, a^{4}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 3 \, {\left (d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{2}\right )} {\rm integral}\left (\frac {5 i \, \sqrt {2} a^{4} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{d e^{2}}, x\right )}{3 \, {\left (d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{2}\right )}} \]