\[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3 \, dx \]
Optimal antiderivative \[ \frac {6 i a^{3} \sqrt {e \sec \left (d x +c \right )}}{d}+\frac {6 a^{3} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {e \sec \left (d x +c \right )}}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 i a \sqrt {e \sec \left (d x +c \right )}\, \left (a +i a \tan \left (d x +c \right )\right )^{2}}{5 d}+\frac {6 i \sqrt {e \sec \left (d x +c \right )}\, \left (a^{3}+i a^{3} \tan \left (d x +c \right )\right )}{5 d} \]
command
integrate((e*sec(d*x+c))^(1/2)*(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (\frac {\sqrt {2} {\left (-15 i \, a^{3} e^{\frac {1}{2}} - 25 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c + \frac {1}{2}\right )} - 36 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c + \frac {1}{2}\right )}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + 15 \, {\left (i \, \sqrt {2} a^{3} e^{\frac {1}{2}} + i \, \sqrt {2} a^{3} e^{\left (4 i \, d x + 4 i \, c + \frac {1}{2}\right )} + 2 i \, \sqrt {2} a^{3} e^{\left (2 i \, d x + 2 i \, c + \frac {1}{2}\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}}{5 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {\sqrt {2} {\left (50 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 72 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 30 i \, a^{3}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 5 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} {\rm integral}\left (-\frac {3 i \, \sqrt {2} a^{3} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{d}, x\right )}{5 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]