\[ \int \frac {(a+i a \tan (c+d x))^3}{\sqrt {e \sec (c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {26 i a^{3}}{3 d \sqrt {e \sec \left (d x +c \right )}}+\frac {14 a^{3} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {\cos \left (d x +c \right )}\, \sqrt {e \sec \left (d x +c \right )}}-\frac {6 a^{3} \tan \left (d x +c \right )}{d \sqrt {e \sec \left (d x +c \right )}}-\frac {2 i a^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{3 d \sqrt {e \sec \left (d x +c \right )}} \]
command
integrate((a+I*a*tan(d*x+c))^3/(e*sec(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (\frac {\sqrt {2} {\left (-9 i \, a^{3} e^{\left (3 i \, d x + 3 i \, c\right )} - 7 i \, a^{3} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + 21 \, {\left (-i \, \sqrt {2} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, \sqrt {2} a^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{3 \, {\left (d e^{\frac {1}{2}} + d e^{\left (2 i \, d x + 2 i \, c + \frac {1}{2}\right )}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {\sqrt {2} {\left (-24 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 18 i \, a^{3} e^{\left (3 i \, d x + 3 i \, c\right )} - 70 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 14 i \, a^{3} e^{\left (i \, d x + i \, c\right )} - 42 i \, a^{3}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 3 \, {\left (d e e^{\left (3 i \, d x + 3 i \, c\right )} - d e e^{\left (2 i \, d x + 2 i \, c\right )} + d e e^{\left (i \, d x + i \, c\right )} - d e\right )} {\rm integral}\left (\frac {\sqrt {2} {\left (-7 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 14 i \, a^{3} e^{\left (i \, d x + i \, c\right )} - 7 i \, a^{3}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{d e e^{\left (3 i \, d x + 3 i \, c\right )} - 2 \, d e e^{\left (2 i \, d x + 2 i \, c\right )} + d e e^{\left (i \, d x + i \, c\right )}}, x\right )}{3 \, {\left (d e e^{\left (3 i \, d x + 3 i \, c\right )} - d e e^{\left (2 i \, d x + 2 i \, c\right )} + d e e^{\left (i \, d x + i \, c\right )} - d e\right )}} \]