\[ \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 i a}{3 d \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {2 a \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {2 a \sin \left (d x +c \right )}{d e \sqrt {e \cos \left (d x +c \right )}} \]
command
integrate((a+I*a*tan(d*x+c))/(e*cos(d*x+c))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (2 \, \sqrt {\frac {1}{2}} {\left (3 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + i \, a e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} + 3 \, {\left (i \, \sqrt {2} a e^{\left (4 i \, d x + 4 i \, c\right )} + 2 i \, \sqrt {2} a e^{\left (2 i \, d x + 2 i \, c\right )} + i \, \sqrt {2} a\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{3 \, {\left (d e^{\frac {3}{2}} + d e^{\left (4 i \, d x + 4 i \, c + \frac {3}{2}\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c + \frac {3}{2}\right )}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {\sqrt {\frac {1}{2}} {\left (-12 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 4 i \, a e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} + 3 \, {\left (d e^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{2}\right )} {\rm integral}\left (\frac {2 i \, \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} a e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{2}}, x\right )}{3 \, {\left (d e^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{2}\right )}} \]