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