\[ \int (e \cos (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx \]
Optimal antiderivative \[ -\frac {2 i a \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d}+\frac {10 a \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \cos \left (d x +c \right )^{\frac {7}{2}}}+\frac {2 a \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}} \tan \left (d x +c \right )}{7 d}+\frac {10 a \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}} \left (\sec ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{21 d} \]
command
integrate((e*cos(d*x+c))^(7/2)*(a+I*a*tan(d*x+c)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (-20 i \, \sqrt {2} a e^{\left (i \, d x + i \, c + \frac {7}{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \sqrt {\frac {1}{2}} {\left (7 i \, a e^{\frac {7}{2}} - 3 i \, a e^{\left (4 i \, d x + 4 i \, c + \frac {7}{2}\right )} - 16 i \, a e^{\left (2 i \, d x + 2 i \, c + \frac {7}{2}\right )}\right )} \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{42 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {{\left (\sqrt {\frac {1}{2}} {\left (-3 i \, a e^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 16 i \, a e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i \, a e^{3}\right )} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} + 42 \, d e^{\left (i \, d x + i \, c\right )} {\rm integral}\left (-\frac {10 i \, \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} a e^{3} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{21 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}}, x\right )\right )} e^{\left (-i \, d x - i \, c\right )}}{42 \, d} \]