\[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{7 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} d \cos \left (d x +c \right )^{\frac {7}{2}}}+\frac {2 \cos \left (d x +c \right ) \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}} \sin \left (d x +c \right )}{15 a^{2} d}+\frac {6 \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}} \tan \left (d x +c \right )}{35 a^{2} d}+\frac {2 \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}} \left (\sec ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{7 a^{2} d}+\frac {4 i \left (\cos ^{2}\left (d x +c \right )\right ) \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}{15 d \left (a^{2}+i a^{2} \tan \left (d x +c \right )\right )} \]
command
integrate((e*cos(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (\sqrt {\frac {1}{2}} {\left (7 i \, e^{\frac {7}{2}} - 15 i \, e^{\left (10 i \, d x + 10 i \, c + \frac {7}{2}\right )} - 185 i \, e^{\left (8 i \, d x + 8 i \, c + \frac {7}{2}\right )} + 430 i \, e^{\left (6 i \, d x + 6 i \, c + \frac {7}{2}\right )} + 162 i \, e^{\left (4 i \, d x + 4 i \, c + \frac {7}{2}\right )} + 49 i \, e^{\left (2 i \, d x + 2 i \, c + \frac {7}{2}\right )}\right )} \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} - 480 i \, \sqrt {2} e^{\left (7 i \, d x + 7 i \, c + \frac {7}{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )} e^{\left (-7 i \, d x - 7 i \, c\right )}}{1680 \, a^{2} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {{\left (1680 \, a^{2} d e^{\left (7 i \, d x + 7 i \, c\right )} {\rm integral}\left (-\frac {2 i \, \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{3} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{7 \, {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}}, x\right ) + \sqrt {\frac {1}{2}} {\left (-15 i \, e^{3} e^{\left (10 i \, d x + 10 i \, c\right )} - 185 i \, e^{3} e^{\left (8 i \, d x + 8 i \, c\right )} + 430 i \, e^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 162 i \, e^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 49 i \, e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i \, e^{3}\right )} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right )} e^{\left (-7 i \, d x - 7 i \, c\right )}}{1680 \, a^{2} d} \]