\[ \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {2 i a}{7 d \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}-\frac {6 a \left (\cos ^{\frac {7}{2}}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}+\frac {2 a \cos \left (d x +c \right ) \sin \left (d x +c \right )}{5 d \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}+\frac {6 a \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{5 d \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}} \]
command
integrate((a+I*a*tan(d*x+c))/(e*cos(d*x+c))^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (2 \, \sqrt {\frac {1}{2}} {\left (21 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} + 77 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 23 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 7 i \, a e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} + 21 \, {\left (i \, \sqrt {2} a e^{\left (8 i \, d x + 8 i \, c\right )} + 4 i \, \sqrt {2} a e^{\left (6 i \, d x + 6 i \, c\right )} + 6 i \, \sqrt {2} a e^{\left (4 i \, d x + 4 i \, c\right )} + 4 i \, \sqrt {2} a e^{\left (2 i \, d x + 2 i \, c\right )} + i \, \sqrt {2} a\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{35 \, {\left (d e^{\frac {7}{2}} + d e^{\left (8 i \, d x + 8 i \, c + \frac {7}{2}\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c + \frac {7}{2}\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c + \frac {7}{2}\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c + \frac {7}{2}\right )}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {\sqrt {\frac {1}{2}} {\left (-84 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} - 308 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} - 92 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 28 i \, a e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} + 35 \, {\left (d e^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{4}\right )} {\rm integral}\left (\frac {6 i \, \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} a e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{5 \, {\left (d e^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{4}\right )}}, x\right )}{35 \, {\left (d e^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{4}\right )}} \]