\[ \int \frac {(e \sin (c+d x))^{7/2}}{a+a \sec (c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 e \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{5 a d}+\frac {4 e^{4} \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\sin }\left (d x +c \right )\right )}{21 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) a d \sqrt {e \sin \left (d x +c \right )}}-\frac {2 e^{3} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}{21 a d}+\frac {2 e^{3} \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {e \sin \left (d x +c \right )}}{7 a d} \]
command
integrate((e*sin(d*x+c))^(7/2)/(a+a*sec(d*x+c)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (5 \, \sqrt {2} \sqrt {-i} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} \sqrt {i} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - {\left (15 \, \cos \left (d x + c\right )^{3} e^{\frac {7}{2}} - 21 \, \cos \left (d x + c\right )^{2} e^{\frac {7}{2}} - 5 \, \cos \left (d x + c\right ) e^{\frac {7}{2}} + 21 \, e^{\frac {7}{2}}\right )} \sqrt {\sin \left (d x + c\right )}\right )}}{105 \, a d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (e^{3} \cos \left (d x + c\right )^{2} - e^{3}\right )} \sqrt {e \sin \left (d x + c\right )} \sin \left (d x + c\right )}{a \sec \left (d x + c\right ) + a}, x\right ) \]