\[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^2} \, dx \]
Optimal antiderivative \[ \frac {4 e \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}}{d \left (a^{2}+a^{2} \sin \left (d x +c \right )\right )}+\frac {10 e^{4} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} d \sqrt {e \cos \left (d x +c \right )}}+\frac {10 e^{3} \sin \left (d x +c \right ) \sqrt {e \cos \left (d x +c \right )}}{3 a^{2} d} \]
command
integrate((e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c))^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-5 i \, \sqrt {2} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 i \, \sqrt {2} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (e^{\frac {7}{2}} \sin \left (d x + c\right ) - 6 \, e^{\frac {7}{2}}\right )} \sqrt {\cos \left (d x + c\right )}}{3 \, a^{2} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {e \cos \left (d x + c\right )} e^{3} \cos \left (d x + c\right )^{3}}{a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}}, x\right ) \]