\[ \int \frac {\cos ^2(e+f x) (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 c^{2} \cos \left (f x +e \right ) \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{a f \sqrt {a +a \sin \left (f x +e \right )}}+\frac {2 c \cos \left (f x +e \right ) \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}}}{3 a f \sqrt {a +a \sin \left (f x +e \right )}}+\frac {\cos \left (f x +e \right ) \left (c -c \sin \left (f x +e \right )\right )^{\frac {7}{2}}}{4 a f \sqrt {a +a \sin \left (f x +e \right )}}+\frac {16 c^{4} \cos \left (f x +e \right ) \ln \left (1+\sin \left (f x +e \right )\right )}{a f \sqrt {a +a \sin \left (f x +e \right )}\, \sqrt {c -c \sin \left (f x +e \right )}}+\frac {8 c^{3} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}}{a f \sqrt {a +a \sin \left (f x +e \right )}} \]
command
integrate(cos(f*x+e)^2*(c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, \sqrt {2} \sqrt {a} c^{\frac {7}{2}} {\left (\frac {12 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {3 \, \sqrt {2} a^{6} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 4 \, \sqrt {2} a^{6} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 6 \, \sqrt {2} a^{6} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 12 \, \sqrt {2} a^{6} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{a^{8}}\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{3 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________