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