\[ \int \frac {(c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^2} \, dx \]
Optimal antiderivative \[ \frac {256 c^{2} \left (\sec ^{3}\left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{3 a^{2} f}-\frac {64 c \left (\sec ^{3}\left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}}}{a^{2} f}+\frac {8 \left (\sec ^{3}\left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{\frac {7}{2}}}{a^{2} f}+\frac {2 \left (\sec ^{3}\left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{\frac {9}{2}}}{3 a^{2} c f} \]
command
integrate((c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {128 \, \sqrt {2} {\left (c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - \frac {3 \, c^{3} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}}\right )} \sqrt {c}}{3 \, a^{2} f {\left (\frac {{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} - 1\right )}^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________