\[ \int (g \cos (e+f x))^{5-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx \]
Optimal antiderivative \[ -\frac {8 a^{3} \left (g \cos \left (f x +e \right )\right )^{6-2 m} \left (a +a \sin \left (f x +e \right )\right )^{-3+m} \left (c -c \sin \left (f x +e \right )\right )^{n}}{f g \left (3-m +n \right ) \left (4-m +n \right ) \left (5-m +n \right )}-\frac {4 a^{2} \left (g \cos \left (f x +e \right )\right )^{6-2 m} \left (a +a \sin \left (f x +e \right )\right )^{-2+m} \left (c -c \sin \left (f x +e \right )\right )^{n}}{f g \left (4-m +n \right ) \left (5-m +n \right )}-\frac {a \left (g \cos \left (f x +e \right )\right )^{6-2 m} \left (a +a \sin \left (f x +e \right )\right )^{-1+m} \left (c -c \sin \left (f x +e \right )\right )^{n}}{f g \left (5-m +n \right )} \]
command
integrate((g*cos(f*x+e))^(5-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________