\[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx \]
Optimal antiderivative \[ \frac {256 a^{3} c^{7} \left (\cos ^{7}\left (f x +e \right )\right )}{3003 f \left (c -c \sin \left (f x +e \right )\right )^{\frac {7}{2}}}+\frac {64 a^{3} c^{6} \left (\cos ^{7}\left (f x +e \right )\right )}{429 f \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {24 a^{3} c^{5} \left (\cos ^{7}\left (f x +e \right )\right )}{143 f \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {2 a^{3} c^{4} \left (\cos ^{7}\left (f x +e \right )\right )}{13 f \sqrt {c -c \sin \left (f x +e \right )}} \]
command
integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {2} {\left (60060 \, a^{3} c^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 15015 \, a^{3} c^{3} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 9009 \, a^{3} c^{3} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 2574 \, a^{3} c^{3} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2002 \, a^{3} c^{3} \cos \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 273 \, a^{3} c^{3} \cos \left (-\frac {11}{4} \, \pi + \frac {11}{2} \, f x + \frac {11}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 231 \, a^{3} c^{3} \cos \left (-\frac {13}{4} \, \pi + \frac {13}{2} \, f x + \frac {13}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {c}}{96096 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________