\[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx \]
Optimal antiderivative \[ \frac {256 a^{3} \left (15 A -B \right ) c^{7} \left (\cos ^{7}\left (f x +e \right )\right )}{45045 f \left (c -c \sin \left (f x +e \right )\right )^{\frac {7}{2}}}+\frac {64 a^{3} \left (15 A -B \right ) c^{6} \left (\cos ^{7}\left (f x +e \right )\right )}{6435 f \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {8 a^{3} \left (15 A -B \right ) c^{5} \left (\cos ^{7}\left (f x +e \right )\right )}{715 f \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {2 a^{3} \left (15 A -B \right ) c^{4} \left (\cos ^{7}\left (f x +e \right )\right )}{195 f \sqrt {c -c \sin \left (f x +e \right )}}-\frac {2 a^{3} B \,c^{3} \left (\cos ^{7}\left (f x +e \right )\right ) \sqrt {c -c \sin \left (f x +e \right )}}{15 f} \]
command
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(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 (3003 \, B a^{3} c^{3} \cos \left (-\frac {15}{4} \, \pi + \frac {15}{2} \, f x + \frac {15}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 225225 \, {\left (8 \, A a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - B a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 75075 \, {\left (6 \, A a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) + 27027 \, {\left (10 \, A a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 3 \, B a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) + 19305 \, {\left (4 \, A a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, B a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) - 5005 \, {\left (12 \, A a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 5 \, B a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right ) - 4095 \, {\left (2 \, A a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 5 \, B a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {11}{4} \, \pi + \frac {11}{2} \, f x + \frac {11}{2} \, e\right ) + 3465 \, {\left (2 \, A a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - B a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {13}{4} \, \pi + \frac {13}{2} \, f x + \frac {13}{2} \, e\right )\right )} \sqrt {c}}{2882880 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________