\[ \int \frac {(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (c -d \right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{2}}{2 f \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {\left (c -d \right )^{2} \left (c +11 d \right ) \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \sin \left (f x +e \right )}}\right ) \sqrt {2}}{4 a^{\frac {3}{2}} f}+\frac {d \left (3 c^{2}-24 c d +13 d^{2}\right ) \cos \left (f x +e \right )}{3 a f \sqrt {a +a \sin \left (f x +e \right )}}+\frac {\left (3 c -7 d \right ) d^{2} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}}{6 a^{2} f} \]
command
integrate((c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {3 \, \sqrt {2} {\left (\sqrt {a} c^{3} + 9 \, \sqrt {a} c^{2} d - 21 \, \sqrt {a} c d^{2} + 11 \, \sqrt {a} d^{3}\right )} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {3 \, \sqrt {2} {\left (\sqrt {a} c^{3} + 9 \, \sqrt {a} c^{2} d - 21 \, \sqrt {a} c d^{2} + 11 \, \sqrt {a} d^{3}\right )} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {6 \, \sqrt {2} {\left (\sqrt {a} c^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, \sqrt {a} c^{2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, \sqrt {a} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {a} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {16 \, \sqrt {2} {\left (2 \, a^{\frac {9}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 9 \, a^{\frac {9}{2}} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, a^{\frac {9}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{24 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________