\[ \int \frac {(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^3} \, dx \]
Optimal antiderivative \[ -\frac {a^{\frac {3}{2}} \left (c +7 d \right ) \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {d}}{\sqrt {c +d}\, \sqrt {a +a \sin \left (f x +e \right )}}\right )}{4 d^{\frac {3}{2}} \left (c +d \right )^{\frac {5}{2}} f}+\frac {a^{2} \left (c -d \right ) \cos \left (f x +e \right )}{2 d \left (c +d \right ) f \left (c +d \sin \left (f x +e \right )\right )^{2} \sqrt {a +a \sin \left (f x +e \right )}}-\frac {a^{2} \left (c +7 d \right ) \cos \left (f x +e \right )}{4 d \left (c +d \right )^{2} f \left (c +d \sin \left (f x +e \right )\right ) \sqrt {a +a \sin \left (f x +e \right )}} \]
command
integrate((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {2} \sqrt {a} {\left (\frac {\sqrt {2} {\left (a c \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 7 \, a d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \arctan \left (\frac {\sqrt {2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c d - d^{2}}}\right )}{{\left (c^{2} d + 2 \, c d^{2} + d^{3}\right )} \sqrt {-c d - d^{2}}} + \frac {2 \, {\left (2 \, a c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 14 \, a d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + a c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 8 \, a c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 9 \, a d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (c^{2} d + 2 \, c d^{2} + d^{3}\right )} {\left (2 \, d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )}^{2}}\right )}}{8 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________