\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(a+a \sin (e+f x))^2} \, dx \]
Optimal antiderivative \[ -\frac {c \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}}{3 a^{2} \left (c -d \right ) f \left (1+\sin \left (f x +e \right )\right )}-\frac {\cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}}{3 f \left (a +a \sin \left (f x +e \right )\right )^{2}}+\frac {c \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticE \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {c +d \sin \left (f x +e \right )}}{3 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) a^{2} \left (c -d \right ) f \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}-\frac {\left (c +d \right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}{3 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) a^{2} f \sqrt {c +d \sin \left (f x +e \right )}} \]
command
integrate((c+d*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (\sqrt {2} {\left (2 \, c^{2} - 3 \, d^{2}\right )} \cos \left (f x + e\right )^{2} - \sqrt {2} {\left (2 \, c^{2} - 3 \, d^{2}\right )} \cos \left (f x + e\right ) - {\left (\sqrt {2} {\left (2 \, c^{2} - 3 \, d^{2}\right )} \cos \left (f x + e\right ) + 2 \, \sqrt {2} {\left (2 \, c^{2} - 3 \, d^{2}\right )}\right )} \sin \left (f x + e\right ) - 2 \, \sqrt {2} {\left (2 \, c^{2} - 3 \, d^{2}\right )}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) + {\left (\sqrt {2} {\left (2 \, c^{2} - 3 \, d^{2}\right )} \cos \left (f x + e\right )^{2} - \sqrt {2} {\left (2 \, c^{2} - 3 \, d^{2}\right )} \cos \left (f x + e\right ) - {\left (\sqrt {2} {\left (2 \, c^{2} - 3 \, d^{2}\right )} \cos \left (f x + e\right ) + 2 \, \sqrt {2} {\left (2 \, c^{2} - 3 \, d^{2}\right )}\right )} \sin \left (f x + e\right ) - 2 \, \sqrt {2} {\left (2 \, c^{2} - 3 \, d^{2}\right )}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) - 3 \, {\left (-i \, \sqrt {2} c d \cos \left (f x + e\right )^{2} + i \, \sqrt {2} c d \cos \left (f x + e\right ) + 2 i \, \sqrt {2} c d + {\left (i \, \sqrt {2} c d \cos \left (f x + e\right ) + 2 i \, \sqrt {2} c d\right )} \sin \left (f x + e\right )\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) - 3 \, {\left (i \, \sqrt {2} c d \cos \left (f x + e\right )^{2} - i \, \sqrt {2} c d \cos \left (f x + e\right ) - 2 i \, \sqrt {2} c d + {\left (-i \, \sqrt {2} c d \cos \left (f x + e\right ) - 2 i \, \sqrt {2} c d\right )} \sin \left (f x + e\right )\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right ) + 6 \, {\left (c d \cos \left (f x + e\right )^{2} + c d - d^{2} + {\left (2 \, c d - d^{2}\right )} \cos \left (f x + e\right ) + {\left (c d \cos \left (f x + e\right ) - c d + d^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{18 \, {\left ({\left (a^{2} c d - a^{2} d^{2}\right )} f \cos \left (f x + e\right )^{2} - {\left (a^{2} c d - a^{2} d^{2}\right )} f \cos \left (f x + e\right ) - 2 \, {\left (a^{2} c d - a^{2} d^{2}\right )} f - {\left ({\left (a^{2} c d - a^{2} d^{2}\right )} f \cos \left (f x + e\right ) + 2 \, {\left (a^{2} c d - a^{2} d^{2}\right )} f\right )} \sin \left (f x + e\right )\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {d \sin \left (f x + e\right ) + c}}{a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}}, x\right ) \]