\[ \int \frac {a+b \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-a d +b c \right ) \cos \left (f x +e \right )}{5 \left (c^{2}-d^{2}\right ) f \left (c +d \sin \left (f x +e \right )\right )^{\frac {5}{2}}}-\frac {2 \left (-8 a c d +3 b \,c^{2}+5 b \,d^{2}\right ) \cos \left (f x +e \right )}{15 \left (c^{2}-d^{2}\right )^{2} f \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {2 \left (-23 a \,c^{2} d -9 a \,d^{3}+3 b \,c^{3}+29 b c \,d^{2}\right ) \cos \left (f x +e \right )}{15 \left (c^{2}-d^{2}\right )^{3} f \sqrt {c +d \sin \left (f x +e \right )}}+\frac {2 \left (-23 a \,c^{2} d -9 a \,d^{3}+3 b \,c^{3}+29 b c \,d^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticE \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {c +d \sin \left (f x +e \right )}}{15 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d \left (c^{2}-d^{2}\right )^{3} f \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}-\frac {2 \left (-8 a c d +3 b \,c^{2}+5 b \,d^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}{15 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d \left (c^{2}-d^{2}\right )^{2} f \sqrt {c +d \sin \left (f x +e \right )}} \]
command
integrate((a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (\frac {{\left (b \sin \left (f x + e\right ) + a\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{d^{4} \cos \left (f x + e\right )^{4} + c^{4} + 6 \, c^{2} d^{2} + d^{4} - 2 \, {\left (3 \, c^{2} d^{2} + d^{4}\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left (c d^{3} \cos \left (f x + e\right )^{2} - c^{3} d - c d^{3}\right )} \sin \left (f x + e\right )}, x\right ) \]