\[ \int \frac {A+B \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3} \, dx \]
Optimal antiderivative \[ -\frac {\left (A -B \right ) \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \sin \left (f x +e \right )}}\right ) \sqrt {2}}{\left (c -d \right )^{3} f \sqrt {a}}+\frac {\left (A d \left (15 c^{2}+10 c d +7 d^{2}\right )-B \left (3 c^{3}+6 c^{2} d +19 c \,d^{2}+4 d^{3}\right )\right ) \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {d}}{\sqrt {c +d}\, \sqrt {a +a \sin \left (f x +e \right )}}\right )}{4 \left (c -d \right )^{3} \left (c +d \right )^{\frac {5}{2}} f \sqrt {a}\, \sqrt {d}}-\frac {\left (-A d +B c \right ) \cos \left (f x +e \right )}{2 \left (c^{2}-d^{2}\right ) f \left (c +d \sin \left (f x +e \right )\right )^{2} \sqrt {a +a \sin \left (f x +e \right )}}+\frac {\left (A d \left (7 c +d \right )-B \left (3 c^{2}+c d +4 d^{2}\right )\right ) \cos \left (f x +e \right )}{4 \left (c^{2}-d^{2}\right )^{2} f \left (c +d \sin \left (f x +e \right )\right ) \sqrt {a +a \sin \left (f x +e \right )}} \]
command
integrate((A+B*sin(f*x+e))/(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Exception raised: TypeError} \]_______________________________________________________