\[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx \]
Optimal antiderivative \[ -\frac {\left (A -B \right ) \cos \left (f x +e \right )}{2 \left (c -d \right ) f \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \left (c +d \sin \left (f x +e \right )\right )^{2}}-\frac {\left (A \left (c -13 d \right )+3 B \left (c +3 d \right )\right ) \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \sin \left (f x +e \right )}}\right ) \sqrt {2}}{4 a^{\frac {3}{2}} \left (c -d \right )^{4} f}-\frac {\left (A d \left (35 c^{2}+42 c d +19 d^{2}\right )-3 B \left (5 c^{3}+10 c^{2} d +13 c \,d^{2}+4 d^{3}\right )\right ) \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {d}}{\sqrt {c +d}\, \sqrt {a +a \sin \left (f x +e \right )}}\right ) \sqrt {d}}{4 a^{\frac {3}{2}} \left (c -d \right )^{4} \left (c +d \right )^{\frac {5}{2}} f}+\frac {d \left (B \left (2 c +d \right )-A \left (c +2 d \right )\right ) \cos \left (f x +e \right )}{2 a \left (c -d \right )^{2} \left (c +d \right ) f \left (c +d \sin \left (f x +e \right )\right )^{2} \sqrt {a +a \sin \left (f x +e \right )}}+\frac {d \left (3 B \left (3 c^{2}+3 c d +2 d^{2}\right )-A \left (2 c^{2}+15 c d +7 d^{2}\right )\right ) \cos \left (f x +e \right )}{4 a \left (c -d \right )^{3} \left (c +d \right )^{2} f \left (c +d \sin \left (f x +e \right )\right ) \sqrt {a +a \sin \left (f x +e \right )}} \]
command
integrate((A+B*sin(f*x+e))/(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
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Exception raised: TypeError} \]_______________________________________________________