JEE Math Practice Question

If $$\int \frac{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}{\sqrt{\sin ^3 x \cos ^3 x \sin (x-\theta)}} d x=A \sqrt{\cos \theta \tan x-\sin \theta}+B \sqrt{\cos \theta-\sin \theta \cot x}+C$$, where $$C$$ is the integration constant, then $$A B$$ is equal to

1. A.$$2 \sec \theta$$
2. B.$$8 \operatorname{cosec}(2 \theta)$$
3. C.$$4 \operatorname{cosec}(2 \theta)$$
4. D.$$4 \sec \theta$$