How to Find the Recurrence Formula for $\int \frac{dx} {(1+\sin x)^n}$?

Question

$$\int \frac{dx} {(1+\sin x)^n}$$

I know that I should use the following step:

$$\int\frac{\sin^2(x)\, dx}{(1+\sin x)^n} +\int\frac{\cos^2(x)\, dx} {(1+\sin x)^n} $$

but here I get stuck. I do not know how shall I proceed.