Evaluate Integral of $\int_0^{\frac{\pi 2^n}{2}}\sin^{2+d}(u)\prod_{i=1}^{n}\cos^{2-d}(u/2^i)du$

Question

Integral of $$\int_0^{\frac{\pi 2^n}{2}}\sin^{2+d}(u)\prod_{i=1}^{n}\cos^{2-d}(u/2^i)du$$

I've tried a number of ways to re-write this in a way that makes sense, but no luck thus far. The integrand looks a bit like the beta function, but i'm not sure if that's leading anywhere.

Any help?