Let $N\in\mathbb{N}$. Integrate: $$\int\frac{\cos x\sin Nx\sin 3Nx}{\sin x}\, \mathrm{d}x$$
We can rewrite the integral as: $$\frac{1}{2}\int\frac{\cos x\sin 2Nx}{\sin x}\, \mathrm{d}x-\frac{1}{2}\int\frac{\cos x\sin Nx}{\sin x}\, \mathrm{d}x.$$
But how do I proceed further? Or could it be expressed as a special function?