$$ I_n = \int \csc^nx dx $$
I find the reduction formula like this: $$ I_n = \frac{n-2}{n-1}I_{n-2} - \frac{\csc^{n-2}x\cot x}{n-1}$$
Now , evaluate $I_{2m}$ and $I_{2m+1}$ in summation notation, I find them seriously ugly
$$I_{2m} = I_2\prod_{k-1}^{m-1}\frac{2k}{2k+1}-\frac{\csc^{2m-2}x\cot x}{2m-1}-\sum_{k=1}^{m-2}\left(\frac{\csc^{2k}x\cot x}{2k+1}\prod_{i=k+1}^{m-1}\frac{2i}{2i+1}\right)$$ and $$I_{2m+1} = I_1\prod_{k=1}^{m}\frac{2k-1}{2k}-\frac{\csc^{2m-1}x\cot x}{2m}-\sum_{k=1}^{m-1}\left(\frac{\csc^{2k-1}x\cot x}{2k}\prod_{i=k+1}^{m}\frac{2i-1}{2i}\right)$$
I think there exists a simpler and clear expression, I have tried my best to solve still have no other good idea. Is there any clever and simpler way to present reduction formula with summation notation?
