A special binomial identity in need of a proof

I've encountered a curious identity as a codicil in some work. Is there a proof or reference?

$$\sum_{k=-n}^n\frac{2k+1}{n+k+1}\binom{2n}{n-k}\frac{x^k}{1+x^{2k+1}}=\frac{x^n}{1+x^{2n+1}}.$$

The $k=j-1$ and $k=-j$ terms cancel, so all that's left is the $k=n$ term.