Based on the comments from @Nemo, it suffices to show that $$ \sum_{k=1}^{2n+1}\left(\frac{(-y)^k}{1-y^{3k}}+ \frac{y^{k}}{1+y^{3k}}\right)=-n-1, $$ where $y=e^{\frac{2\pi i}{6n+4}}$ is the primitive $(6n+4)$th root of unity.

@Seva, It seems to be difficult to prove this sum is real.