sequences and series - Identical Summation and Integration of specific functions

A strange coincidence which I discovered recently, is that

$$\int_{-\infty}^{\infty}{\tan^{-1}{\frac{1}{(x-\alpha)^2+\frac{3}{4}}}}dx = \sum_{x=-\infty}^{\infty}{\tan^{-1}{\frac{1}{(x-\alpha)^2+\frac{3}{4}}}}$$
Are there any other functions for which this characteristic holds true, and if so, what do they all have in common, and how can we determine them (if possible)?