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)?
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