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This is a standard exercise on residue theory. If $f$ is a rational function with zero of order $\geq 2$ at infinity and no poles at integers, then $$\sum_{-\infty}^\infty f(n)=-\sum{\mathrm{res}}_af(z)\pi\cot\pi z,$$ where the summation is over all poles of $f$. For $f(z)=1/(1+z^2)$ we obtain $$\sum_0^\infty\frac{1}{1+n^2}=1+\sum_{1}^\infty\frac{1}{1+n^2}=(\pi\coth\pi+1)/2,$$ so you copied the result from Wolfram incorrectly.