An Alternating Sum

Calculus Level 5

$S=\sum_{n=-\infty}^{\infty} \dfrac{(-1)^n}{1+n^2}=\dfrac{a\pi e^{b\pi}}{e^{c\pi}-1},$

where $a,b,c$ are natural numbers. Find $a+b+c$ .

Note: Please avoid using wolfram alpha.

Hint: Think about the Fourier transform of $f(x)=e^{-|ax|}$ .