Simple looking Integral?

Calculus Level 5

\[\large {\displaystyle \int^{\infty}_{0} \frac{dx}{(e^{\pi x}+e^{-\pi x})(16+x^2)}}\]

If the integral above equals to \({\frac{\pi}{a}-\frac{b}{c}}\) where \(a,b,c\) are positive integers and with \(b,c\) are coprime, find \(a+b+c\).

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