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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