# What does e have to do with this integral?

Calculus Level 5

$\large \displaystyle \int_{-\infty}^{\infty} \dfrac{\cos ^3 {(x)}}{1+x^2} \, dx$

If the value of the integral above equals to $$\dfrac{\pi^p}q \left( \dfrac r{e^s} + \dfrac t{e^u} \right)$$ for positive integers $$p,q,r,s,t$$ and $$u$$ with $$\gcd (r, q) = \gcd(t, q) = 1$$. Find the value of $$p+q+r+s+t+u$$.

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