Integrate the Functional

Calculus Level 2

A function \(f(x)\) satisfies the functional equation \[f(\tan \theta)=\frac{\sin^2 2\theta}{4}\] for all real \(\theta\). If \(\displaystyle \int_{0}^{\infty}f(x) \ dx\) is equal to \(\dfrac{\pi^a}{b}\) for positive integers \(a,b\), then what is the value of \(a+b\)?

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