Integrate the Functional

Calculus Level 1

A function f(x)f(x) satisfies the functional equation f(tanθ)=sin22θ4f(\tan \theta)=\frac{\sin^2 2\theta}{4} for all real θ\theta. If 0f(x) dx\displaystyle \int_{0}^{\infty}f(x) \ dx is equal to πab\dfrac{\pi^a}{b} for positive integers aa and bb, then what is the value of a+ba+b?

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