A calculus problem by Jose Sacramento

Calculus Level 3

Let \(f: [0,1] \to \mathbb R\) be a continuous and positive function satisfying \(f(x) f(1-x) = 1\) for \(0 \leq x \leq 1\).

Compute the closed form of \( \displaystyle \int_0^1 \dfrac1{1+f(x)} \, dx \).