When You're High on Math

Calculus Level 5

$\prod_{k=1}^{\infty}\left(\frac{1+2 \cos \left(\frac{2x}{3^k}\right)}{3}\right)$

Define the expression above as $f(x)$ and if

$\int_{0}^{\infty}(f(x)+f^2(x)+f^3(x)+f^4(x)) \ \mathrm dx=\frac{A}{B}\pi$

where $A$ and $B$ are coprime positive integers. Find the value of $A+B$ .