According to the paper "Ten Problems in Experimental Mathematics",
$$\int_0^\infty \cos(2x)\prod_{n=1}^\infty \cos\left(\frac{x}{n}\right)dx \quad = \quad \frac{\pi}{8}\color{blue}{-7.407 \times 10^{-43}}$$
The article goes into some detail on how to compute the integral numerically in order to verify that the LHS is not strictly equal to $\pi/8$, but no theoretical explanation is given for why they are so close.
The extremely high accuracy to which this relation holds leaves a strong feeling that it is more than a mere "mathematical coincidence", in the same sense that it is not a coincidence that $e^{\pi\sqrt{163}}$ is almost an integer.
I am looking for an insight that can support that feeling.