Another interesting integral related to the Omega constant is the following $$\int^\infty_0 \frac{1 + 2\cos x + x \sin x}{1 + 2x \sin x + x^2} dx = \frac{\pi}{1 + \Omega}.$$ Here $\Omega = {\rm W}_0(1) = 0.56714329\ldots$ is the Omega constant while ${\rm W}_0(x)$ is the principal branch of the Lambert W function. A similar interesting integral can be found here.
When attempting to check the above answer numerically, Mathematica 9.0 gives $2.0101 \pm 0.0005$ (with a complaint about failure to converge to prescribed accuracy after 9 recursive bisections...) compared to the exact answer of $2.004662032\ldots$
Plotting the integrand two observations can be made: (i) it tends to $\infty$ as $x \rightarrow 0^+$, and (ii) for $x$ greater than about 3 it is highly oscillatory with the amplitude of the oscillations tending to zero as $x \rightarrow \infty$.
Given this, what would be the best way to evaluate the integral numerically if an accuracy of say $10^{-9}$ is needed? This accuracy should be achieved in a "reasonable" amount of time.