real analysis - How to prove $frac{pi^2}{6}le int_0^{infty} sin(x^{log x}) mathrm dx $?

I want to prove the inequality
$$\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $$

There are some obstacles I face: the indefinite integral cannot be expressed in terms of elementary functions,
Taylor series leads to a another function that cannot be expressed in terms of elementary functions. What else to try?