Been struggling with this problem for quite some time now and can't seem to be able to find the solution by myself.
The said integral is
$$ \int\limits_{0}^{+\infty} \sin(x\ln^{1/3}(x))\ \mathrm{d}x $$
I managed to proof it's only convergent using Abel - Dirichlet's test but that's pretty much everything. Cannot prove it's not absolutely convergent. Any suggestions are welcomed.