convergence of $\int_{0}^{\infty} \cos(x^3-x)\, dx$

Question

What can we say about the convergence of the integral $$\int_{0}^{\infty}\cos(x^3-x) dx$$ A related integral would be $\int_{0}^{\infty} \cos x^3 \,dx$. Integrating by parts, it is easy to see that the integral converges. But I couldn't quite relate these two integrals. What are some general methods to try when posed with testing convergence of integrals? Another interesting question to consider might be to replace $x^3-x$ by any polynomials in $x$. What happens then?