Is $\int_0^{\infty} \frac{\prod_{n-1}^m \sin (x^n)}{f(x)}dx$ convergent if $f(x) \to \infty$ is strictly positive increasing?

Question

This question is motivated by this question How to show whether these integrals converge or not? where exact integral reformulation and Dirichlet's test is used to give an answer. However suppose we just have a positive increasing continuous function $f(x)$ such that $f(x) \to \infty$. Then is $\int_0^{\infty} \frac{\prod_{n-1}^m \sin (x^n)}{f(x)}dx$ convergent? It seems like it should be, but the alternation of the numerator from positive to negative seems so unpredictable I can't see how to show it, except for the case $m = 1$.