Asymptotic behaviour of athe function $\displaystyle\exp\left( \beta \int_{0}^{at}\frac{\cos(u)-1}{u}du \right)$ for $t\to\infty$

I am struggling to get from the last step of the proof of Lemma 2.4 to equation 2.17 of this paper.

The problem is the following:let $a>0$ and $\beta>0$ be two positive numbers and $I_\beta$ the integral defined as:$$\forall t\in\mathbb{R},\quad I_\beta(t) = \exp\left(\beta \int_{0}^{at}\frac{\cos(u)-1}{u}du\right). $$

How can one prove that we have $I_\beta(t) = O(t^{-\beta})$ for $t\to\infty$ ?

Thank you very much!