The oscillatory integral $\sup_{b, z > 0} \left| \int_0^b \frac{\cos(z \sqrt{b + x}) - \cos(z \sqrt{b - x})}{x} dx \right| < \infty$

I am trying to prove the boundedness of certain oscillatory integrals and I can not deal with the following situation, which I have reduced to a specific example. I claim that$$\sup_{b, z > 0} \left| \int_0^b \frac{\cos(z \sqrt{b + x}) - \cos(z \sqrt{b - x})}{x} dx \right| < \infty$$

More generally I am looking for strategies that allow us to prove$$\sup_{b, z > 0} \left| \int_0^b \frac{\cos(z f(b + x)) - \cos(z f(b - x))}{x} dx \right| < \infty$$for functions $f(x) \approx x^\alpha$, $\alpha \in (0, 1)$.

I think Van-der-Corput's lemma allows us to bound$$\sup_{b, z > 0} \left| \int_{\frac{1}{z}}^b \frac{\cos(z \sqrt{b \pm x})}{x} dx \right| < \infty,$$so the integral over $\left(0, \frac{1}{z}\right)$ is the actual hard part.Thank you!