I would like to prove that the following function is integrable $$f(x) = \begin{cases} \frac{1}{|x| (\log(|x|^{-1})^2} & |x| \leq \frac{1}{2} \\ 0 & \mathrm{o.w.} \\ \end{cases}$$ i.e. I would like to show $\int_{\mathbb{R}^d} |f(x)| \, dx < \infty$. Unfortunately, I have no clue where to start. The exercise comes from Stein & Shakarchi's Measure Theory, Integration, and Hilbert Spaces text (pg. 146).
Unfortunately, with a messy function like this, I am not sure how to proceed. I could start with
$$\int_{\mathbb{R}^d} |f(x)| \, dx = \int_{|x| \leq \frac{1}{2}} |f(x)| \, dx + 0 = \int_{|x| \leq \frac{1}{2}} \Big| \frac{1}{|x| (\log(|x|^{-1})^2} \Big| \, dx$$
$$= \int_{|x| \leq \frac{1}{2}} \frac{1}{|x| (\log(|x|^{-1})^2} \, dx \leq \int_{|x| \leq \frac{1}{2}} \frac{1}{|x|} \, dx$$
But it feels wrong to omit the log function entirely like this. Any suggestions or hints on how to proceed would be great. Thanks.