Showing that $\log (\log 1+\frac{1}{|x|})$ belongs to $W^{1,p}(\Omega)$ for $p \geq 2$.

Question

I want to show that the function $f$ belongs to $W^{1,p}(\mathbb{R}^n)$ for $p \geq 2$, where $f$ is defined as $$ f(x)=\log\left(\log \left(1+\frac{1}{|x|}\right) \right)$$

Note: This is an example of a function that is not in $L^{\infty}$, I want it as an example of a function that belongs to $W^{1,p}(\mathbb{R}^n)$ but not to $L^{\infty}(\Omega)$.