Sobolev embedding fails for $p=n$

Question

As everyone knows, the Sobolev embedding fails fails for $n\ge 2$ if we assume $p=n$. The standard example is the function $u(x)=\log \log \bigl(1+\tfrac{1}{x}\bigr)$. This function is obviously unbounded. But how can I quickly check that it is in $W^{1,1}(B(0,1))$? I started out calculating, but it gets quite messy, so I was wondering if there is a short way? Thanks!