Why is $f(x) = \frac{1}{x(|\ln x|+1)^2}\in\mathscr{L}^1$ but not $\mathscr{L}^p$ for $p>1$?

So I am reading this post for an example of a function $f$ being integrable but $|f|^p$ is not integrable for $p>1$. But I couldn't see why$$f(x) = \frac{1}{x(|\ln x|+1)^2}$$is an example. Could someone please explain it? Thanks a lot!

Also, is there a nice example for a function $f$ being in $\mathscr{L}^1(X,\mathscr{A},\mu)$ but not in $\mathscr{L}^p(X,\mathscr{A},\mu)$ for $p>1$ under general definition of integration (not Riemann integration).

I realy appreciate any help!