I am looking for an example of a random variable $X > 0$ with finite expectation but infinite $E[X \log_+(X)]$.
I considered the heavy tail distributions that I know. But none of them seems to satisfy this condition.
This is equivalent of finding an example of density function $f:[0,\infty) \to [0,\infty)$ such that $$ \int_{0}^\infty f(x) \mathbb d x = 1; $$ $$ \int_{0}^\infty x f(x) \mathbb d x < \infty; $$ but $$ \int_{1}^\infty x \log(x) f(x) \mathbb d x = \infty. $$
Take for example $$ f(x)= \begin{cases} c\frac 1{x^2(\log x)^2}&\mbox{ for }x>e; \\ 0&\mbox{ for }x\leqslant e, \end{cases} $$ where $c$ is such that $\int_{\mathbb R}f(x)dx=1$.