$\sum_{n=1}^{\infty}\frac{1}{n(\ln{n})^2+n}$ Does the following series converge or diverge $$\sum_{n=1}^{\infty}\frac{1}{n(\ln{n})^2+n}$$
I know that $\sum_{n=1}^{\infty}\frac{1}{(\ln{n})^2}$ diverges.
$\sum_{n=1}^{\infty}\frac{1}{n(\ln{n})^2}$ dominates the series in question, and converges. What comparison can I use to understand this?

Note that
$$\frac{1}{n\log^2(n)+n}\le \frac{1}{n\log^2(n)}$$
and 
$$\int_2^\infty \frac{1}{x\log^2(x)}\,dx=\frac{1}{\log(2)}$$
