I had attempted to evaluate
$\int_2^\infty (\zeta(x)-1)\, dx \approx 0.605521788882.$
Upon writing out the zeta function as a sum, I got
$\int_2^\infty \left(\frac{1}{2^x}+\frac{1}{3^x}+\cdots\right)\, dx = \sum_{n=2}^\infty \frac{1}{n^2\log n}.$
This sum is mentioned in the OEIS.
All my attempts to evaluate this sum have been fruitless. Does anyone know of a closed form, or perhaps, another interesting alternate form?