Calculate the sum: $\sum_{x=2}^\infty (x^2 \operatorname{arcoth}(x) \operatorname{arccot} (x) -1)$ $${\color\green{\sum_{x=2}^\infty (x^2 \operatorname{arcoth} (x) \operatorname{arccot} (x) -1)}}$$
This is an impressive sum that has bothered me for a while. Here are the major points behind the sum...
Believing on a closed form:
I believe that this has a closed form because a very similar function
$$\sum_{x=1}^\infty x\operatorname{arccot}(x)-1$$ also holds a closed form. (Link to that sum: Calculate the following infinite sum in a closed form $\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$?)
I think that the asked question has a closed form because of the fact that:
$\operatorname{arcot} x \lt \frac{1}{x} \lt \operatorname{arcoth} x$
for $x\ge 1$
Attempts:
I put this summation into W|A, which returned nothing.
I have tried also, to use the proven identity above (same as the link), but to no avail. I have made very little progress on the sum for that reason.
Thank you very much. Note that there is no real use for this sum, but I am just curious as it looks cool.

The lazy way.
According to Wolfy,
the sum to 1000 terms is
0.024493435956841733223540110946...
The Inverse Symbolic Calculator
doesn't recognize it.
