Integral form of $2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$

Question

Being inspired by this post, I've wondered if the infinite series below may be expressed as
an intregral. I'm very curious about that.

$$2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$$

I'd appreciate your feedback here. How should I ponder over this case as a simple student?
Where should I start from and what techniques I need to think of? Thanks!