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

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!

-
 I finally solve it :) – Cortizol Feb 20 at 21:21 @Cortizol: really??? – Chris's wise sister Feb 20 at 21:21 I was thinking on my problem. I add answer (in my topic). – Cortizol Feb 20 at 21:22 I'd also like to see if there is solution via integrals :) – Cortizol Feb 20 at 21:30