If \( S=\left\{x\in \mathbb{R}:\sin^{-1}\!\left(\dfrac{x+1}{\sqrt{x^{2}+2x+2}}\right)-\sin^{-1}\!\left(\dfrac{x}{\sqrt{x^{2}+1}}\right)=\dfrac{\pi}{4}\right\} \), then \( \sum\limits_{x\in S}\left(\sin\!\left(\left(x^{2}+x+5\right)\dfrac{\pi}{2}\right)-\cos\!\left(\left(x^{2}+x+5\right)\pi\right)\right) \) is equal to _____.
