The Laplace Transforms

You just answered my question on cramster about laplace transforms
how did you get

$$\frac{1}{2}(\frac{1+s^2}{(1+s^2)^2} - \frac{ s^2-1}{(s^2+1)^2})$$

from

$$\frac{1}{(s^2+1)^2}$$

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