How to rearrange this fraction so it matches a Laplace Transform table identity? I have the fraction: 
$$\frac{s}{s^2+2s+2}$$
I want to rearrange the fraction so that I can solve find the inverse Laplace of it using the following identities from a Laplace Transform table:
$$f(t).................F(s)$$
$$d(t)....................1$$
$$tut(t)..................\frac{1}{s}$$
$$t^nu(t)..............\frac{n!}{s^1+1}$$
$$e^{-at}u(t).............\frac{1}{s+a}$$
$$\sin \omega tu(t)............\frac{\omega}{s^2+\omega^2}$$
$$\cos\omega tu(t).............\frac{s}{s^2+\omega^2}$$
I have tried to pull the two out from the bottom however it only makes it more complex and doesn't get me closer to one of the identities. I feel like this should be easy and I am forgetting some high school maths somewhere.

One approach is to note that the denominator is $s^2+2s+2=(s+1)^2+1$, so to define $t=s+1$ and then $$\frac s{s^2+2s+2}=\frac s{(s+1)^2+1}=\frac {t-1}{t^2+1}=\frac t{t^2+1}-\frac 1{t^2+1}$$
and the last two are in your table.  If you don't mind complex variables, you can do partial fractions
$$\frac s{s^2+2s+2}=\frac s{(s+1)^2+1}=\frac s{(s+1-i)(s+1+i)}$$
and the result will be in the table.
