How do I calculate $\mathcal{L}^{-1}\left[\frac{s}{s^{\alpha}(s^{2}+1)}\right]$?

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I was solving some problems of fractional calculus, and needed this intermediate result. How do I calculate $\mathcal{L}^{-1}\left[\frac{s}{s^{\alpha}(s^{2}+1)}\right]$?