# What is the inverse laplace transform of (s^3−a^2s)/((s^2+a^2)^2)

What is the inverse laplace transform of $\frac{{s}^{3}-{a}^{2}s}{\left({s}^{2}+{a}^{2}{\right)}^{2}}$
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Partial fraction expansion
$\frac{s}{{a}^{2}+{s}^{2}}-\frac{2{a}^{2}s}{{\left({a}^{2}+{s}^{2}\right)}^{2}}$
Notice that
$\frac{d}{ds}\frac{{a}^{2}}{{a}^{2}+{s}^{2}}=-\frac{2{a}^{2}s}{{\left({a}^{2}+{s}^{2}\right)}^{2}}$
Use the transform table and the derivative rule to find the result as
$\mathrm{cos}\left(at\right)-at\mathrm{sin}\left(at\right)$
This is a somewhat difficult problem I think