$$F(s) = \frac{s^2-4s-4}{s^4+8s^2+16}$$

My work is as follows,


The inverse laplace of the first term is, $\frac{1}{2} \sin(2t)$ The second one has no direct transform, perhaps using the convolution theorem would do. The third and final term is -$t\sin(2t)$. Is there a faster way to solve this instead of using the convolution theorem for the second term.


Note that $$ \mathcal{L}^{-1}\left[\frac{a}{s^2+a^2}\right]=\sin at $$ and $$ \mathcal{L}^{-1}\left[\frac{s^2-a^2}{(s^2+a^2)^2}\right]=t\cos at, $$ then $$ \frac{a}{s^2+a^2}-a\cdot\frac{s^2-a^2}{(s^2+a^2)^2}=\frac{2a^3}{(s^2+a^2)^2}. $$ Hence $$ \mathcal{L}^{-1}\left[\frac{2a^3}{(s^2+a^2)^2}\right]=\sin at-at\cos at $$ and $$ \mathcal{L}^{-1}\left[\frac{8}{(s^2+4)^2}\right]=\frac12(\sin 2t-2t\cos 2t). $$


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