how to find inverse laplace transform of

Question

how to find the inverse laplace transform of $\frac{s}{s^4+s^2+1}$. I tried to do it via partial fraction and reached $\frac{s}{(s^2-s+1)(s^2+s+1)}$

Answer 1

HINT:

Decompose $s$ as $$s^2+s+1-(s^2-s+1)=2s$$

and $$s^2\pm s+1=\left(s\pm\frac12\right)^2+\left(\frac{\sqrt3}2\right)^2$$

Finally from this, $$L\left[e^{at}\sin bt\right]=\frac b{(s-a)^2+b^2}$$

Answer 2

Hint: use complex roots for the denominator and then perform a partial-fraction decomposition. The result must remain real.