It is well-known that the Laplace integrals
$$ \int_0^{+\infty} \frac {\cos (ax)}{b^2 + x^2} \mathrm dx, \quad \int_0^{+\infty} \frac {x\sin (ax)}{b^2 + x^2} \mathrm dx $$
are computable to get a closed form via various kinds of methods [diff. wrt. parameters, contour integration, etc.]. What if we "shift" the "center" of the denominators? For example, could we get a closed form of $$ \int_0^{+\infty} \frac {\cos (ax)}{x^2 + x+1} \mathrm dx $$ for $a > 0$? Thanks in advance. Any references, comments and solutions are appreciated
$$(x^2+x+1)=(x+\frac 12)^2+\frac 34$$ and so: $$\int_0^\infty\frac{\cos(ax)}{x^2+x+1}dx=\int_{1/2}^\infty\frac{\cos\left[a(u-\frac 12)\right]}{u^2+(\frac{\sqrt{3}}{2})^2}du$$ this lower limit is what causes the problems