Non-Elementary Integration · Superchallenge problem

Given $\displaystyle\int_{-\infty}^{\infty}\left(\dfrac{\sin x}{x}\right)^2\cos(ax)\,dx=\pi\left(1-\dfrac{|a|}2\right)$ for $|a|\le2$, evaluate $\displaystyle\int_{-\infty}^{\infty}\left(\dfrac{\sin x}{x}\right)^2\cos x\,dx$.