Stationary-phase asymptotic for the generalized Bessel cosine integral ($Q_{\alpha }(x)$)

Forfixed $\alpha$ and $x\to +\infty$, the stationary point near $\pi /2$ givesWhen $\alpha =x$, the stationary point coalesces cubically with the endpoint at zero and the leading order changes to $x^{-1/3}$.