Integral Representation for the Square of the Airy Fuction

Question

Formula 9.11.3 in DLMF states that$$\text{Ai}(x)^2=\frac{1}{4 \pi \sqrt{3}} \int_0^{\infty } J_0\left(\frac{t^3}{12}+x t\right) t \, dt$$ where $J_0 $ is the Bessel function of the first kind of order zero. I am looking for a derivation of this result. Perhaps one can obtain this result from a related formula $$\text{Ai}(x)^2=\frac{1}{2 \pi ^{3/2}} \int_0^{\infty } \frac{1}{\sqrt{t}} \cos \left(\frac{t^3}{12}+x t+\frac{\pi }{4}\right) \, dt$$