I am trying to find a closed form solution for the following integral: $$\phi(t) = \int_0^\infty J_0(r\cdot t) \; e^{-\frac{1}{2}(r^2-1)^2} r \; dr $$ The function $J_0$ denotes the Bessel function of the first kind (https://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html). I tried to solve it with Mathematica but that failed.
Question: Is there a way to solve this integral and to find a better representation for $\phi(t)$?