# Evaluating function

I'm trying to evaluate this function, which is gradient of integral with variable limit of integration: $$u(x,y,t)=\nabla(\int_{v(x,y)}^\infty \frac{e^{-b(t-\tau-c)^2}}{\sqrt{\tau^2-{v(x,y)}^2}} d\tau )$$ where $v(x,y)={\sqrt{x^2 + y^2}\over a}$, and $a,b,c$ are arbitrary constants. This was my attempt:

ClearAll;
a=1.;
b=2.;
c=3.;

g[x_,y_,t_]:=Integrate[Exp[-(b (t-τ-c)^2)]/Sqrt[τ^2-(Sqrt[x^2+y^2]/a)^2],
{τ,Sqrt[x^2+y^2]/a,Infinity},Assumptions->{t>0}];

(*try to evaluate function at point [20,30,1]*)

f[x,y,t]/.{x->20,y->30,t->1}


but is not working. I am not expecting, that I will get function $u(x,y,t)$ in explicit form as a function of time $(t)$ and spatial coordinates $(x,y)$,but the values of function at some dicrete time points for given $x,y$ would be great. Any advice and suggestions will be greatly appreciated.

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A solution can be obtained for the specific case t -> c.
Grad[Integrate[(Exp[-(b (t - τ - c)^2)]/Sqrt[τ^2 - (Sqrt[x^2 + y^2]/a)^2]) /. t -> c,

Integrate returns unevaluated for other values of t, unfortunately.