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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}];

f[x_,y_,t_]:=Grad[g[x,y,t],{x,y}];

(*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,
    {τ, Sqrt[x^2 + y^2]/a, Infinity}, GenerateConditions -> False], {x, y}] // FullSimplify
(* {-((b E^(-((b (x^2 + y^2))/(2 a^2))) x Sqrt[a^2/(x^2 + y^2)] Sqrt[(x^2 + y^2)/a^2] 
   (BesselK[0, (b (x^2 + y^2))/(2 a^2)] + BesselK[1, (b (x^2 + y^2))/(2 a^2)]))/(2 a^2)), 
   -((b E^(-((b (x^2 + y^2))/(2 a^2))) y Sqrt[a^2/(x^2 + y^2)] Sqrt[(x^2 + y^2)/a^2] 
   (BesselK[0, (b (x^2 + y^2))/(2 a^2)] + BesselK[1, (b (x^2 + y^2))/(2 a^2)]))/(2 a^2))} *)

Integrate returns unevaluated for other values of t, unfortunately.

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