recently I have been trying to calculate the following numerical integral
\begin{equation} \frac{4 t}{\text{DD}}\frac{1}{2 \pi } \int_{0}^{\infty} dx \int_{0}^{\infty} d\epsilon \frac{\left(\frac{\sin \left(\frac{x}{2}\right)}{x}\right)^2 \left(\left(\epsilon ^3 \exp (-2 \epsilon )\right) \left(\frac{\text{DD} t}{d^2}\right)^2\right)}{\left(\text{DD} x^2\right) \left(\frac{\epsilon ^4 \left(\frac{\text{DD} t}{d^2}\right)^2}{x^2}+1\right)}. \end{equation}
which in Mathematica reads
exponent[(t_)?NumericQ, (DD_)?NumericQ, (d_)?NumericQ] := ((4*t)/DD)*(1/(2*Pi))*NIntegrate[(Sin[x/2]/x)^2*(((t*(DD/d^2))^2/(DD*x^2))*((\[Epsilon]^3*Exp[-2*\[Epsilon]])/(1 + ((DD/d^2)*t)^2*(\[Epsilon]^4/x^2)))), {x, zero, inf}, {\[Epsilon], zero, inf}]
Unfortunately, when I plot something like
plotexact = LogLogPlot[{exponent[1, 1, d], exponent[.1, 1, d], exponent[10, 1, d]}, {d, 10^-3, 10^3}, PlotLegends -> "Expressions", PlotPoints -> 10]
The Mathematica does not give me any plot after hours. I have tried to change the integral do main from $10^{-n}$ to $10^n$ with $n\approx5$ but that did not help at all. Any thoughts?
Thanks!