Note: This is an extended comment.
This integral can be expressed in terms of the
MeijerG special function:
Plotting using this expression takes about 6 seconds, compared to 12 seconds using OPs code. To obtain this expression, I first made a change of variables, replacing
u is a new variable, and then evaluated the resulting integral
which takes about 10 seconds and gives the
Having a symbolic result has advantages besides plotting, for example if OP wanted to know about $x \to \infty$ asymptotics, they could use
(* (-EulerGamma+Log+Log[x]+2 PolyGamma[0,1/2])/x+O[1/x]^(3/2) *)