Just an addition to @HenrikSchumacher`s nice answer.

Without compilation (incl. fast C-compiler) Mathematica & Gaussian quadrature evaluates ~25times faster than QP's result `aa` . Not so bad I think. 

 aa = modelLN2[1., 1., 1., 1., 1., 1., 1.]; // AbsoluteTiming 
 (* 3.81s *)

 Needs["NumericalDifferentialEquationAnalysis`"];

 xi = Subdivide[0. , 4.0, 10 ]; 
 xwGauss =Flatten[Map[GaussianQuadratureWeights[9(*5*), #[[1]], #[[2]]] & ,Partition[xi, 2, 1]], 1] ;

 int = Module[{AN = 1., tN = 1., tr = 1., ALN = 1. , xmf = 1., w = 1., 
 y0 = 1. }, 
 Table[AN*Exp[-(i*0.128 - 0.128)/tN] + 
 Exp[-(i*0.128 - 0.128)/tr]*
 Total@Map[(ALN*Exp[-(1/w^2)*(Log[#[[1]]/xmf])^2]*
 Exp[-#[[1]]*(i*0.128 - 0.128)]) #[[2]] &, xwGauss], {i, 
 1.0, 5000.0, 1.0}] 
 ] (* 0.14s *)