Fitting data with NIntegration related function

Question

I am trying to fit experimental data with data file given below function, but I am not able to find the best fit. Constant value is A=6.26 and B=0.0041684

data = Import["https://pastebin.com/raw/WB4RX3Bi", "RawJSON"];

γ1 = 72*10^-3; d = 1.0*10^3; g = 9.8; w = 175*10^-6;
lc = Sqrt[γ1/(g*d)];
B = (w/lc)^2;
c = 3*10^8;
n1 = 1;
n2 = 1.33;
P1 = ((4*3)/(c*Pi*w^2))*(n1)*((n2 - n1)/(n2 + n1));
cd[(az_)?NumericQ, (t_)?NumericQ] := 
  1*10^9*P1*w^2/(4*γ1)*
   Quiet[NIntegrate[
     Exp[-(x^2)/
        8]*(1/(1 + B/x^2))*(1 - Exp[-(t/az)*x (1 + B/x^2)/(1 + B)])/
       x, {x, 0, 10}]];
fit3 = NonlinearModelFit[data, {cd[az, t], az > 0}, az, t]
Show[DiscretePlot[fit3[t], {t, 0, .15, .005}, PlotRange -> All, 
  Frame -> True, ImageSize -> 560, PlotStyle -> Red, 
  PlotRange -> All], 
 ListLinePlot[d5r, PlotStyle -> Blue, Frame -> True, 
  PlotRange -> All]]

Answer 1

This problem can be solved in three seconds (on my laptop). But as Ulrich Neumann correctly noted, the function is chosen incorrectly. Pre-calculate and interpolate the integral depending on the parameter (this takes 2.9 seconds):

 \[Gamma]1 = 72*10^-3; d = 1.0*10^3; g = 9.8; w = 175*10^-6;
 lc = Sqrt[\[Gamma]1/(g*d)]; B = (w/lc)^2; c = 
  3*10^8; n1 = 1; n2 = 1.33;
  P1 = ((4*3)/(c*Pi*w^2))*(n1)*((n2 - n1)/(n2 + n1));
lst = Table[{s, 
      NIntegrate[
       Exp[-(x^2)/
          8]*(1/(1 + B/x^2))*(1 - Exp[-s*x (1 + B/x^2)/(1 + B)])/
         x, {x, 10^-6, 10}]}, {s, 0, 50, .1}]; // Quiet // AbsoluteTiming

f = Interpolation[lst];

Then we find the parameter az (this takes 0.01 s):

data = {{0, 0.01172}, {0.001, 6.23548}, {0.002, 8.3244}, {0.003, 
    9.2106}, {0.004, 9.37318}, {0.005, 9.9102}, {0.006, 
    10.1176}, {0.007, 10.3073}, {0.008, 10.6963}, {0.009, 
    10.874}, {0.01, 11.034}, {0.011, 11.0045}, {0.012, 
    11.2776}, {0.013, 11.305}, {0.014, 11.4793}, {0.015, 
    11.6099}, {0.016, 11.691}, {0.017, 11.718}, {0.018, 
    12.0219}, {0.019, 11.8255}, {0.02, 12.0732}, {0.021, 
    12.0837}, {0.022, 12.1648}, {0.023, 12.074}, {0.024, 
    12.2191}, {0.025, 12.3367}, {0.026, 12.5222}, {0.027, 
    12.5023}, {0.028, 12.536}, {0.029, 12.5234}, {0.03, 
    12.5596}, {0.031, 12.8146}, {0.032, 12.9112}, {0.033, 
    12.7478}, {0.034, 12.825}, {0.035, 12.8779}, {0.036, 
    12.8668}, {0.037, 12.9696}, {0.038, 12.8714}, {0.039, 
    13.0408}, {0.04, 13.1484}, {0.041, 13.1499}, {0.042, 
    13.1902}, {0.043, 13.2082}, {0.044, 13.0781}, {0.045, 
    13.255}, {0.046, 13.1519}, {0.047, 13.3301}, {0.048, 
    13.3384}, {0.049, 13.2736}, {0.05, 13.3463}, {0.051, 
    13.3512}, {0.052, 13.4537}, {0.053, 13.4734}, {0.054, 
    13.4174}, {0.055, 13.4653}, {0.056, 13.7339}, {0.057, 
    13.5058}, {0.058, 13.6605}, {0.059, 13.6965}, {0.06, 
    13.5999}, {0.061, 13.7753}, {0.062, 13.6785}, {0.063, 
    13.9492}, {0.064, 13.8257}, {0.065, 13.9978}, {0.066, 
    13.9168}, {0.067, 14.0393}, {0.068, 13.9264}, {0.069, 
    13.8798}, {0.07, 14.0839}, {0.071, 14.0549}, {0.072, 
    13.9781}, {0.073, 13.9947}, {0.074, 13.9951}, {0.075, 
    14.0768}, {0.076, 14.1712}, {0.077, 13.9664}, {0.078, 
    14.1032}, {0.079, 14.2659}, {0.08, 14.1357}, {0.081, 
    14.2042}, {0.082, 14.2916}, {0.083, 14.2316}, {0.084, 
    14.3284}, {0.085, 14.1925}, {0.086, 14.3797}, {0.087, 
    14.1594}, {0.088, 14.2352}, {0.089, 14.2466}, {0.09, 
    14.2859}, {0.091, 14.233}, {0.092, 14.3774}, {0.093, 
    14.4911}, {0.094, 14.5253}, {0.095, 14.2847}, {0.096, 
    14.4617}, {0.097, 14.3361}, {0.098, 14.4019}, {0.099, 
    14.3737}, {0.1, 14.351}, {0.101, 14.4912}, {0.102, 
    14.4534}, {0.103, 14.6539}, {0.104, 14.6208}, {0.105, 
    14.4896}, {0.106, 14.6649}, {0.107, 14.5448}, {0.108, 
    14.5012}, {0.109, 14.6028}, {0.11, 14.608}, {0.111, 
    14.5389}, {0.112, 14.6703}, {0.113, 14.5858}, {0.114, 
    14.7303}, {0.115, 14.6637}, {0.116, 14.831}, {0.117, 
    14.7325}, {0.118, 14.8648}, {0.119, 14.8289}, {0.12, 
    14.9094}, {0.121, 14.855}, {0.122, 14.9639}, {0.123, 
    14.9003}, {0.124, 14.813}, {0.125, 14.9014}, {0.126, 
    14.8026}, {0.127, 14.81}, {0.128, 15.0311}, {0.129, 
    14.8367}, {0.13, 15.0784}, {0.131, 14.8689}, {0.132, 
    14.9648}, {0.133, 14.9325}, {0.134, 14.9264}, {0.135, 
    14.9104}, {0.136, 14.9641}, {0.137, 14.915}, {0.138, 
    14.9932}, {0.139, 15.1191}, {0.14, 15.1196}, {0.141, 
    15.0484}, {0.142, 15.1241}, {0.143, 15.0201}, {0.144, 
    15.0168}, {0.145, 15.0083}, {0.146, 15.1844}, {0.147, 
    15.2427}, {0.148, 15.1382}};
p = 1*10^9*P1*w^2/(4*\[Gamma]1);
fit = NonlinearModelFit[data, p f[k t], {k}, t]
Show[DiscretePlot[fit[t], {t, 0, .15, .005}, PlotRange -> All, 
  Frame -> True, ImageSize -> 560, PlotStyle -> Red, 
  PlotRange -> All], 
 ListLinePlot[data, PlotStyle -> Blue, Frame -> True, 
  PlotRange -> All]]

I recommend using the function

fit3 = NonlinearModelFit[data, a p f[k t] + b, {a, b, k}, t]
Show[DiscretePlot[fit3[t], {t, 0, .15, .005}, PlotRange -> All, 
  Frame -> True, ImageSize -> 560, PlotStyle -> Red, 
  PlotRange -> All], 
 ListLinePlot[data, PlotStyle -> Blue, Frame -> True, 
  PlotRange -> All]]

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