MATLAB: Fitting data to custom integral function

curving fittingdata fitintegral

Hi everyone,
I struggle with fitting my measured data to an integral function. I already tried this with the curve fitting toolbox and "lsqnonlin" from the Optimization Toolbox.
Let's say I have the following data set:
x = [228.1194 179.7485 149.5914 121.6736 91.7255 60.6427 40.8913 23.9615 14.4217 11.9658 9.7682 7.4930 5.4940 3.8771 2.6096];
y = [0.7440 0.7349 0.7276 0.7049 0.6939 0.6607 0.6417 0.6069 0.5868 0.5818 0.5781 0.5748 0.5704 0.5606 0.5611];
I want to fit the the following equation to the data set:
y =
where a,b,c,d are the coeffients to fit. i think my problem is the integration part of the fitting function
I tried different approaches.
1) With Curve Fitting Toolbox
%% data set
x = [228.1194  179.7485  149.5914  121.6736   91.7255   60.6427   40.8913   23.9615   14.4217   11.9658    9.7682    7.4930    5.4940    3.8771 2.6096];
y = [0.7440    0.7349    0.7276    0.7049    0.6939    0.6607    0.6417    0.6069    0.5868    0.5818    0.5781    0.5748    0.5704    0.5606 0.5611];
plot(x,y,'x'); 
fun= @(a,b,c,d,x) 1/2/x*(integral(b * (log(1 + a/b * (exp(x/c) - 1))) *exp(-x/d) * (1/(x/c)), 0, 2*x));
g = fittype('(1/2/x*integral(b * (log(1 + a/b * (exp(x/c) - 1))) *exp(-x/d) * (1/(x/c)), 0, 2*x))',...
            'dependent', {'y'},'independent', {'x'}, 'coefficients', {'a','b','c','d'});     
[fitobject,gof] = fit(x, y, g);
Thats the error message:
Error using fittype/testCustomModelEvaluation (line 12)
Expression (integral(b * (log(1 + a/b * (exp(x/c) – 1))) *exp(-x/d) * (1/(x/c)), 0, 2*x)) is not a valid MATLAB expression, has non-scalar
coefficients, or cannot be evaluated:
Error in fittype expression ==> (integral(b .* (log(1 + a./b .* (exp(x./c) – 1))) .*exp(-x./d) .* (1./(x./c)), 0, 2.*x))
??? First input argument must be a function handle.
[…]
2) "lsqnonlin"-approach
xdata = [228.1194  179.7485  149.5914  121.6736   91.7255   60.6427   40.8913   23.9615   14.4217   11.9658    9.7682    7.4930    5.4940    3.8771 2.6096];
ydata = [0.7440    0.7349    0.7276    0.7049    0.6939    0.6607    0.6417    0.6069    0.5868    0.5818    0.5781    0.5748    0.5704    0.5606 0.5611];
plot(xdata,ydata,'x'); 
fun= @(x,xdata) 1/2/xdata*(integral(x(2) * (log(1 + x(1)/x(2) * (exp(xdata/x(3)) - 1))) *exp(-xdata/x(4)) * (1/(xdata/x(3))), 0, 2*xdata)) - ydata;
x0 = [0.65, 0.8, 8, 100000];
x = lsqnonlin(fun, x0);
Error message:
Not enough input arguments.
Error in try2>@(x,xdata)1/2/xdata*(integral(x(2)*(log(1+x(1)/x(2)*(exp(xdata/x(3))-1)))*exp(-xdata/x(4))*(1/(xdata/x(3))),0,2*xdata))-ydata
Error in lsqnonlin (line 196)
initVals.F = feval(funfcn{3},xCurrent,varargin{:});
Error in try2 (line 18)
x = lsqnonlin(fun, x0);
Caused by:
Failure in initial objective function evaluation. LSQNONLIN cannot continue.
Another way I tried first was this, where I solve the problem with a loop and simply guess the parameters: 
xdata = [228.1194  179.7485  149.5914  121.6736   91.7255   60.6427   40.8913   23.9615   14.4217   11.9658    9.7682    7.4930    5.4940    3.8771 2.6096];
ydata = [0.7440    0.7349    0.7276    0.7049    0.6939    0.6607    0.6417    0.6069    0.5868    0.5818    0.5781    0.5748    0.5704    0.5606 0.5611];
plot(xdata,ydata,'x'); 
hold on
x = [0.558, 0.78, 25, 100000];
fun = @(xdata) (x(2) .* (log(1 + x(1)./x(2) .* (exp(xdata./x(3)) - 1))) .* exp(-xdata./x(4)) .* (1./(xdata./x(3))));
for i=1:length(xdata)
    y_fit(i) = integral(fun, 0, 2*xdata(i));   
    y_fit2(i) = 1/(2*xdata(i)) *y_fit(i);         
end
plot(xdata,y_fit2,'o');
I found some similar questions in the forum but I wasn't able to adapt these to my case. I'm very thankful for any tip / hint.
Thanks in advance

Best Answer

Try the following code. Compare the equations with your code to see the differences. Also, the reason for using arrayfun() is a bit intricate, and I am feeling a bit lazy to write a long explanation :P, so I leave it to you to read the documentation and closely study this code to find out why everything is being done like this. If you have any confusion, then you can ask in the comment.
xdata = [228.1194  179.7485  149.5914  121.6736   91.7255   60.6427   40.8913   23.9615   14.4217   11.9658    9.7682    7.4930    5.4940    3.8771 2.6096];
ydata = [0.7440    0.7349    0.7276    0.7049    0.6939    0.6607    0.6417    0.6069    0.5868    0.5818    0.5781    0.5748    0.5704    0.5606 0.5611];
integral_term = @(a,b,c,d,X) integral(@(x) b.*(log(1 + a./b.*(exp(x./c)-1))).*exp(-x/d).*(1./(x/c)), 0, 2*X);
fun = @(x,xdata) 1/(2*xdata)*integral_term(x(1),x(2),x(3),x(4), xdata);
fun2 = @(x,Xdata) arrayfun(@(xdata) fun(x,xdata), Xdata);
x0 = [0.65, 0.8, 8, 100000];
x = lsqcurvefit(fun2, x0, xdata.', ydata.');
plot(xdata, ydata, '+', xdata, fun2(x, xdata), '-');
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