Can you provide values for all your variables? S what is EI,k0 etc.
What normally helps to speed integration up is to use MATLAB Coder to generate a MEX file out of the integration part. Integral itself is not supported for code generation but quadgk is. If you can provide some values for your variable I could test it and post the timing result.
What do you set 'ArrayValued' to true? As far as I see its not a vector valued function. fun seems like a R^1 => C^1 function.
Have you tested other integral functions like quadgk?