Looking for fast integration method?

I am trying to do an integration 10000*5 times, it seems like a incomplete gamma function, but it is not, because after i did some changes, the lower limit, in the incomplete gamma function, is (M-i*phi)*epsilon, which is not a real number. Please see the attached file for details,

and please help me to find a fast way to do the integration! Thanks!

[Walter Roberson]

For complex integrals you need to define the path of integration.

[Liqun]

originally, the complex number is not in the integration limits. it is in the integrand function. I have tried trapezoidal numerical integration, and Gauss-Legendre numerical integration, they both can get the correct results, but very slow. After rewriting the integration function, it seems like incomplete function, but the lower integrate limit is a complex number. so I cannot use gammainc function in matlab.