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I need to evaluate the following 3-dimensional integral in closed-form (if possible)

[tex]\int_{y_1=0}^{\infty}\int_{y_2=0}^{\infty}\int_{x_2=0}^{zy_2}\exp\left(-\min(x_2,\,y_1(z-\frac{x_2}{y_2}))\right)e^{-(K-1)x_2}e^{-y_1}e^{-y_2}\,dx_2dy_2dy_1[/tex]

where ##z## is real positive number, and ##K \geq 1## is an integer.

I need to find a way to divide the limits of the integrals such that the resulting integrals can be evaluated in closed-form, or at least in compact form in terms of functions that are implemented in major scientific software like MATLAB.

What I did was that I divided the inner most integral into two interval as following

[tex]

\int_{y_1=0}^{\infty}\int_{y_2=0}^{\infty}\int_{x_2=0}^{\frac{z\,y_1y_2}{y_1+y_2}}e^{-Kx_2}e^{-y_1}e^{-y_2}\,dx_2dy_2dy_1 + \int_{y_1=0}^{\infty}\int_{y_2=0}^{\infty}\int_{x_2=\frac{z\,y_1y_2}{y_1+y_2}}^{zy_2}\exp\left(-y_1(z-\frac{x_2}{y_2})\right)e^{-(K-1)x_2}e^{-y_1}e^{-y_2}\,dx_2dy_2dy_1

[/tex]

but this gives a very complicated integrals to be evaluated at some point later. Is there any other way I can divide the integrals (over ##y_1## and/or ##y_2## and/or ##x_2##) in a way that would make the integrals easier to evaluate?

Thanks in advance