Is there a close form solution for the integration of the dirichlet distribution over finite intervals of each random variable set up over the simplex. For example

$$\int_{a}^{b}\int_{c}^{d} \frac{\Gamma\left(\alpha_1 + \alpha_2 + \alpha_3\right)}{\Gamma\left(\alpha_1\right)\Gamma\left(\alpha_2\right)\Gamma\left(\alpha_3\right)}x_1^{\alpha_1-1}x_2^{\alpha_2-1}(1-x_1-x_2)^{\alpha_3-1} \rm \, \, dx_2 \, \rm dx_1.$$

If not, how do you use montecarlo or numerical ways to integrate the integral.


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