Consider the following integral $$\int_0^1 dx_1 \int_0^{1-x_1} dx_2 \, (1-x_1-x_2)^{-\epsilon-1} (-sx_2 - x_1p_1^2)^{-\epsilon-1}$$ where $s$ and $p_1^2$ are to be treated as constants throughout the integration and $\epsilon$ is a real number.
This calculation cropped up midway in a physics calculation and I could not immediately see a way to integrate it. I subsequently put it in to Mathematica and, much to my surprise, after fifteen mins it is still running a finished evaluation of it. I have kept Mathematica running in the meantime but was wondering if anyone had any tips on how to progress with the integral on a more analytic level.
I tried $$\int _0^1 dx_1\int _0^{1-x_1}dx_2 \to \int _0^1 dx_1\int _0^{1-x_1}dx_3$$ with $x_3=1-x_1-x_2$, and change the order of integral of $x_1$ and $x_3$ so that it is $$\int _0^1 dx_3\int _0^{1-x_3}dx_1 \left(x_3(-s(1-x_1-x_3)-x_1p_1^2)\right)^{-\epsilon-1}$$and Mathematica can do it now$$=-\int _0^1dx_3\left(\frac{(x_3-1)\left(s(x_3(x_3-1)s)^{-\epsilon-1}-p_1^2(x_3(x_3-1)p_1^2)^{-\epsilon-1}\right)}{(s-p_1^2)(-\epsilon)}\right)$$ Hope it will help