I'm trying to compute two integrals involving the Dirac delta, namely \begin{align} I_1&=\int^{1}_0\!\!\!\! dx_1\!\cdots\!\int^{1}_0\!\!\!\! dx_{8}\,\delta(x_1-x_2+x_3-x_4+x_5-x_6+x_7-x_8)\,,\\ I_2&=\int^{1}_0\!\!\!\! dx_1\!\cdots\!\int^{1}_0\!\!\!\! dx_{8}\,\delta(x_1-x_2+x_4-x_5)\delta(x_3-x_4+x_6-x_7)\,\delta(x_5-x_6+x_8-x_1)\,, \end{align} but I don't seem to have the right approach. I try to do case differentiations to find the individual contributions, but I havne't made much progress this way.
Is there a systematic method to evaluate such integrals? I also tried to evaluate them in Mathematica, but I didn't succeed with getting the exact fraction - however, I could approximate the integrals numerically and found $I_1\approx .50\pm.02$ and $I_2\approx .38\pm.02$.
I'd be happy about any suggestions!