I want to calculate the integral $$ \int_0^1\int_0^1\int_0^1 {\rm d}r_1{\rm d}r_2{\rm d}r_3 \, r_1 r_2 r_3\int_0^\pi \int_0^\pi \int_0^\pi {\rm d}\phi_1{\rm d}\phi_2{\rm d}\phi_3 \\ \left| r_1r_2\sin(\phi_2-\phi_1) + r_2r_3\sin(\phi_3-\phi_2) + r_3r_1\sin(\phi_1-\phi_3)\right| $$ The problem obviously is the absolute value. I have tried for ages but don't get anywhere :-(
I tried to split the integral, but that leads to non-integrable terms, but maybe there is a trick using the symmetry?
I'm thinking to somehow transform the $\phi$ integrals and do them first.
The absolute value actually is two times the area of a triangle with vertices $P_1,P_2,P_3$ inside a semi-circle.
The transform mentioned by Yuriy seems a bit complicated:
