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We present a new analytic framework for phase-space integrals, central to precision collider physics. Angular components are computed through multifold Mellin–Barnes representations, producing results in terms of Goncharov polylogarithms for up to four denominators. We establish recursion relations that systematically reduce higher-power denominators. The angular part is combined with the radial integration via a careful treatment of singularities. Applications to NNLO QCD corrections in semi-inclusive deep inelastic scattering will be discussed.