<p>We study multi-propagator angular integrals, a class of phase-space integrals relevant to processes with multiple observed final states and a test-bed for transferring loop-integral technology to phase-space integrals without reversed unitarity. We present an Euler integral representation similar to the Lee-Pomeransky representation and explicitly describe a recursive IBP reduction and dimensional shift relations for the general case of <i>n</i> denominators. On the level of master integrals, applying a differential equation approach, we explicitly calculate the previously unknown angular integrals with four denominators for any number of masses to finite order in <i>ε</i>. Extending the idea of dimensional recurrence, we explore the decomposition of angular integrals into branch integrals reducing the number of scales in the master integrals from (<i>n</i> + 1)<i>n</i>/2 to <i>n</i> + 1. To showcase the potential of this method, we calculate the massless three denominator integral and establish all-order results in <i>ε</i>, including a resummation of soft logarithms.</p>