Invariant and Group Theoretical Integrations over the U(n) Group

In a previous article, an `invariant method' to calculate monomial integrals over the U(n) group was introduced. In this paper, we study the more traditional group-theoretical method, and compare its strengths and weaknesses with those of the invariant method. As a result, we are able to introduce a `hybrid method' which combines the respective strengths of the other two methods. There are many examples in the paper illustrating how each of these methods works.