Meromorphic Reduction in Integration

It is argued that for certain meromorphic functions $u:\mathcal{R}\to \mathcal{R}$ and analytic function $A_1$ and for any integrable function $F$, as long as it converges as a Cauchy Principal Value, $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{A}_1(x)F[u(x)]dx={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{A}_2(x)F(x)dx,$$ where $A_2$ is also analytic.