An integral with respect to the Haar measure on a unitary group

Question

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary matrices and $H^\dagger$ is the conjugate transpose of $H$. (If $A=I$ this is very easy to solve, but I want the answer for $A\neq I$ in terms of $A$ and $D$.)