I'm trying to compute the following Haar integral over the unitary group: $\int_{\mathbb{U}(d)}\dfrac{1}{\sum_{k,l=1}^d u_{ik}\overline{u_{il}}c_{kl}}dU $. Is there anything known about the value of such integrals? I haven't been able to find anything that studies the integrals of either non polynomial functions or non class functions of unitaries.

I'm trying to compute the following Haar integral over the unitary group: $$ \int\limits_{\mathbb{U}(d)}\dfrac{1}{\sum_{k,l=1}^d u_{ik}\overline{u_{il}}c_{kl}}dU. $$ Is there anything known about the value of such integrals? I haven't been able to find anything that studies the integrals of either non polynomial functions or non class functions of unitaries.