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.



Haar integral of rational function of unitaries
There exist no closed-form expressions for arbitrary $d$ for the integral over the unitary group $\mathbb{U}(d)$ of a rational function of the matrix elements.  

There are asymptotic results for large $d$, see for example [J. Math. Phys. 37, 4904 (1996)](https://arxiv.org/abs/cond-mat/9604059). The leading order term for $\text{tr}\,C$ of order $d$ is
$$\int\limits\_{\mathbb{U}(d)}\dfrac{1}{\sum\_{k,l}u\_{ik}\overline{u\_{il}}c\_{kl}}dU=\frac{d}{\text{tr}\,C}+{\cal O}(1/d),$$
independent of the index $i$.


Alternatively, if $d$ is not large but the matric $C$ is close to the unit matrix, $c\_{kl}=\delta\_{kl}+\epsilon\_{kl}$, one can expand
$$
\begin{split}
\int\limits\_{\mathbb{U}(d)}\dfrac{1}{\sum\_{k,l}u\_{ik}\overline{u\_{il}}c\_{kl}}dU & =1-\int\limits\_{\mathbb{U}(d)}\sum\_{k,l}u\_{ik}\overline{u\_{il}}\epsilon\_{kl}\,dU+{\cal O}(\epsilon^2)\\
 & =1-\frac{1}{d}\sum\_{k}\epsilon\_{kk}+{\cal O}(\epsilon^2).
\end{split}$$


