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There are 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). The leading order term is $$\int_{\mathbb{U}(d)}\dfrac{1}{\sum_{k,l}u_{ik}\overline{u_{il}}c_{kl}}dU\rightarrow \frac{d}{\sum_{k}c_{kk}},$$ independent of the index $i$.