This integral plays a central role in a physics problem (Casimir effect)${}^\ast$ 
$$\Omega(\phi,L)=-\frac{1}{\pi}\operatorname{Re}\int_0^\infty \ln\bigl[1+\beta(\omega)^2 e^{i\phi-2\omega L}\bigr],$$
$$\beta(\omega)=\omega-\sqrt{1+\omega^2},\quad L>0,\quad 0<\phi<\pi.$$

I know the small-$L$ asymptotics and the large-$L$ asymptotics, but for the full $L$-dependence I must resort to a numerical evaluation. Does there exist a closed-form expression, perhaps in terms of some special function? The large-$L$ Taylor expansion${}^{\ast\ast}$ suggests a representation as a series of Bernoulli polynomials.

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$^\ast$ $\Omega$ is the energy of a Josephson junction of length $L$, as a function of the superconducting phase difference $\phi$. The derivative with respect to $\phi$ gives the supercurrent through the junction, the derivative with respect to $L$ gives the Casimir force, see <A HREF="https://arxiv.org/abs/2310.06672">arXiv:2310.06672</A>.

$^{\ast\ast}$ 
The Taylor series expansion for large $L$,
$$\Omega=\frac{1}{2\pi}\operatorname{Re}\left[(L^{-1}-L^{-2}+L^{-3}-L^{-4}+L^{-5}-L^{-6})\operatorname{Li}_2(-e^{i\phi})+(\tfrac{1}{4}L^{-4}-L^{-5}+\tfrac{5}{2}L^{-6})\operatorname{Li}_4(-e^{i\phi})-\tfrac{9}{16}L^{-6}\operatorname{Li}_6(-e^{i\phi})+{\cal O}(L^{-7})\right],$$
is a polynomial in $\phi$ in view of an identity relating poly-logarithms and Bernoulli polynomials,
$$\operatorname{Re}\operatorname{Li}_{2n}(-e^{i\phi})=-\tfrac{1}{2}(-1)^n\frac{(2\pi)^{2n}}{(2n)!}B_{2n}\left(\frac{\phi+\pi}{2\pi}\right),\;\;0<\phi<\pi.$$
The small-$L$ expansion has terms that depend logarithmically on $L$.


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