This integral plays a central role in a physics problem (Casimir effect)${}^\ast$ $$\Omega(\phi,L)=-\frac{1}{\pi}\operatorname{Re}\int_0^\infty \ln\bigl[1+\beta(\omega)^2 e^{i\phi-2\omega L}\bigr]\,d\omega,$$ $$\beta(\omega)=\omega-\sqrt{1+\omega^2},\quad L>0,\quad 0<\phi<\pi.$$
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.
$^\ast$ $\Omega$ 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 arXiv:2310.06672.
$^{\ast\ast}$ The Taylor series expansion of $\Omega$ for large $L$, $$\Omega=\frac{1}{2\pi}\operatorname{Re}\biggl[(L^{-1}-L^{-2}+L^{-3}-L^{-4}+L^{-5}-L^{-6})\operatorname{Li}_2(-e^{i\phi})+{}$$ $$\qquad{}+(\tfrac{1}{4}L^{-4}-L^{-5}+\tfrac{5}{2}L^{-6})\operatorname{Li}_4(-e^{i\phi})-\tfrac{9}{16}L^{-6}\operatorname{Li}_6(-e^{i\phi})+{\cal O}(L^{-7})\biggr],$$ 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 of $\Omega$ has terms that depend logarithmically on $L$, $$\Omega=\frac{1}{\pi}-\frac{1}{2}\cos(\phi/2)-\frac{1}{2\pi}(L\ln L)\cos\phi+{\cal O}(L),\;\;0<\phi<\pi.$$
