Prove that $\int_0^1\log\left(\Lambda-\sqrt{\Lambda^2+C}\sin(\pi x+D)\right)\,dx\le\frac{\Lambda-1}2+\frac{\Lambda\sinh\pi\sqrt C}{2\pi\sqrt C}$

Question

Consider the generalised Poincaré inequality^[1] $$\left\||u|^2\right\|_{L^1}-\left\||u|^2\right\|_{L^{1/p}}\le(p-1)C_p(\mu)\int|\nabla u|^2\,d\mu$$ for $u\in H^1(\mu)$, $p\in(1,2]$ and some positive constant $C_p(\mu)$.

The special case $p=2$ over the finite interval $[0,1]\in\Bbb R^1$ yields the convex Sobolev inequality^[2] $$\int_0^1\phi(u)\,dx-\phi\left(\int_0^1u\,dx\right)\le\frac1{2\pi^2}\int_0^1\left(\frac{\partial\psi(u)}{\partial x}\right)^2\,dx$$ where $\phi$ is convex, $(\phi'')^{-1/2}$ is concave and $\psi'^2=\phi''$. In particular, we can take $\psi=-\phi=\log$ to obtain the result $$F(x,u,u’):=\int_0^1\log u\,dx+\frac1{2\pi^2}\int_0^1\frac{u'^2}{u^2}\,dx\ge0$$ which is shown in Prove that $0\le\int_0^1\log(u){\rm d}x+\frac1{2\pi^2}\int_0^1\frac1{u^2}\left(\frac{{\rm d}u}{{\rm d}x}\right)^2{\rm d}x$. This is of course, an easy consequence of a rather advanced result so I tried a calculus of variations approach.

With the additional constraint that $u(x)$ is a positive, differentiable density function, we define the functional $F(x,u,u’)+\lambda\int_0^1u\,dx$ which yields the stationary path $$u_s(x)=\frac1{\Lambda-\sqrt{\Lambda^2+C}\sin(\pi x+D)}$$ where $C,\Lambda=-\lambda>0$ are subject to $\sqrt{\Lambda^2+C}\cos D\tanh(\pi\sqrt C/2)=-\sqrt C$. Thus the inequality becomes (details in my answer to linked question) $$\int_0^1\log\left(\Lambda-\sqrt{\Lambda^2+C}\sin(\pi x+D)\right)\,dx\le\frac{\Lambda-1}2+\frac{\Lambda\sinh\pi\sqrt C}{2\pi\sqrt C}\tag1$$ for all $C,D$ whenever $u_s(x)$ is continuous on $[0,1]$ and $F(x,u_s,u_s')$ is defined and finite.

Is there a way to prove $(1)$ directly?

_References

_{[1] Bartier, J-P., Doubault, J. (2006). Convex Sobolev inequalities and spectral gap. Comptes Rendus Mathematique. 342(5):307-312.}

_{[2] Matthes, D. Lecture Notes on the Course "Entropy Methods and Related Functional Inequalities". Available from https://www2.karlin.mff.cuni.cz/~kaplicky/pages/pages/2020l/matthes_entropymethods.pdf.}