How to prove $\int_0^\pi\frac{\ln(2+\cos\phi)}{\sqrt{2+\cos\phi}}d\phi=\frac{\ln3}{\sqrt3}K\left(\sqrt{\frac23}\right)$?

$\int^{\pi}_0\cos^{2k+1}\phi d\phi=0$, and $$\int^{\pi}_0\cos^{2k}\phi d\phi=\sqrt{\pi}\Gamma(k+1/2)/\Gamma(k+1)=2^{-2k}\pi\binom{2k}{k}.$$

Therefore $$ \begin{align*} I(n) &=\int^{\pi}_0\sum_{m=0}^{\infty}2^{n-m}\binom{n}{m}\cos^m\phi~d\phi\\ &=2^n\sum_{m=0}^{\infty}2^{-m}\binom{n}{m}\int^{\pi}_0\cos^m\phi~d\phi\\ &=2^n\pi\sum_{k=0}^{\infty}2^{-2k}\binom{n}{2k}2^{-2k}\binom{2k}{k}\\ &=2^n\pi~{}_2F_1\left(\frac{1-n}2,-\frac{n}{2};1;\frac14 \right)\\ &=2^n\pi\left(\frac23\right)^{-n}~{}_2F_1\left(-n,\frac{1}{2};1;\frac23\right)\\ &=3^n\pi~{}_2F_1\left(-n,\frac{1}{2};1;\frac23\right)\\ \end{align*} $$ Using DLMF 15.8.13 with $a=-n$, $b=1/2$ and $z=2/3$.

We note that $I(-1/2)=\frac{2}{\sqrt{3}}K(\sqrt{2/3})$.

Edit: We have $$ I(n)=3^n\pi~{}_2F_1\left(-n,\frac{1}{2};1;\frac23\right)=\frac{\pi}{\sqrt{3}}~{}_2F_1\left(n+1,\frac{1}{2};1;\frac23\right)=3^{n+1/2}I(-n-1).$$

Therefore, If we write $J(n)=3^{-n/2}I(n)$, then $J(n)=J(-n-1)$, and consequently $J'(-1/2)=0$.

Thus $$ \begin{align*} I'(-1/2) &=\left.\frac{d}{dn}3^{n/2}J(n)\right|_{n=-1/2}\\ &=\left.\left(3^{n/2}J'(n)+3^{n/2}\frac{\log 3}{2}J(n)\right)\right|_{n=-1/2}\\ &=3^{-1/4}\frac{\log 3}{2}J(-1/2)\\ &=\frac{\log 3}{2}I(-1/2)\\ &=\frac{\log 3}{\sqrt{3}}K(\sqrt{2/3}). \end{align*} $$

$$I=\int_0^\pi \frac{\ln(2+\cos x)}{\sqrt{2+\cos x}}dx\overset{2+\cos x=t}=\int_1^3\frac{\color{blue}{\ln t}}{\sqrt{t}}\frac{dt}{\sqrt{1-(2-t)^2}}\overset{t\to \frac{3}{t}}=\int_1^3\frac{\color{red}{\ln\left(\frac{3}{t}\right)}}{\sqrt t}\frac{dt}{\sqrt{1-(2-t)^2}}$$ $$\require{cancel}\Rightarrow 2I= \int_1^3\frac{\color{blue}{\cancel{\ln 3}}+\color{red}{\ln 3-\cancel{\ln t}}}{\sqrt t}\frac{dt}{\sqrt{1-(2-t)^2}}\overset{t=2+\cos x}=\ln 3\int_0^\pi\frac{dx}{\sqrt{2+\cos x}}$$ $$\Rightarrow I\overset{x\to 2x}=\frac{\ln 3}{\sqrt 3}\int_0^\frac{\pi}{2}\frac{dx}{\sqrt {\frac{2+\cos(2x)}{3}}}=\frac{\ln 3}{\sqrt 3}\int_0^\frac{\pi}{2} \frac{dx}{\sqrt{1-\frac{2}{3}\sin^2 x}}=\frac{\ln 3}{\sqrt 3}K\left(\sqrt{\frac23}\right)$$