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Trying to compute Integral $\int_{0}^{\frac{\pi}4} \ln(\sin{x}+\cos{x}+\sqrt{\sin{2x}})dx$

I was facing: \begin{align}J=\int_0^\infty \frac{\ln\left(1+x-\sqrt{2x}\right)}{1+x^2}\,dx\end{align}

I want to prove that $\displaystyle J=0$, or equivalently, that, \begin{align}\int_0^1 \frac{\ln\left(1+x-\sqrt{2x}\right)}{1+x^2}\,dx=-\dfrac{1}{2}\text{G}\end{align}

$\text{G}$ being the Catalan constant.

Read carefully please.

I know, using so-called Feynman's trick, how to prove this. I would like to obtain a proof, using only integration by parts and change of variable in simple integrals (that is, no multiple integrals) I don't know, if, under these restrictions, such computation is possible.

NB: You're probably wondering what is the link between : $\int_{0}^{\frac{\pi}4} \ln(\sin{x}+\cos{x}+\sqrt{\sin{2x}})dx$ and $J$.

The link is, for $x\in\left[0;\frac{\pi}{2}\right]$, $(\sin{x}+\cos{x}+\sqrt{\sin{2x}})(\sin{x}+\cos{x}-\sqrt{\sin{2x}})=1$ and $\sin(2x)=2\sin x\cos x$