Performine the change of variables: $z = e^{ix}$, then , $x =\frac{1}{i}\log(z)$. The integral takes the form:
$ I = \Re \int_{|z|=1 \arg(z)=0}^{|z|=1 \arg(z)=\frac{\pi}{2}} \log \big(-(\log(z))^2 +(\log(\frac{z^2+1}{2z}))^2\big ) \frac{dz}{iz} $
The real part is added, since the logarithm of the cosine is singular at $x = \frac{\pi}{2}$ and can pick up an imaginary part. Expressing the difference of squares as a product we obtain:
$ = \Re\int_{|z|=1 \arg(z)=0}^{|z|=1\arg(z)=\frac{\pi}{2}}\big ( (\log(\log(\frac{z^2+1}{2})) + (\log(\log(\frac{z^{-2}+1}{2})) \frac{dz}{iz}$.
The second part of the integral can be brought to the form of the first part by the transformation $z\rightarrow z^{-1}$ , thus
$ I = \Re\int_{|z|=1 \arg(z)=0}^{|z|=1\arg(z)=\pi}\big ( (\log(\log(\frac{z^2+1}{2})) \frac{dz}{iz}$.
The integrand is invariant under the transformation $z\rightarrow -z$, thus:
$ I = \Re\frac{1}{2}\oint_{|z|=1 } \big ( (\log(\log(\frac{z^2+1}{2})) \frac{dz}{iz}$.
The numerator has no poles in the unit disc, thus using the residue theorem:
$I = \Re \frac{2\pi i}{2 i}\log(\log(-2)) = \pi \log(\log(2))$.
Another way to show $$\int_{0}^{1} \log \Gamma(x+ \alpha) \, dx = \int_{0}^{1} \log \Gamma(x) \, dx + \alpha \log \alpha - \alpha $$
is to rewrite the integral as
$$ \begin{align} \int_{0}^{1} \log \Gamma (x+\alpha) \, dx &= \int_{\alpha}^{\alpha+1} \log \Gamma(u) \, du \\ &= \int_{0}^{1} \log \Gamma (u) \, du + \int_{1}^{\alpha+1} \log \Gamma (u) \, du - \int_{0}^{\alpha} \log \Gamma (u) \, du \\ &= \int_{0}^{1} \log \Gamma (u) \, du + \int_{0}^{\alpha} \log \Gamma (w+1) \, dw - \int_{0}^{\alpha} \log \Gamma (u) \, du \end{align}$$
and then combine the 2nd and 3rd integrals and use the functional equation $\frac{\Gamma(x+1)}{\Gamma (x)} = x.$
Best Answer
$\log$ is concave. this is just Jensen's inequality. See http://en.wikipedia.org/wiki/Jensen's_inequality Look at the measure theoretic form. Please check that this makes sense to you.