Evaluating $\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \,dx$
What tools would you recommend me for evaluating this integral?
$$\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \,dx$$
My first thought was to use the beta function, but it's hard to get such a form because
of $\cos(2x)$. What other options do I have?
The function $g(x)=\left(1-\left(\frac{4x}{\pi }\right)^4\right)^{\frac{1}{4}}-1$ is a good approximation of the integrand. And its integral between $[0,\frac{\pi}{4}]$ is equal to $\frac{1}{8}(\sqrt{\pi}\Gamma(\frac{1}{4})\Gamma(\frac{5}{4})-2\pi)$ which approximatly equals to $-0.0573047$.