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?

Mathematica cannot find an expression of this integral in terms of elementary functions. However, it can be integrated numerically, like this
 NIntegrate[Log[Sin[x]] Log[Cos[x]] Log[Cos[2 x]], {x, 0, Pi/4}]

to yield the result
 -0.05874864

Edit: I'm told an analytic expression might still exist, even though it cannot be found with Mathematica. Regardless, the numeric value is the one above.
