integration - Evaluate $int_0^{frac{π}{2}}tan (x)ln (sin x)ln (cos x)dx$

Evaluate: $$I=\int_0^{\frac{π}{2}}\tan (x)\ln (\sin x)\ln (\cos x)dx$$

My ideas is to use the Fourier series of log sin and log cos:

$$\ln (2\sin x)=-\sum_{k=1}^{\infty}\frac{\cos (2kx)}{k}$$ $$\ln (2\cos x)=-\sum_{k=1}^{\infty}\frac{(-1)^{k}\cos (2kx)}{k}$$

But my problem is that I find difficult integrals like:

$$\int\tan (x)\cos (2kx)dx$$

My another idea is:

Use the substation : $y=\tan x$ then $dx=\frac{dy}{1+y^2}$

Then where $x=0 \Rightarrow y=0$ and for $x=\frac{π}{2} \Rightarrow y=\infty$

So:

$$I=\frac{1}{2}\int_0^{\infty}\frac{y\ln \left(\frac{y}{\sqrt{1+y^2}}\right)\ln (1+y^2)}{1+y^2}dy$$

But now I don't know how to complete.