MHB Simple closed form for integral

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The discussion revolves around finding a closed form for the integral involving trigonometric and logarithmic functions. The integral in question is $$\int_{0}^{1}t\cos(2t\pi)\tan(t\pi)\ln[\sin(t\pi)]\mathrm dt$$ and is proposed to equal $$\frac{1}{\pi}\cdot\frac{\ln 2}{2}(1-\ln 2)$$. Participants express a desire for hints to approach the solution, indicating a lack of progress over time. The request for assistance remains unanswered after multiple follow-ups. The conversation highlights the challenge of solving complex integrals in mathematical analysis.
Tony1
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How may we go about to show that,

$$\int_{0}^{1}t\cos(2t\pi)\tan(t\pi)\ln[\sin(t\pi)]\mathrm dt=\color{green}{1\over \pi}\cdot\color{blue}{{\ln 2\over 2}(1-\ln 2)}$$
 
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A hint is requested ... (Blush)
 
lfdahl said:
A hint is requested ... (Blush)
can I get the hint?
 
64 days after my 1st request:

A hint is still requested ... (Wave)
 

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