MHB How Do You Integrate $\int^1_0 \log^2(1-x) \log^2(x) \, dx$?

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Solve the following

$$\int^1_0 \log^2(1-x) \log^2(x) \, dx$$​
 
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Hint

$$\sum H_k x^k = -\frac{\log(1-x)}{1-x}$$
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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