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

[alyafey22]

Solve the following

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

 

[alyafey22]

Hint

Spoiler

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