About $\int_0^1\bigg(\frac{x^a-x^b}{\log(x)}\bigg)^2 \text{d}x=\log\bigg(\frac{(2a+1)^{2a+1}(2b+1)^{2b+1}}{(a+b+1)^{2(a+b+1)}}\bigg)$

For $a>-\frac{1}{2}, b>-\frac{1}{2}$, we have

$$\int_0^1\bigg(\frac{x^a-x^b}{\log(x)}\bigg)^2 \text{d}x=\log\bigg(\frac{(2a+1)^{2a+1}(2b+1)^{2b+1}}{(a+b+1)^{2(a+b+1)}}\bigg)$$

For example, choosing $a=9$ and $b=6$, we have

$$\int_0^1\bigg(\frac{x^9-x^6}{\log(x)}\bigg)^2 \text{d}x=\log\bigg(\frac{(19)^{19}(13)^{13}}{(16)^{32}}\bigg)$$

I could not prove the result by any method actually.

I am interested in the proof of this result, using both real method and complex method, not only one.

Is there a known resource for proofs of such amazing results?

Your help would be appreciated. THANKS!