Recently, I found this identity on a mathematical site(seems true):
$$\int_{0}^{1}K(x)^2\text{d}x -\int_{0}^{1} \frac{x\sqrt{1-x^2} }{2-x^2}K(x)^2\text{d}x =\frac{\Gamma\left ( \frac{1}{4} \right )^4 }{64}$$ where $K(x)=\int_{0}^{1} \frac{1}{\sqrt{1-t^2}\sqrt{1-x^2t^2} }\text{d}t$.
A problem that resemble the above identity is here. Which has the power $3$. But so far, I still don't really know how to relate those two integrals.