How to evaluate the following integral$$\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$$It looks like beta function but Wolfram Alpha cannot evaluate it. So, I computed the numerical value of integral above to 70 digits using Wolfram Alpha and I used the result to find its closed-form. The possible candidate closed-form from Wolfram Alpha is$$\pi\sqrt{\frac{1+\sqrt{2}}{2}}-\pi$$Is this true? If so, how to prove it?
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