Recently a friend who is writing a book on integrals added this problem to his book:

$$\int_{0}^{1}\arcsin{\sqrt{1-\sqrt{x}}}\ dx=\frac{3\pi}{16}$$

After a while, when trying to generalize, I was able to find the following integrals:

$$\int_{0}^{1}\arcsin{\sqrt{1-\sqrt{1-\sqrt{x}}}}\ dx=\frac{61\pi}{256}$$ $$\int_{0}^{1}\arcsin{\sqrt{1-\sqrt{1-\sqrt{1-\sqrt{x}}}}}\ dx=\frac{12707\pi}{65536}$$ For this last integral, simply apply a u-substitution with the term inside the arcsin: $$\int_{0}^{1}16u(1-u^2)(1-(1-u^2)^2)(1-(1-(1-u^2)^2)^2)\arcsin{u}\ du$$ Applying integration by parts and some algebra: $$\int_{0}^{1}\frac{(1-(1-(1-u^2)^2)^2)^2}{\sqrt{1-u^2}}\ du$$ With $\sin{t} = u$, the integral becomes: $$\int_{0}^{\frac{\pi}{2}}(1-(1-\cos^4{t})^2)^2\ dt$$ Which can be easily calculated with some formulas for the beta function. However, when trying to solve the problem for n radicals, I was not able to solve this last step in a more general way, furthermore, the integrals with more radicals quickly become too large to calculate manually. Is it possible to calculate integrals for n radicals with some more simplified expression?