Can anyone help with these integrals?

Question

$$1.\int{x\over \sqrt{1-\sqrt{1-x^2}}}dx$$ $$2.\int{\sqrt{x+1}+2\over (x+1)^2- \sqrt{x+1}}dx$$ $$3.\int{\sqrt[3]{1-\sqrt[4]{x}}\over \sqrt{x}}dx$$

Just need the general idea, im not expecting someone to do each one individually in detail. For the first one i was thinking Euler's substitution in the case a>0. With the second im having trouble choosing a substitution, because of the different roots, also goes for the first.

Answer 1

For the first one, $x=\sin\theta$ plus the sine duplication/bisection formulas.

For the second one, $x=z^2-1$.

For the third one, $x=(1-t^3)^{4}$.

Answer 2

In general if you have a radical mixed together with other stuff, it sometimes helps to "wish away" the most complicated radical. In this case, letting $u=1-\sqrt{1-x^2}$ in 1), $u=\sqrt{x+1}$ in 2), and $u=1-\sqrt[4]x$ in 3) works wonders.

This "wishful thinking" doesn't always work, of course. But it never hurts to try.