<s>[INST] How to find this simplification integral involving products of roots? Consider the following integral
$$
f = \int_0^1 \frac{1}{\sqrt{-\frac{1}{2} \, t^{2} + 1} \sqrt{-t^{2} + 1}} \mathrm \,dt.
$$
If we change variable by letting $x^2=t^2/(2-t^2)$, then we have
$$
f = \int_0^1 \sqrt{2} \cdot\sqrt{\frac{1}{1-x^{4}}} \mathrm \,dx,
$$
which is a simpler form.
I read this in a book and wonder how can we come up with this sort of simplification? Is it just experience or is there systematic way to do it?
Note: The integral is the complete elliptic integral of the first kind $K(1/\sqrt{2})$.
 [/INST] $$x^2=\frac{t^2}{2-t^2}$$
$$t=x \sqrt{\frac{2}{x^2+1}}$$
$$dt = \sqrt{2} \sqrt{\frac{1}{x^2+1}} \left(1-\frac{x^2}{x^2+1}\right) dx$$
So the integrand becomes is 
$$\frac{\sqrt{2} \left(\sqrt{\frac{1}{x^2+1}} \left(1-\frac{x^2}{x^2+1}\right)\right)}{\sqrt{1-\frac{2 x^2}{x^2+1}} \sqrt{1-\frac{x^2}{x^2+1}}}$$
and finally
$$\frac{\sqrt{2}}{\sqrt{1-x^4}}$$
</s>