I've a doubt regarding the substitution $t=x^{2}$ in indefinite integrals. I know that it is not bijective and therefore it is not possible to obtain an explicit expression of $x$ and of $dx$ But there are some cases in which it is not necessary to express the $x$ For example
$\int \frac{x}{x^{4}-4x^{2}+3} dx$
Let $t=x^{2}$ then $dt=2x dx$ and so
$\int \frac{1}{2(t^{2}-4t+3)} dt$
While in other situations it is not possible to do it
$\int \sqrt{x^2-4} dx$
Is there a particular reason for that? I can't understand completely why sometimes it is allowed and other times is not
Thanks a lot