Clever way of evaluating $\frac{1}{\sqrt{1-2xt+x^2}} dt$ How would you spot a good substitution here?
Or perhaps there is a better way of solving it (i.e. without a substitution).
It is misleadingly (unless you can in fact arrive at useful result) attractive to factorize it: $$\int_{-1}^1\frac{1}{\sqrt{(x-t)^2-(t^2-1)}}dt$$ but the factorization does not seem to lead anywhere.
I shall not say what the substitution is that is used in the result in hope that there is a clever way of arriving at it that you will inform me of.

Hint Use the Change  of variable  $u=1-2xt+x^2$, then $du=-2xdt$. I think you can take it from here. 
Indeed, since in the denomenator you have under redical  $t$ in degree one, so directly think about a linear change of variable , as its derivative would be constant. 
