On what properties is the substitution $t= a\cot x+\csc x$ based and how reliable is it in integrals?

Question

I remember that the change of variable $t= a\cot x+\csc x$ was mentioned briefly as a casual gimmick in "Analyse MPSI PCSI - 1re année - Costantini, Gilles", which I don't have access to anymore. And then I see it today used in Quanto's answer here while lurking. But otherwise I can't find it anywhere using Google. So I'm curious to know what makes it interesting and on what nice properties it is based. Like the universal trigonometric substitution is based on these nice properties, assume $t=\tan(\frac{x}{2})$: $$\cos(x)=\frac{1-t^2}{1+t^2}$$ $$\sin(x)=\frac{2t}{1+t^2}$$ $$\mathrm dx=\frac{2\mathrm dt}{1+t^2}$$