I want to integrate $f:=(1+\langle x,x\rangle)^{-2}$ over $\mathbb{R}^2$. Geometrically this problem is finding the volume of the stack of disks $$\frac{1}{\sqrt{z}}-1\ge x^2+y^2$$ I'm sure there is some substitution into polar coordinates to simplify this question, simply because we are talking about circles. My problem is that any diffeomorphism $\varphi:\mathbb{R}^2\rightarrow\mathbb{R}^2$ we could use must itself depend on two variables, so I can't simplify.
Is there a substitution to simplify the integral $$\int_{\mathbb{R}^2}f(x)d\lambda$$?
Write $x=(r\cos t, r\sin t)$ for $x\in\mathbb R^2$. Then $\langle x,x\rangle=r^2$, your integrand is $f=1/(1+r^2)^2$, and your integral is $$\iint_{\mathbb R^2} \frac 1{(1+\langle x,x\rangle)^2}dx = \int_0^{2\pi}\int_0^\infty \frac 1{(1+r^2)^2}\,rdrdt=\pi\int_1^\infty \frac 1 {s^2}\,ds=\pi,$$ using the change of variables $s=1+r^2$.