How to prove $\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x - \frac{1}{x}\right)dx?$ If $f(x)$ is a continuous function on $(-\infty, +\infty)$ and $\int_{-\infty}^{+\infty} f(x) \, dx$ exists. How can I prove that
$$\int_{-\infty}^{+\infty} f(x) \, dx = \int_{-\infty}^{+\infty} f\left( x - \frac{1}{x} \right) \, dx\text{ ?}$$

Reference:
Glaisher 1876 On a formula of Cauchy's for the evaluation of a class of definite integrals, Proc. Camb. Phil. Soc. III, p. 5-12
This has the original reference to Cauchy, and an added solution by Cayley using a very original change of variable,  "weirder" than those used in previous answers.
