I was thinking about $\int_0^{\infty} A( f(B(x)) ) + C(x) ) dx = \int_0^{\infty} f(x) dx$
The inspiration came from the following 3 integrals :
Lemma
If $f(x)$ is a bounded non-negative function, then \begin{equation}\int_0^\infty f\left(x+\frac{1}{x}\right)\,\frac{\ln x}{x}\,dx=0\end{equation}
proved here : Show $\int_0^\infty f\left(x+\frac{1}{x}\right)\,\frac{\ln x}{x}\,dx=0$ if $f(x)$ is a bounded non-negative function
and
\begin{align} \int_{-\infty}^{\infty}f\left(x-\frac{1}{x}\right)dx&=\int_{0}^{\infty}f\left(x-\frac{1}{x}\right)dx+\int_{-\infty}^{0}f\left(x-\frac{1}{x}\right)dx=\\ &=\int_{-\infty}^{\infty}f(2\sinh T)\,e^{T}dT + \int_{-\infty}^{\infty}f(2\sinh T)\,e^{-T}dT=\\ (collecting\space terms ) &=\int_{-\infty}^{\infty}f(2\sinh T)\,2\cosh T\,d T=\\ &=\int_{-\infty}^{\infty}f(x)\,dx. \end{align}
as discussed here : Why is this integral $\int_{-\infty}^{+\infty} F(f(x)) - F(x) dx = 0$?
(or the more general in the link)
And the trivial case valid for $-\infty < y < +\infty$ :
$$\int_{-\infty}^{+\infty}f(x)dx = \int_{-\infty}^{+\infty} f(x+y)dx$$
Im intrested in seeing generalizations of this such as
$\int_0^{\infty} A( f(B(x)) ) + C(x) ) dx = \int_0^{\infty} f(x) dx$
or other similar improper integrals.