Let X be a chi-square variable with two degrees of freedom. Let A and B be to arbitrary constants, with $A>B>0$.
I need the variance of $Y=\log(1+AX)-\log(1+BX).$
The mean is, maybe not simple, but doable, since it is the ergodic channel capacity of a Reyleigh fading channel, well known in communication theory, and is given by a sum of Exponential integrals, unless I recall incorrectly.