Linear Second Order ODE/y'' - 2 y' + y = 1 over 1 + e^x

Theorem
The second order ODE:
 * $(1): \quad y'' - 2 y' + y = \dfrac 1 {1 + e^x}$

has a particular solution:
 * $y = 1 + e^x \displaystyle \int \map \ln {1 + e^{-x} } \rd x$

Proof
Then we need to solve:

and so on.