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\end{equation}  I am not yet sure if But  this one is correct. May need to numerically check. would then imply \begin{equation}  \frac{e^x(x-3)+3)}{(e^x-1)}=1  \end{equation}  In theory for any function which ends up as a polynomial, the differentiation and multiplication with $x$ should leave the form unchanged except for a numeric factor, which will drop out in some circumstances.