Rearranging exponential functions

Using two different strategies, I've derived an equation for a particular function $f(\varphi )$. That equation is

$f(\varphi )=\frac{1}{1-{e}^{-T\gamma}}(1-{e}^{-T\gamma \varphi}).$

However, the paper whose result I'm trying to replicate is telling me that the function is

$f(\varphi )=(1-{e}^{-\gamma})(1-{e}^{-\gamma \varphi}).$

Is there a way to rearrange the result I got so that it matches the result in the paper?

EDIT:

What's confusing me about this is that the function needs to be such that $f(0)=0$ and $f(1)=1$. This is true of the equation I derived. But for the equation in the paper, $f(1)=1$ only if $\gamma =\mathrm{ln}(1/2)$, which is not the case in the specific example given later in the paper. So something weird is afoot (either on my part or on the paper's).