The integral in question
$\int e^{-t}\cdot (1+e^{-t})^{-2}dt$
can be solved with an easy $u$-substitution. I'm getting $(1+e^{-t})^{-1}$ which I can verify is correct by derivation. I'm rusty though and I wanted to verify the solution I obtained with W.A.:
It says $-\frac{1}{1+e^t}$ which can't be right since that's a negative function for all t. The integrand is nonnegative, so no way that's the answer.
What is going on here? Anyone have an explanation?
EDIT: Mathematica 5.2 is outputting the same thing as WA.
The indefinite integral is only defined up to addition of a constant. Now
$$1 - \frac{1}{1+e^t} = \frac{e^t}{1+e^t} = \frac{1}{e^{-t}+1}$$
is precisely your result.