First expand the integrand: \((1+e^{-x})^2=1+2e^{-x}+e^{-2x}\).

An antiderivative is \(F(x)=x-2e^{-x}-\frac12e^{-2x}\). Apply the limits \(a\) and \(2a\):

\[\begin{aligned}\int_a^{2a}(1+e^{-x})^2\,\mathrm{d}x&=\left[x-2e^{-x}-\frac12e^{-2x}\right]_a^{2a}\\&=2a-2e^{-2a}-\frac12e^{-4a}-\left(a-2e^{-a}-\frac12e^{-2a}\right)\\&=\boxed{a+2e^{-a}-\frac32e^{-2a}-\frac12e^{-4a}}.\end{aligned}\]