Maple 2015 produces a simple answer by
with(Statistics): X := RandomVariable(Normal(0, sigma)):
Mean(c+exp(X))^(-n) assuming c>0, n::posint;
$$ \left( c+{{\rm e}^{1/2\,{\sigma}^{2}}} \right) ^{-n} $$
Maple 2015 produces a simple answer by
with(Statistics): X := RandomVariable(Normal(0, sigma)):
Mean(c+exp(X))^(-n) assuming c>0, n::posint;
$$ \left( c+{{\rm e}^{1/2\,{\sigma}^{2}}} \right) ^{-n} $$