Answer
$${e^{{e^z}}} + C $$
Work Step by Step
$$\eqalign{
& \int {{e^{z + {e^z}}}} dz \cr
& {\text{use the property }}{e^{m + n}} = {e^m}{e^n} \cr
& = \int {{e^z}{e^{{e^z}}}} dz \cr
& {\text{integrate by the substitution method}} \cr
& {\text{set }}u = {e^z}{\text{ then }}du = {e^z}dz \cr
& {\text{write the integrand in terms of }}u \cr
& \int {{e^z}{e^{{e^z}}}} dz = \int {{e^u}} du \cr
& {\text{integrating}}{\text{,}} \cr
& = {e^u} + C \cr
& {\text{replace }}{e^z}{\text{ for }}u \cr
& = {e^{{e^z}}} + C \cr} $$
