The same topic of the OP of this question is posted here seeking for closed form of $I(a)=\int_{0}^a {(e^{-x²})}^{\operatorname{erf}(x)}dx $ and it's behavior compared to error function, the question is answered nicely and numerically by Turiy , but in this question I want to know the asymptotic series of :
$I(a)=\int_{0}^a {(e^{-x²})}^{\operatorname{erf}(x)}dx $ ? and if it does have a complementary function as error function ? Does it satisfies any functional equation ? . Only what i have deduced about Bondedness of the titled integral is :
$0\leq I(a)=\int_{0}^a {(e^{-x²})}^{\operatorname{erf}(x)}dx \leq \operatorname{erf}(x) $ , for $x \geq 0$