This question is related to my question [here](https://math.stackexchange.com/questions/2803363/behavior-of-int-ll-x2k-operatornameerfxk-dx) such that i want to find a closed form of $\int\_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx $ , for $k$ is even integer because for odd integer is $0$ as we have integrand of odd function , Really for $k=2$ it gives a nice closed form as shown [here in wolfram alpha](https://www.wolframalpha.com/input/?i=integral+(+x%5E(4)+erf(x)%5E2)),+x%3D-1+to+1), Now my question here is :



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> >  Is it possible to get a Closed form of :$\int\_{-1}^1 x^{2k} \operatorname{erf}(x)^k \,dx $ for $ k$ is even integer?
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**Note:** The Motivation of this question is to find a Series representation of 
 :$\int \_{0}^{t}\exp(-x^2 \operatorname{erf}(x))dx$



Closed form of :$\int_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx $ for $ k$ is even integer and :$\int _{0}^{t}\exp(-x^2 \operatorname{erf}(x))dx$
I understand from the OP that the motivation for this question is to find a series expansion in powers of $t$ of 
$$I(t)=\int \_{0}^{t}\exp(-x^2 \operatorname{erf}(x))dx=\sum\_{p=1}^\infty c\_p t^p.$$
The coefficients $c\_p=p^{-1}d\_{p-1}$ follow from the series expansion
$e^{-x^2\,{\rm erf}\,x}=\sum\_{p=0}^\infty d\_p x^p$, resulting in
$$I(t)=\sum\_{p=1}^\infty c\_p t^p=t-\frac{t^4}{2 \sqrt{\pi }}+\frac{t^6}{9 \sqrt{\pi }}+\frac{2 t^7}{7 \pi }-\frac{t^8}{40 \sqrt{\pi }}-\frac{4 t^9}{27 \pi }+\frac{(\pi -28) t^{10}}{210 \pi ^{3/2}}+O\left(t^{11}\right).$$
The series $I(1)=\sum\_{p=1}^\infty c\_p$ seems to converge:


![](https://ilorentz.org/beenakker/MO/errorfunction_3.png)


Plot of $I\_N=\sum\_{p=0}^{N} c\_p $ as a function of $N$ up to $N=25$. The value of $I\_{25}=0.8162$ agrees with $I(1)=0.816377$ to three decimal places. For $N=50$ the agreement is up to six decimal places.


