Closed form of :$\int_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx $ for $ k$ is even integer and :$\int _{0}^{t}\exp(-x^2 \operatorname{erf}(x))dx$

Question

This question is related to my question here 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, Now my question here is :

Question:
Is it possible to get a Closed form of :$\int_{-1}^1 x^{2k} \operatorname{erf}(x)^k \,dx $ for $ k$ is even integer?

Note: The Motivation of this question is to find a Series representation of :$\int _{0}^{t}\exp(-x^2 \operatorname{erf}(x))dx$

Answer 1

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:

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.