# numerical integration – Integrating a B-Spline basis function with respect to the standard normal PDF

I am looking for ways to evaluate exactly (i.e. analytically or semi-analytically) integrals of the type:
$$int_{-infty}^{+infty}B_{i}^k(u)e^{-frac{(u-mu)^2}{2sigma^2}}du,$$
where $$B_i^k$$ is a spline of order $$k$$, an element of the B-Spline basis for the linear space of splines of order $$k$$ on knots $${t_i}$$, defined as usual recursively by:
$$B_i^k(x)=frac{x-t_i}{t_{i+k}-t_i}B_i^{k-1}(x)+frac{t_{i+k+1}-x}{t_{i+k+1}-t_{i+1}}B_{i+1}^{k-1}(x),$$
with
$$B_i^0(x)=begin{cases} 1 & xin (t_i;t_{i+1}) \ 0 & text{otherwise } end{cases}$$
Of particular interest would be the case of $$mu=0, sigma=1$$.

I am aware of the Gauss-Hermite quadrature :
$$int_{-infty}^{+infty}f(x)e^{-frac{x^2}{2}}approx sum_{i=1}^n w_i f(x_i),$$
where $$x_i$$ are the roots of a Hermite polynomial of order $$n$$ and $$w_i$$ are the associated weights. Importantly, the approximation sign can be replaced by an exact equality when $$f$$ is a polynomial of degree $$leq 2n-1$$. (There are versions where the integral is with respect to $$e^{-x^2}$$ instead of $$e^{-frac{x^2}{2}}$$, by changing the type of Hermite polynomial employed).

My question is : is there such an exact equality formula for B-spline basis functions?
I am looking to express the integral at the beginning of this question as a sum analogously to the Gauss-Hermite quadrature.

The problem seems to be that even though $$B_i^k$$ is known to have finite support, it is not itself a polynomial: each of the restrictions $$B_i^k|_{(t_j;t_{j+1})}$$ is a polynomial, without the full function being a polynomial. Otherwise, the answer would have been a trivial application of the Gauss-Hermite quadrature. Is is possible that there is a Gauss-Hermite-type quadrature for integration domains that are compact intervals (as opposed to integration domains that are $$mathbb{R}$$) ?