Analytic formula for $\int_0^{2\pi} \cos^N\phi H_n(r\cos\phi)\sin^M\phi H_m(r\sin\phi)d\phi$, where $H_n$ is $n$th Hermite polynomial

Question

This question is related to Find coefficient of $t^k$ in $\int_0^{2\pi}H_n(t\cos\phi+(1-t^2)^{1/2}\sin\phi)H_m(r\cos\phi)d\phi$, where $H_n$ is $n$th Hermite polynomial

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Let $n,m,N,M \ge 0$ be integers, $r \ge 0$ be a real number, and define $I=I(n,m,N,M,r)$ by $$ I:= \int_0^{2\pi} \cos^N\phi H_n(r\cos\phi)\sin^M\phi H_m(r\sin\phi)d\phi, $$ where $H_n$ is the $n$th Hermite polynomial.

Question. Is there an analytic formular for $I$, perhaps in terms of special functions (Gamma function, etc.)?