Let  $SRMI_q(2n)$ denote the number of self-reciprocal
irreducible monic polynomials of even degree $2n$ over the finite
field $\mathbf{F}_q$ with $q$ elements. According to  Thm 3.1.20 of
[Handbook of Finite Fields][1] by
Mullen and Panario we have that
\begin{equation*}
  SRMI_q(2n) = \frac{1}{2n}\sum_{\text{odd $d \mid n$}}
  \mu(d)(q^{n/d}-1), \qquad
  SRMI_q(2n) = \frac{1}{2n}\sum_{\text{odd $d \mid n$}} \mu(d)q^{n/d}
\end{equation*}
depending on whether $q$ is odd or even. The sums range over all odd
divisors of $n$.  Now comes my question. Is it possible to write the
infinite product
\begin{equation*}
   \prod_{n \geq 1} \frac{1}{(1-x^{2n})^{SRMI_q(2n)}} 
\end{equation*}
as a rational function?

What I have in mind is an analogue of the well-known formula
\begin{equation*}
  \prod_{n \geq 1} \frac{1}{(1-x^n)^{I_q(n)}} = \frac{1}{1-qx}
\end{equation*}
where $I_q(n)$ is the number of irreducible monic polynomials over $\mathbf{F}_q$.


  [1]: https://www.crcpress.com/Handbook-of-Finite-Fields/Mullen-Panario/p/book/9781439873786