Is there an integral for $\frac{1}{\zeta(3)} $?

There are many integral representations for $\zeta(3)$

Some lesser known are for instance :

$$\int_0^1\frac{x(1-x)}{\sin\pi x}\text{d}x= 7\frac{\zeta(3)}{\pi^3} $$

$$\int_0^1 \frac{\operatorname{li}(x)^3 \space (x-1)}{x^3} \text{d}x = \frac{\zeta(3)}{4} $$

$$\int_0^\pi x(\pi - x) \csc(x) \space \text{d}x = 7 \space \zeta(3) $$

$$ \int_0^{\infty} \frac{\tanh^2(x)}{x^2} \text{d}x = \frac{ 14 \space \zeta(3)}{\pi^2} $$

$$\int_0^{\frac{\pi}{2}} x \log\tan x \;\text{d}x=\frac{7}{8}\zeta(3)$$

$\zeta(2) $ also has many integral representations as does $ \frac{1}{\zeta(2)} $ , although this probability is because $\frac{1}{\pi}$ and $\frac{1}{\pi^2} $ have many. Well I suspect that because I know no simple integral expression for $\frac{1}{\zeta(3)} $.

My question is: is there some interesting integral $^*$ whose result is simply $\frac{1}{\zeta(3)}$?

Note

$^*$ Interesting integral means that things like

$$\int\limits_0^{+\infty} e^{- \zeta(3) \space x}\ \text{d}x = \frac{1}{\zeta(3)} $$

are not a good answer to my question.

You could find an integral representation similar to one for the regular zeta function by using the mobius function. $$\Gamma(s)\frac{\mu(n)}{n^s}=\int_0^\infty x^{s-1}\mu(n)e^{-nx}dx$$

Summing over both sides we get $$\frac{\Gamma(s)}{\zeta(s)}=\int_0^\infty x^{s-1}\rho(x)dx$$

where $\rho(x)=\sum_{n=1}^\infty \mu(n)e^{-nx}$. However, I don't think that $\rho(x)$ has an expression in closed form.