In this MSE-question I've asked about three, similarly shaped, integrals for integer vales of $\zeta(s)$ that I found numerically:

$$\zeta \left( 3 \right) =\frac12{\int_{0}^{1} \frac{1}{x}\big(\zeta(2)-{\it Li_2} \left(1-x\right)\big) \,{\rm d}x} \tag{1}$$

$$\zeta \left( 4 \right) =\frac{4}{5}{\int_{0}^{1} \frac{1}{x}\big(\zeta(3)-{\it Li_3} \left(1-x\right)\big) \,{\rm d}x} \tag{2}$$

$$\zeta \left( 5 \right) =\frac12{\int_{0}^{1} \frac{1}{x}\big(\zeta(2)-{\it Li_2} \left(1-x\right)\big)^2 \,{\rm d}x} \tag{3}$$

**ADDED:** found one more:

$$\zeta \left( 3 \right) = \frac32 - \frac14{\int_{0}^{1} \big(\zeta(2)-{\it Li_2} \left(1-x\right)\big)^2 \,{\rm d}x} \tag{4}$$

where ${\it Li_n}(z)$ is the PolyLogarithm.

I have not found any similar expressions at other integer values.

The answer to the MSE-question helped reducing the integral for $\zeta(3)$ to a known integral, however still curious whether the other two could be reduced to something known as well.