calculus - Integral ${largeint}_0^1left(-frac{operatorname{li} x}xright)^adx$

Let $\operatorname{li} x$ denote the logarithmic integral

$$\operatorname{li} x=\int_0^x\frac{dt}{\ln t}.$$

Consider the following parameterized integral:
$$I(a)=\int_0^1\left(-\frac{\operatorname{li} x}x\right)^adx.$$

Can we find a closed form for this integral?

We can find some special values of this integral:

$$I(0)=1,\,\,I(1)=1,\,\,I(2)=\frac{\pi^2}6,\,\,I(3)\stackrel?=\frac{7\zeta(3)}2$$
The last value was suggested by numeric computations, and I do not yet have a proof for it.

Can we prove the conjectured value of $I(3)$?

One could expect that $I(4)$ might be a simple rational (or at least algebraic) multiple of $\pi^4$ but I could not find such a form.

Can we find closed forms for $I(4),I(5)$ and other small integer arguments?