The integrals \begin{align} I_{7} = \int_{0}^{1} \ln(x) \ \ln^{2}\left( \ln \left(\frac{1}{x}\right) \right) \ \frac{dx}{1-x} \end{align} and \begin{align} I_{8} = \int_{0}^{1} \ln(x) \ \ln^{2}\left( \ln \left(\frac{1}{x}\right) \right) \ \ln\left(\frac{1}{x}\right) \ \frac{dx}{1-x} \end{align} can be expressed in terms of $\partial_{s}^{2}\zeta(s)|_{s=2}$ and $\partial_{s}^{2}\zeta(s)|_{s=3}$, respectively. Can these integrals be evaluated and provide a closed form expression without the use of derivatives of the Zeta function?
If the integrals can be evaluated in such a way what is the resulting value?