Evaluating $\int_0^1\frac{\ln(1+x^2)\text{Li}_2(x)}{x}dx$ without using $\sum_{n=1}^\infty\frac{H_n}{n^3}x^n$

Question

I am trying to evaluate

$$I=\int_0^1\frac{\ln(1+x^2)\text{Li}_2(x)}{x}dx$$

Integration by parts yields

$$I=\frac58\zeta(4)-\frac12\int_0^1\frac{\ln(1-x)\text{Li}_2(-x^2)}{x}dx$$

Another related integral is

$$\int_0^1 \frac{x\ln^2x\ln(1-x)}{1+x^2}dx$$

Any idea now to evaluate any of the integrals above without using the generating function $\sum_{n=1}^\infty\frac{H_n}{n^3}x^n$ ?

Thanks