Evaluate $\int_{0}^{1} \operatorname{Li}_3\left [ \left ( \frac{x(1-x)}{1+x} \right ) ^2 \right ] \text{d}x$

Question

Possibly evaluate the integral? $$ \int_{0}^{1} \operatorname{Li}_3\left [ \left ( \frac{x(1-x)}{1+x} \right ) ^2 \right ] \text{d}x. $$ I came across this when playing with Legendre polynomials, and I am curious about the existence of this closed-form of the simple-looking integral. The relevant functional equations are much easier to arrive at, but leaving a massive calculation to reform the integral, out of my reach. I am appreciated for your help.
I somehow believe its existence, as I have derived the following expression:
Denoting $\chi_2(x)=\sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)^2}$ and defining $$ f(x)=\operatorname{Li}_3(x^2) +4\ln\left ( 1-x^2 \right ) +8\left ( \frac{\text{artanh}(x)+\chi_2(x)}{x} -1 \right ), $$ one have $$ \int_{0}^{1} f\left ( \frac{x(1-x)}{1+x} \right ) \text{d}x =\frac{21}{2}\zeta(3)-4\ln(2)^2-16+\pi^2-\frac{2\pi^2}3\ln(2). $$ The idea is to find a function, whose Legendre-Fourier coefficients consisting of $$ \int_{0}^{1} \left ( \frac{x\left ( 1-x \right ) }{1+x} \right )^n \text{d} x, $$ and therefore it's an obvious task.