Closed form for ∫10ln2x√1−x+x2dx{\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx

I want to find a closed form for this integral:

Mathematica and Maple cannot evaluate it directly, and I was not able to find it in tables. A numeric approximation for it is

Mathematica is able to find a closed form for a parameterized integral in terms of the Appell hypergeometric function:

I suspect this expression could be rewritten in a simpler form, but I could not find it yet.

It’s easy to see that $I=I''(0),$ but it’s unclear how to find a closed-form derivative of the Appell hypergeometric function with respect to its parameters.

Could you help me to find a closed form for $I$?

Update: Numerical calculations suggest that for all complex $z$ with $\Re(z)>0$ the following functional equation holds:

Edited for a more concise derivation:

Define $\mathcal{I}$ to be the value of the definite integral,

The definite integral $\mathcal{I}$ is found to have as an approximate numerical value

We begin by transforming the integral via an Euler substitution of the first kind:

Under the transformation $(3)$, the integral $\mathcal{I}$ becomes

Next, using the algebraic identity

with $a=\ln{\left(1-t\right)}\land b=\ln{\left(1+t\right)}\land c=\ln{\left(1+2t\right)}$, we may expand the integral $\mathcal{I}$ as a sum of five simpler logarithmic integrals:

Now, we’ll find it convenient to introduce the following two auxiliary functions:

and

As we shall see, the integral $\mathcal{I}$ can be expressed entirely in terms of the auxiliary functions $J$ and $H$, and hence, entirely in terms of elementary functions and the standard polylogarithms:

The function $H$ can be expressed in terms of $J$, dilogarithms, and elementary functions. Define $\gamma:=\frac{a-c}{c}$. For $0, we have $-1<\gamma$ and

The function $J$ reduces to the trilogarithm. For $-1\le z\land z\neq0$,

Thus, continuing from where we left off at the last line of $(7)$,

For $z\notin[1,\infty)$,

For $0,

Setting $z=\frac13$,