Recently, in connection with the probem of calculating generating functions of the antisymmetric harmonic number (https://math.stackexchange.com/a/3526006/198592, and What's the generating function for $\sum_{n=1}^\infty\frac{\overline{H}_n}{n^3}x^n\ ?$) I stumbled on the beautiful integral
$$f(z) = \int_{0}^z \frac{\log(1-x)\log(x)\log(1+x)}{x}\,dx\tag{1}$$
which seems to be hard.
I tried the common procedure of partial integrations, variable transformations and antiderivatives hunting with Mathematica which generated a multitude of different variants of the integral but finally I could not solve it.
Question Can you calculate the integral $(1)$?
Notice that we are looking here for the integral as a function of the upper limit $z$, or equvalently, for an antiderivative. The problem counts as solved once $f(z)$ is expressed through known functions, we also say that $f(z)$ has a "closed functional form".
On the other hand there are myriads of integrations problems in this forum which are similar but have fixed limits, i.e. they are definite integrals which define a constant, and the question is then if this constant is expressible by known constants - has a "closed form".
Our problem also has a compagnion in the constant species ( Evaluating $\int^1_0 \frac{\log(1+x)\log(1-x) \log(x)}{x}\, \mathrm dx$) which provided the closed form for $f(1)$.
from Hot Weekly Questions - Mathematics Stack Exchange
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