Challenging integral $\int_{0}^{1}\frac{x\operatorname{li}(x)}{x^2+1}dx$

Long Comment:

@L. Milla's integral can be written:

$$I=-\frac{1}{2}\int_0^1 \frac{\log \left(\frac{1}{2} \left((1-x)^2+1\right)\right)}{ \log (1-x)} \, dx\tag{1}$$

The first few terms series expansion approximation of $\frac{1}{\log (1-x)}$ are

$$\frac{1}{\log (1-x)}\approx-\frac{1}{x}+\frac{1}{2}+\frac{x}{12}+\frac{x^2}{24}+\frac{19 x^3}{720}+\frac{3 x^4}{160}+\frac{863 x^5}{60480}+...$$

https://oeis.org/A002206 and https://oeis.org/A002207 can be used to find the full series expansion, modifying the signs to fit the present purpose:

$$\frac{1}{\log (1-x)}=-\frac{1}{x}+\sum _{n=0}^{\infty } \frac{(-1)^{n-1} x^n }{n!}\left(\sum _{j=1}^{n+1} \frac{B_j S_n^{(j-1)}}{j}\right)$$

with $B_j$ being Bernoulli Numbers and $S_n^{(j-1)}$ being Stirling number of the first kind.

Quickly using Mathematica to integrate the first few terms of this expansion, perhaps results in a "generalised algebraic form" after the first two terms, rather than a closed form as such. i.e.

$$I\approx\left(\frac{\log ^2(2)}{8}-\frac{5 \pi ^2}{96}\right)+\left(\frac{4-\pi }{8}\right)$$

for the first two terms, with terms after that being of the general algebraic form

$$+\left(\frac{a-b \,\pi +c\, \log (2)}{d}\right)$$

with $a$,$b$,$c$ and $d$ being integers.

Not a complete answer, but too long for a comment.

First, let’s use the definition of the Logarithmic Integral Function. Then, let’s switch the order of integration.

$$I=\int_0^1 \frac{xli(x)}{1+x^2}dx=\int_0^1\int_0^x\frac{x}{(1+x^2)log(y)}dydx=\int_0^1\int_y^1\frac{x}{(1+x^2)log(y)}dxdy$$

$$I=\frac{1}{2} \int_0^1 \underbrace{\frac{log(2)-log(1+y^2)}{log(y)}}_{y=e^{-t}}dy=\frac{1}{2} \int_0^\infty \frac{log(1+e^{-2t})-log(2)}{t}e^{-t}dt$$

Now, let’s make use of Laplace Transform.

$$I=\frac{1}{2}\mathscr{L}\left[\frac{log(1+e^{-2t})-log(2)}{t}\right]_{s=1}=\frac{1}{4}\int_1^\infty \frac{\psi^{(0)}(\frac{s}{4}+\frac{1}{2})-\psi^{(0)}(\frac{s}{4}+1)}{s}ds$$

Rewritting the expression using some basic properties of Digamma Function. $$I=\frac{1}{4} \lim_{R\rightarrow \infty}\int_1^R \frac{2\psi^{(0)}(\frac{s}{2})-2log(2)-\psi^{(0)}(\frac{s}{4})-\psi^{(0)}(\frac{s}{4})-\frac{4}{s}}{s}ds$$

Now, let's integrate the functions that doesn't involve Digamma and let's apply some substitutions using the limit of each integral. $$I=-1-\frac{log(2)}{2}\lim_{R\rightarrow \infty}log(R)+\frac{1}{2}\lim_{R\rightarrow \infty}\int_1^R \underbrace{\frac{\psi^{(0)}(\frac{s}{2})}{s}}_{s\rightarrow2z}ds-\frac{1}{2}\lim_{R\rightarrow \infty}\int_1^R \underbrace{\frac{\psi^{(0)}(\frac{s}{4})}{s}}_{s\rightarrow4z}ds$$

$$I=-1-\frac{log(2)}{2}\lim_{R\rightarrow \infty}log(R)+\frac{1}{2}\lim_{R\rightarrow \infty}\int_{1/2}^{R/2} \frac{\psi^{(0)}(z)}{z}dz-\frac{1}{2}\lim_{R\rightarrow \infty}\int_{1/4}^{R/4} \frac{\psi^{(0)}(z)}{z}dz$$

$$I=-1-\frac{log(2)}{2}\lim_{R\rightarrow \infty}log(R)+\frac{1}{2}\lim_{R\rightarrow \infty}\int_{R/4}^{R/2} \frac{\psi^{(0)}(z)}{z}dz-\frac{1}{2}\int_{1/4}^{1/2} \frac{\psi^{(0)}(z)}{z}dz$$

Let's use the asymptotic expansion of $\psi^{(0)}(z)$ to evaluate the integral with limits at infinity and let's use Taylor Series to evaluate the integral with finite limits.

$$I=-1-\frac{log(2)}{2}\lim_{R\rightarrow \infty}log(R)+\frac{1}{2}\lim_{R\rightarrow \infty}\left[\frac{log^2(z)}{2}\right]^{R/2}_{R/4}+ \frac{1}{2}\left[-\frac{1}{s}+\gamma\log(s)+\sum_{k=1}^{\infty}\frac{\left(-1\right)^k\zeta\left(k+1\right)}{k}s^k\ \right]_{1/4}^{1/2}$$

$$I=-1-\frac{log(2)}{2}\lim_{R\rightarrow \infty}log(R)+\lim_{R\rightarrow\infty}\left[\frac{\log{\left(2\right)}}{2}\log{\left(R\right)}-\frac{3}{4}\log^2{\left(2\right)}\right]+\frac{1}{2}\left[-\left(2-4\right)-\left(\gamma\log{\left(2\right)}-2\gamma\log{\left(2\right)}\right)+\sum_{k=1}^{\infty}\frac{\left(-1\right)^k\zeta\left(k+1\right)}{k}\left(\frac{1}{2^k}-\frac{1}{2^{2k}}\right)\right]$$

$$I=-\frac{3}{4}\log^2{\left(2\right)}+\frac{\gamma\log{\left(2\right)}}{2}+\frac{1}{2}\sum_{k=1}^{\infty}{\frac{\left(-1\right)^k}{2^kk}\eta\left(k+1\right)}$$

$\eta(z)$ is the Dirichlet Eta Function.