# Integral \int_0^1\frac{\ln x}{\left(1+x\right)\left(1+x^{-\left(2+\sqrt3\right)}\right)}dx\int_0^1\frac{\ln x}{\left(1+x\right)\left(1+x^{-\left(2+\sqrt3\right)}\right)}dx

There is a curious known integral:

If we consider $\alpha=2+\sqrt{3\vphantom{\large3}}$ as a parameter and take a derivative w.r.t. $\alpha$ at this point, we get the following:

Is it possible to express the integral $I$ in a closed form?

Here is a partial progress report. I am basically repeating Jim Belk’s analysis from the previous answer.

Set $F(a) = \int_{x=0}^1 \frac{\log(1+x^a)}{1+x} dx$. Then

so

(In order to combine the integrals, first switch the names of $x$ and $y$ in the second one.)

So

This gives a linear relation between $F'(2 + \sqrt{3})$ and $F'(2-\sqrt{3})$. If we find a second one, we can solve the linear equations and be done.

Notice that

Integrating by parts, $\int_{x=0}^1 x^b \log x dx = \frac{-1}{(b+1)^2}$. So, ignoring issues of convergence, we should have

In the last step, we turned $m+1$ and $n+1$ into $m$ and $n$ to make things pretty. My guess is that the convergence issues can be dealt with for any $a>0$, but I haven’t thought much about it.

So

Putting $a=2 + \sqrt{3}$, this is

Here is where I run out of ideas. The first sum is basically the one at the end of Jim Belk’s post, but I have no ideas for the second one.