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### A difficult logarithmic integral ${\Large\int}_0^1\log(x)\,\log(2+x)\,\log(1+x)\,\log\left(1+x^{-1}\right)dx$

A friend of mine shared this problem with me. As he was told, this integral can be evaluated in a closed form (the result may involve polylogarithms). Despite all our efforts, so far we have not ...
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### Evaluating $\int_0^1 \frac{\ln^m (1+x)\ln^n x}{x}\; dx$ for $m,n\in\mathbb{N}$

Evaluate $\displaystyle \int\limits_0^1 \dfrac{\ln^m (1+x)\ln^n x}{x}\; dx$ for $m,n\in\mathbb{N}$ I was wondering if the above had some kind of a closed form, here some of the special cases have ...
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### Interesting Logarithmic Integral: $\int_{0}^{1} \frac{\ln^2 x \ln^2(1+x)}{x} \;dx$ [closed]

Other than numerical approximation, how can we calculate the closed form of this integral? $$\int_{0}^{1} \frac{\ln^2 x \ln^2(1+x)}{x} \;dx$$
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### On a certain integral that involves a product of powers of logarithms.

This is a follow-up question to the following questions: Evaluating $\int_0^1 \frac{\ln^m (1+x)\ln^n x}{x}\; dx$ for $m,n\in\mathbb{N}$ Closed form for ${\large\int}_0^1\frac{\ln^4(1+x)\ln x}x \, dx$...
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### The closed form representations of Integrals of logarithm functions

I wish to find a closed form representations of the following integral $$\int\limits_{0}^1\frac{\log^p(x)\log^r\left(\frac{1-x}{1+x}\right)}{x}dx=?$$ Here $p\ge 1$ and $r\ge 0$ are nonnegative ...
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