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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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(This summarizes my posts on Nielsen polylogs.) I. Question 1: How to complete the table below? Consider the special cases $z=-1$ and $z=\frac12$. Given the Nielsen generalized polylogarithm, $$S_{n,... • 48.4k 10 votes 2 answers 327 views ### Infinite Series \sum_{n=1}^\infty\frac{H_n}{n^5 2^n} Given the nth harmonic number  H_n = \sum_{j=1}^{n} \frac{1}{j}, we get from this post that apparently,$$\sum_{n=1}^{\infty}\frac{H_n}{n^k}z^n= S_{k-1,2}(z) + \rm{Li}_{\,k+1}(z)$$for -1\leq z\... • 48.4k 4 votes 2 answers 173 views ### How to evaluate \int_{0}^{\infty}\ln^2(x)\ln(1+x)\ln^2\left(1+\frac{1}{x}\right)\frac{dx}{x}$$I=\int_{0}^{\infty}\ln^2(x)\ln(1+x)\ln^2\left(1+\frac{1}{x}\right)\frac{\mathrm dx}{x}$$\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots$$\int_{0}^{\infty}\left(1-\frac{x}{2}+\frac{x^2}{3}+\cdots\... 228 views

### 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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