Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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8
votes
0answers
206 views
+50

Closed-form of $\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$

Is there a possibility to find a closed-form for $$\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$$ We have $$I=\int_0^1\frac{Li_2^3(-x)+x^4Li_2^3(-\frac{1}{x})}{x^3}\,dx$$ After repeatedly ...
18
votes
1answer
232 views
+200

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 ...