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I'm interested in the following definite integral: $$I=\int_0^1\frac{\ln^2\!\left(1+x+x^2\right)}x\,dx.\tag1$$ The corresponding antiderivative can be evaluated with Mathematica, but even after simplification is quite clumsy. It matches results of numerical integration, and its correctness can potentially be verified by hand using differentiation. So, we are assured that a closed form exists for $I$, albeit complicated one.

My program for a numerical search for closed forms found a much simpler candidate:

$$I\stackrel{\color{gray}?}=\frac{2\pi}{9\sqrt3}\psi^{\small(1)}\!\left(\tfrac13\right)-\frac{4\pi^3}{27\sqrt3}-\frac23\zeta(3).\tag2$$

Note that the trigamma value here can be expressed in terms of the dilogarithm of complex argument (see formula $(5)$ here) or of the $2^{nd}$ order harmonic number of fractional argument: $$\begin{align}\psi^{\small(1)}\!\left(\tfrac13\right)&=\frac{2\pi^2}3+2\sqrt3\,\Im\,\operatorname{Li}_2\!\left[(-1)^{\small1/3}\right],\tag3\\\psi^{\small(1)}\!\left(\tfrac13\right)&=\frac{\pi^2}6+9-H^{\small(2)}_{\small1/3}.\tag4\end{align}$$ Can we prove $(2)$, preferably not going through the huge intermediate antiderivative?

One possible direction that I thought of is to factor the polynomial under the logarithm: $$I=\int_0^1\Big[\ln\!\left(x+(-1)^{\small1/3}\right)+\ln\!\left(x-(-1)^{\small2/3}\right)\Big]^2x^{-1}\,dx.\tag5$$ After expanding the square brackets, Mathematica can find a simpler antiderivative for it. Can we reach $(2)$ following this direction manually?

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    $\begingroup$ As a bonus point, we can also try to prove $$\int_0^1\frac{\ln^2\!\left(1+x+x^2\right)}{x^2}\,dx=\frac{4\pi^2}9-\frac32 \ln^2 3+\frac\pi{\sqrt3}\ln3-\frac13\psi^{\small(1)}\!\left(\tfrac13\right). \tag{$\diamond$}$$ $\endgroup$
    – Vladimir Reshetnikov
    Commented Sep 9, 2015 at 0:03
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    $\begingroup$ Bonus point is easier. Integrating by parts kills one of the logarithms; Any expression of the form $R_1(x)\ln R_2(x)$ admits an easily computable antiderivative expressed in terms of dilogarithms (factor $R_2$ and decompose $R_1$ into partial fractions). $\endgroup$
    – Start wearing purple
    Commented Sep 9, 2015 at 0:37
  • $\begingroup$ I knew this was Vladimir before I opened the post... I'm always impressed with the formulas you come up with! $\endgroup$
    – Bruno Joyal
    Commented Sep 9, 2015 at 2:01
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    $\begingroup$ You may want to consider Catalan's constant $K = \rm{Cl}_2\big(\tfrac\pi2\big)$ and its cubic counterpart Gieseking's constant $\color{red}\kappa =\rm{Cl}_2\big(\tfrac\pi3\big)$ as having equal status. Hence, many integrals would have a closed-form like, $$\int_0^1 \frac{\ln^2(x^2+x+1)}{x}dx =\frac{4\pi\,\color{red}\kappa}9-\frac{2\zeta(3)}3 $$ $$\int_1^\infty \frac{\ln(x)}{x^2+x+1}dx =\frac{4\sqrt3\,\color{red}\kappa}9$$ etc. More can be found in this list. $\endgroup$
    – Tito Piezas III
    Commented Jul 1, 2019 at 17:03
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    $\begingroup$ See also here and the related links from that answer. $\endgroup$
    – Zacky
    Commented Feb 13, 2020 at 9:37

2 Answers 2

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  1. Replace $\ln(1+x+x^2)$ by $\ln(1-x^3)-\ln(1-x)$.

  2. Two of the resulting integrals are easy to compute: $$\int_0^1\frac{\ln^2\left(1-x^3\right)dx}{x}=\frac23\zeta(3),\quad \int_0^1\frac{\ln^2\left(1-x\right)dx}{x}=2\zeta(3).$$

  3. Mathematica computes and fullsimplifies the remaining nontrivial integral $\int_0^1\frac{\ln\left(1-x\right)\ln\left(1-x^3\right)dx}{x}$ to a one-line expression containing a sum of two trilogarithms $\operatorname{Li}_3(z_1)+\operatorname{Li}_3(z_2)$.

  4. It so happens that $z_1+z_2=1$, hence thanks to Landen's identity the above sum is equal to $$-\operatorname{Li}_3\left(\frac{z_1}{z_1-1}\right)+\zeta\left(3\right)+\text{elementary}.$$

  5. Finally, it so happens that $\frac{z_1}{z_1-1}=e^{2\pi i/3}$ and the corresponding trilogarithmic value is known in terms of $\zeta(3)$.

Altogether this should lead to your answer.

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  • $\begingroup$ Thanks! 4. I would hesitate to call the term $\zeta(3)$ elementary. I believe, it's still an open question. $\endgroup$
    – Vladimir Reshetnikov
    Commented Sep 9, 2015 at 1:55
  • $\begingroup$ @VladimirReshetnikov Yes, the word "elementary" is maybe not very appropriate but I have not found a better one. I meant that it does not contain polylogs. $\endgroup$
    – Start wearing purple
    Commented Sep 9, 2015 at 7:19
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By some simple manipulations the integral gets reduced to calculating

$$\frac{1}{2}\int_1^3 \frac{\log ^2(x)}{x-1} \ dx+\frac{1}{2}\int_1^3\frac{\log ^2(x)}{ (x-1) \sqrt{4 x-3}} \ dx,$$

and both residual integrals are pretty easy to finish at this point.

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  • $\begingroup$ i thought u became offsharing :D $\endgroup$
    – Abr001am
    Commented Apr 5, 2016 at 8:53
  • $\begingroup$ @Agawa001 or almost offsharing once in a while. :-) $\endgroup$
    – user 1591719
    Commented Apr 5, 2016 at 10:51

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