Una mejor forma cerrada? $\int_0^1 \frac{\log (x) \log \left(x^2-x+1\right)}{x^2-x+2} \, dx$

Question

Mathematica no retorno de un buen resultado de la integral a continuación, tal vez porque tal no existe, o existe pero que depende mucho de una cierta manera de abordar las cosas. ¿Qué te parece? $$\int_0^1 \frac{\log (x) \log \left(x^2-x+1\right)}{x^2-x+2} \, dx$$

$$=\frac{2 i \log ^3(2)}{3 \sqrt{7}}+\frac{i \log \left(\frac{(-1)^{5/6}}{\sqrt{3}-\sqrt{7}}\right) \log ^2(2)}{\sqrt{7}}+\frac{i \log \left(\frac{\left(i+\sqrt{3}\right) \left(1+i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right) \log ^2(2)}{\sqrt{7}}+\frac{i \log \left(\sqrt[6]{-1} \left(1+i \sqrt{7}\right)\right) \log ^2(2)}{\sqrt{7}}+\frac{2 i \log \left(3-i \sqrt{7}\right) \log ^2(2)}{\sqrt{7}}-\frac{i \log \left(\frac{\left(-i+\sqrt{3}\right) \left(1-i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right) \log ^2(2)}{\sqrt{7}}-\frac{i \log \left(i-\sqrt{7}\right) \log ^2(2)}{\sqrt{7}}-\frac{i \log \left(\left(1-i \sqrt{3}\right) \left(-i+\sqrt{7}\right)\right) \log ^2(2)}{\sqrt{7}}-\frac{2 i \log \left(3+i \sqrt{7}\right) \log ^2(2)}{\sqrt{7}}-\frac{\pi \log ^2(2)}{\sqrt{7}}+\frac{i \log ^2\left(\frac{(-1)^{5/6}}{\sqrt{3}-\sqrt{7}}\right) \log (2)}{\sqrt{7}}+\frac{i \log ^2\left(\frac{i-\sqrt{3}}{\sqrt{3}-\sqrt{7}}\right) \log (2)}{\sqrt{7}}+\frac{i \log ^2\left(\frac{\left(-i+\sqrt{3}\right) \left(1-i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right) \log (2)}{\sqrt{7}}+\frac{i \log ^2\left(\left(-1+\sqrt[3]{-1}\right) \left(-i+\sqrt{7}\right)\right) \log (2)}{\sqrt{7}}+\frac{2 i \log (4) \log \left(\frac{i+\sqrt{3}}{\sqrt{3}-\sqrt{7}}\right) \log (2)}{\sqrt{7}}+\frac{2 i \log (4) \log \left(\frac{\left(-i+\sqrt{3}\right) \left(1-i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right) \log (2)}{\sqrt{7}}+\frac{2 \pi \log \left(i-\sqrt{7}\right) \log (2)}{3 \sqrt{7}}+\frac{2 i \log \left(-\frac{1}{\sqrt{3}-\sqrt{7}}\right) \log \left(\left(-1+\sqrt[3]{-1}\right) \left(-i+\sqrt{7}\right)\right) \log (2)}{\sqrt{7}}+\frac{2 i \log \left(i-\sqrt{7}\right) \log \left(\sqrt[6]{-1} \left(1+i \sqrt{7}\right)\right) \log (2)}{\sqrt{7}}+\frac{2 \pi \log \left(\left(i+\sqrt{3}\right) \left(1+i \sqrt{7}\right)\right) \log (2)}{\sqrt{7}}+\frac{2 i \log \left(\frac{\left(i+\sqrt{3}\right) \left(1+i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right) \log \left(3+i \sqrt{7}\right) \log (2)}{\sqrt{7}}+\frac{4 \pi \log \left(7+i \sqrt{7}\right) \log (2)}{\sqrt{7}}+\frac{2 i \log \left(\frac{-i+\sqrt{3}}{\sqrt{3}-\sqrt{7}}\right) \log \left(3-i \sqrt{7}\right) \log (2)}{\sqrt{7}}+\frac{4 \pi \log \left(7-i \sqrt{7}\right) \log (2)}{\sqrt{7}}-\frac{i \log ^2\left(\frac{\left(i+\sqrt{3}\right) \left(1+i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right) \log (2)}{\sqrt{7}}-\frac{i \log ^2\left(-\frac{i+\sqrt{3}}{\sqrt{3}-\sqrt{7}}\right) \log (2)}{\sqrt{7}}-\frac{4 \pi \log (7) \log (2)}{\sqrt{7}}-\frac{8 \pi \log (8) \log (2)}{\sqrt{7}}-\frac{i \log (16) \log \left(i+\sqrt{3}\right) \log (2)}{\sqrt{7}}-\frac{2 i \log (4) \log \left(\frac{-i+\sqrt{3}}{\sqrt{3}-\sqrt{7}}\right) \log (2)}{\sqrt{7}}-\frac{2 i \log (4) \log \left(\frac{\left(i+\sqrt{3}\right) \left(1+i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right) \log (2)}{\sqrt{7}}-\frac{2 i \log \left(i+\sqrt{3}\right) \log \left(-i+\sqrt{7}\right) \log (2)}{\sqrt{7}}-\frac{2 i \log \left(\frac{-i+\sqrt{3}}{\sqrt{3}-\sqrt{7}}\right) \log \left(-i+\sqrt{7}\right) \log (2)}{\sqrt{7}}-\frac{2 i \log \left(-i+\sqrt{3}\right) \log \left(3+i \sqrt{7}\right) \log (2)}{\sqrt{7}}-\frac{2 i \log \left(\frac{i+\sqrt{3}}{\sqrt{3}-\sqrt{7}}\right) \log \left(3+i \sqrt{7}\right) \log (2)}{\sqrt{7}}-\frac{2 i \log \left(\frac{\left(-i+\sqrt{3}\right) \left(1-i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right) \log \left(3-i \sqrt{7}\right) \log (2)}{\sqrt{7}}-\frac{\pi \log (64) \log (2)}{3 \sqrt{7}}-\frac{4 \pi \log \left(-\frac{1}{\sqrt{3}-\sqrt{7}}\right) \log (2)}{3 \sqrt{7}}-\frac{2 \pi \log \left(\left(1-i \sqrt{3}\right) \left(-i+\sqrt{7}\right)\right) \log (2)}{3 \sqrt{7}}+\frac{503 \pi ^3}{648 \sqrt{7}}+\frac{i \log ^3\left(\frac{(-1)^{5/6}}{\sqrt{3}-\sqrt{7}}\right)}{3 \sqrt{7}}+\frac{i \log ^3\left(\frac{i-\sqrt{3}}{\sqrt{3}-\sqrt{7}}\right)}{3 \sqrt{7}}+\frac{i \log ^3\left(-\frac{1}{\sqrt{3}-\sqrt{7}}\right)}{3 \sqrt{7}}+\frac{i \log ^3\left(\left(-1+\sqrt[3]{-1}\right) \left(-i+\sqrt{7}\right)\right)}{3 \sqrt{7}}+\frac{i \log (4) \log ^2\left(-i-\sqrt{3}\right)}{\sqrt{7}}+\frac{i \log (64) \log ^2\left(-i+\sqrt{3}\right)}{3 \sqrt{7}}+\frac{\pi \log ^2\left(\frac{\left(i+\sqrt{3}\right) \left(1+i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right)}{3 \sqrt{7}}+\frac{\pi \log ^2\left(\frac{\left(-i+\sqrt{3}\right) \left(1-i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right)}{3 \sqrt{7}}+\frac{i \log (8) \log ^2\left(-\frac{1}{\sqrt{3}-\sqrt{7}}\right)}{3 \sqrt{7}}+\frac{i \log \left(\frac{\left(i+\sqrt{3}\right) \left(1+i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right) \log ^2\left(-\frac{i+\sqrt{3}}{\sqrt{3}-\sqrt{7}}\right)}{\sqrt{7}}+\frac{i \log \left(-\frac{1}{\sqrt{3}-\sqrt{7}}\right) \log ^2\left(\left(-1+\sqrt[3]{-1}\right) \left(-i+\sqrt{7}\right)\right)}{\sqrt{7}}+\frac{\pi \log ^2\left(\sqrt[6]{-1} \left(1+i \sqrt{7}\right)\right)}{3 \sqrt{7}}+\frac{2 i \pi ^2 \log \left(-i+\sqrt{3}\right)}{3 \sqrt{7}}+\frac{i \log (4) \log (64) \log \left(-i+\sqrt{3}\right)}{3 \sqrt{7}}+\frac{i \log ^2\left(\frac{\left(i+\sqrt{3}\right) \left(1+i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right) \log \left(\frac{\left(-i+\sqrt{3}\right) \left(1-i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right)}{\sqrt{7}}+\frac{i \pi ^2 \log \left(-\frac{1}{\sqrt{3}-\sqrt{7}}\right)}{2 \sqrt{7}}+\frac{5 \pi \log \left(-\frac{1}{\sqrt{3}-\sqrt{7}}\right) \log \left(-\frac{2}{\sqrt{3}-\sqrt{7}}\right)}{3 \sqrt{7}}+\frac{13 i \pi ^2 \log \left(i-\sqrt{7}\right)}{18 \sqrt{7}}+\frac{2 \pi \log (32) \log \left(i-\sqrt{7}\right)}{3 \sqrt{7}}+\frac{2 i \log \left(\frac{-i+\sqrt{3}}{\sqrt{3}-\sqrt{7}}\right) \log \left(\frac{\left(-i+\sqrt{3}\right) \left(1-i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right) \log \left(-i+\sqrt{7}\right)}{\sqrt{7}}+\frac{i \log ^2\left(-\frac{1}{\sqrt{3}-\sqrt{7}}\right) \log \left(\left(-1+\sqrt[3]{-1}\right) \left(-i+\sqrt{7}\right)\right)}{\sqrt{7}}+\frac{i \pi ^2 \log \left(\left(i+\sqrt{3}\right) \left(1+i \sqrt{7}\right)\right)}{\sqrt{7}}+\frac{2 i \log \left(\frac{i+\sqrt{3}}{\sqrt{3}-\sqrt{7}}\right) \log \left(\frac{\left(i+\sqrt{3}\right) \left(1+i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right) \log \left(3+i \sqrt{7}\right)}{\sqrt{7}}+\frac{2 \pi \log (3) \log \left(7+i \sqrt{7}\right)}{\sqrt{7}}+\frac{4 \pi \log \left(-\frac{1}{\sqrt{3}-\sqrt{7}}\right) \log \left(7+i \sqrt{7}\right)}{\sqrt{7}}+\frac{i \log ^2\left(i+\sqrt{3}\right) \log \left(3-i \sqrt{7}\right)}{\sqrt{7}}+\frac{i \log ^2\left(\frac{-i+\sqrt{3}}{\sqrt{3}-\sqrt{7}}\right) \log \left(3-i \sqrt{7}\right)}{\sqrt{7}}+\frac{i \log ^2\left(\frac{\left(-i+\sqrt{3}\right) \left(1-i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right) \log \left(3-i \sqrt{7}\right)}{\sqrt{7}}+\frac{i \log (64) \log \left(i+\sqrt{3}\right) \log \left(3-i \sqrt{7}\right)}{3 \sqrt{7}}+\frac{4 \pi \log \left(-\frac{1}{\sqrt{3}-\sqrt{7}}\right) \log \left(7-i \sqrt{7}\right)}{\sqrt{7}}+\frac{2 \pi \log (3) \log \left(\frac{1}{448} \left(7-i \sqrt{7}\right)\right)}{\sqrt{7}}-\frac{i \log (4) \log ^2\left(i-\sqrt{3}\right)}{\sqrt{7}}-\frac{i \log (4) \log ^2\left(i+\sqrt{3}\right)}{\sqrt{7}}-\frac{i \log \left(i-\sqrt{7}\right) \log ^2\left(-\frac{2}{\sqrt{3}-\sqrt{7}}\right)}{\sqrt{7}}-\frac{i \log \left(\frac{\left(i+\sqrt{3}\right) \left(1+i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right) \log ^2\left(\left(-1+\sqrt[3]{-1}\right) \left(-i+\sqrt{7}\right)\right)}{\sqrt{7}}-\frac{i \log ^2\left(\frac{\left(-i+\sqrt{3}\right) \left(1-i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right) \log \left(\frac{\left(i+\sqrt{3}\right) \left(1+i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right)}{\sqrt{7}}-\frac{i \log ^2\left(-\frac{1}{\sqrt{3}-\sqrt{7}}\right) \log \left(\frac{\left(i+\sqrt{3}\right) \left(1+i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right)}{\sqrt{7}}-\frac{i \log ^2\left(\frac{i-\sqrt{3}}{\sqrt{3}-\sqrt{7}}\right) \log \left(\frac{\left(-i+\sqrt{3}\right) \left(1-i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right)}{\sqrt{7}}-\frac{i \log ^2\left(-\frac{2}{\sqrt{3}-\sqrt{7}}\right) \log \left(-\frac{1}{\sqrt{3}-\sqrt{7}}\right)}{\sqrt{7}}-\frac{4 \pi \log (7) \log \left(-\frac{1}{\sqrt{3}-\sqrt{7}}\right)}{\sqrt{7}}-\frac{8 \pi \log (8) \log \left(-\frac{1}{\sqrt{3}-\sqrt{7}}\right)}{\sqrt{7}}-\frac{i \log ^2\left(i+\sqrt{3}\right) \log \left(-i+\sqrt{7}\right)}{\sqrt{7}}-\frac{i \log ^2\left(\frac{-i+\sqrt{3}}{\sqrt{3}-\sqrt{7}}\right) \log \left(-i+\sqrt{7}\right)}{\sqrt{7}}-\frac{i \log ^2\left(\sqrt[6]{-1} \left(1+i \sqrt{7}\right)\right) \log \left(-(-1)^{2/3} \left(-i+\sqrt{7}\right)\right)}{\sqrt{7}}-\frac{2 i \log \left(-\frac{1}{\sqrt{3}-\sqrt{7}}\right) \log \left(\frac{\left(i+\sqrt{3}\right) \left(1+i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right) \log \left(\left(-1+\sqrt[3]{-1}\right) \left(-i+\sqrt{7}\right)\right)}{\sqrt{7}}-\frac{2 i \log (4) \log \left(-i-\sqrt{3}\right) \log \left(\left(1-i \sqrt{3}\right) \left(-i+\sqrt{7}\right)\right)}{\sqrt{7}}-\frac{i \log ^2\left(-i+\sqrt{3}\right) \log \left(3+i \sqrt{7}\right)}{\sqrt{7}}-\frac{i \log ^2\left(\frac{i+\sqrt{3}}{\sqrt{3}-\sqrt{7}}\right) \log \left(3+i \sqrt{7}\right)}{\sqrt{7}}-\frac{i \log ^2\left(\frac{\left(i+\sqrt{3}\right) \left(1+i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right) \log \left(3+i \sqrt{7}\right)}{\sqrt{7}}-\frac{2 i \log \left(\frac{-i+\sqrt{3}}{\sqrt{3}-\sqrt{7}}\right) \log \left(\frac{\left(-i+\sqrt{3}\right) \left(1-i \sqrt{7}\right)}{\sqrt{3}-\sqrt{7}}\right) \log \left(3-i \sqrt{7}\right)}{\sqrt{7}}-\frac{i \log ^3\left(-\frac{2}{\sqrt{3}-\sqrt{7}}\right)}{3 \sqrt{7}}-\frac{i \log ^3\left(-\frac{i+\sqrt{3}}{\sqrt{3}-\sqrt{7}}\right)}{3 \sqrt{7}}-\frac{i \pi ^2 \log \left(i+\sqrt{3}\right)}{3 \sqrt{7}}-\frac{5 \pi \log \left(i-\sqrt{7}\right) \log \left(\left(-1-i \sqrt{3}\right) \left(-i+\sqrt{7}\right)\right)}{3 \sqrt{7}}-\frac{i \pi ^2 \log \left(\left(1-i \sqrt{3}\right) \left(-i+\sqrt{7}\right)\right)}{3 \sqrt{7}}-\frac{\pi \log \left(i-\sqrt{7}\right) \log \left(\left(i+\sqrt{3}\right) \left(1+i \sqrt{7}\right)\right)}{3 \sqrt{7}}-\frac{5 \pi \log ^2\left(-\frac{2}{\sqrt{3}-\sqrt{7}}\right)}{6 \sqrt{7}}-\frac{4 i \pi ^2 \log \left(\left(-i+\sqrt{3}\right) \left(1-i \sqrt{7}\right)\right)}{9 \sqrt{7}}-\frac{i \pi ^2 \log (45671926166590716193865151022383844364247891968)}{36 \sqrt{7}} ...\text{and so on (that means many other terms)}$$

Answer 1

Después de un montón de simplificaciones tengo este sencillo formulario: $$\int_0^1\frac{\ln(x)\ln\left(x^2-x+1\right)}{x^2-x+2}\, dx\\=\frac1{648\sqrt7}\Big[71\!\;\pi^3-522\!\;\alpha\!\;\pi^2-54\!\;\big(4\!\;\alpha^3-3\!\;\alpha\ln^2\xi-24\!\;\beta\big)\\+27\!\;\pi\!\;\big(28\!\;\alpha^2-2\!\;(16\!\;\gamma+\ln^22)+(4\ln2-9\ln\xi)\cdot\ln\xi\big)\Big]$$ donde

$$\alpha=\arctan\big(\sqrt7\big)$$

$$\beta=\Im\left[2\operatorname{Li}_3\left(\frac{1+i\sqrt7}4\right)-\operatorname{Li}_3\left(\frac{1-2\!\;i\sqrt3+i\sqrt7}2\right)\\+2 \operatorname{Li}_3\left(\frac{i+\sqrt3}\eta\right)-2 \operatorname{Li}_3\left(\frac{i-\sqrt3}\eta\right)-\operatorname{Li}_3\left(\frac{\xi-3\!\;i\sqrt3+i\sqrt7}8\right)\right]$$

$$\gamma=\Re\left[\operatorname{Li}_2\left(\frac{i+\sqrt3}\eta\right)\right]$$

$$\xi=5-\sqrt{21}$$

$$\eta=\sqrt3+\sqrt7$$

Answer 2

Vamos \begin{align}\tag{1} a &= \frac{1+i\sqrt{3}}{2} \hspace{10mm} \alpha = \frac{1 + i \sqrt{7}}{2} \\ b &= \frac{1-i\sqrt{3}}{2} \hspace{10mm} \beta = \frac{1 - i \sqrt{7}}{2} \end{align} para tomar la integral \begin{align}\tag{2} I = \int_0^1 \frac{\log (x) \log \left(x^2-x+1\right)}{x^2-x+2} \, dx \end{align} en el formulario \begin{align}\tag{3} I = \int_0^1 \frac{\log (x) \, [ \ln(x-a) + \ln(x-b)]}{(x-\alpha)(x-\beta)} \, dx \end{align} y descomponer la integral en 4 integrales y ver lo que pasa a partir de ahí.

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Uno puede también considerar la forma \begin{align}\tag{4} J(x,y; a) = \int_{0}^{1} \frac{t^{x} \, (t-a)^{y}}{(t-\alpha)(t-\beta)} \, dt \end{align} y tomar la primera derivada con respecto a $x$$y$. De hecho \begin{align}\tag{5} I = \partial_{x,y} \left[J(x,y; a) + J(x,y;b) \right]_{x,y=0}. \end{align} Una forma fundamental puede ser \begin{align}\tag{6} J_{1}(x,y;a;\alpha) = \int_{0}^{1} \frac{t^{x} \, (t-a)^{y}}{t- \alpha} \, dt \end{align} para que \begin{align} J(x,y;a) = \frac{1}{\alpha - \beta} \left( J_{1}(x,y;a;\alpha) - J_{1}(x,y;a;\beta) \right) \end{align}