Optimal. Leaf size=371 \[ -\left (1+i \sqrt{3}\right ) \text{PolyLog}\left (2,-\frac{2 i x-\sqrt{3}+i}{2 \sqrt{3}}\right )-\left (1-i \sqrt{3}\right ) \text{PolyLog}\left (2,\frac{2 i x+\sqrt{3}+i}{2 \sqrt{3}}\right )+x \log ^2\left (x^2+x+1\right )+\left (1-i \sqrt{3}\right ) \log \left (x^2+x+1\right ) \log \left (2 x-i \sqrt{3}+1\right )-4 x \log \left (x^2+x+1\right )+\left (1+i \sqrt{3}\right ) \log \left (2 x+i \sqrt{3}+1\right ) \log \left (x^2+x+1\right )-2 \log \left (x^2+x+1\right )+8 x-\frac{1}{2} \left (1-i \sqrt{3}\right ) \log ^2\left (2 x-i \sqrt{3}+1\right )-\frac{1}{2} \left (1+i \sqrt{3}\right ) \log ^2\left (2 x+i \sqrt{3}+1\right )-\left (1-i \sqrt{3}\right ) \log \left (-\frac{i \left (2 x+i \sqrt{3}+1\right )}{2 \sqrt{3}}\right ) \log \left (2 x-i \sqrt{3}+1\right )-\left (1+i \sqrt{3}\right ) \log \left (\frac{i \left (2 x-i \sqrt{3}+1\right )}{2 \sqrt{3}}\right ) \log \left (2 x+i \sqrt{3}+1\right )-4 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]
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Rubi [A] time = 0.541176, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 14, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.556, Rules used = {2523, 2528, 773, 634, 618, 204, 628, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\left (1+i \sqrt{3}\right ) \text{PolyLog}\left (2,-\frac{2 i x-\sqrt{3}+i}{2 \sqrt{3}}\right )-\left (1-i \sqrt{3}\right ) \text{PolyLog}\left (2,\frac{2 i x+\sqrt{3}+i}{2 \sqrt{3}}\right )+x \log ^2\left (x^2+x+1\right )+\left (1-i \sqrt{3}\right ) \log \left (x^2+x+1\right ) \log \left (2 x-i \sqrt{3}+1\right )-4 x \log \left (x^2+x+1\right )+\left (1+i \sqrt{3}\right ) \log \left (2 x+i \sqrt{3}+1\right ) \log \left (x^2+x+1\right )-2 \log \left (x^2+x+1\right )+8 x-\frac{1}{2} \left (1-i \sqrt{3}\right ) \log ^2\left (2 x-i \sqrt{3}+1\right )-\frac{1}{2} \left (1+i \sqrt{3}\right ) \log ^2\left (2 x+i \sqrt{3}+1\right )-\left (1-i \sqrt{3}\right ) \log \left (-\frac{i \left (2 x+i \sqrt{3}+1\right )}{2 \sqrt{3}}\right ) \log \left (2 x-i \sqrt{3}+1\right )-\left (1+i \sqrt{3}\right ) \log \left (\frac{i \left (2 x-i \sqrt{3}+1\right )}{2 \sqrt{3}}\right ) \log \left (2 x+i \sqrt{3}+1\right )-4 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
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Rule 2523
Rule 2528
Rule 773
Rule 634
Rule 618
Rule 204
Rule 628
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \log ^2\left (1+x+x^2\right ) \, dx &=x \log ^2\left (1+x+x^2\right )-2 \int \frac{x (1+2 x) \log \left (1+x+x^2\right )}{1+x+x^2} \, dx\\ &=x \log ^2\left (1+x+x^2\right )-2 \int \left (2 \log \left (1+x+x^2\right )-\frac{(2+x) \log \left (1+x+x^2\right )}{1+x+x^2}\right ) \, dx\\ &=x \log ^2\left (1+x+x^2\right )+2 \int \frac{(2+x) \log \left (1+x+x^2\right )}{1+x+x^2} \, dx-4 \int \log \left (1+x+x^2\right ) \, dx\\ &=-4 x \log \left (1+x+x^2\right )+x \log ^2\left (1+x+x^2\right )+2 \int \left (\frac{\left (1-i \sqrt{3}\right ) \log \left (1+x+x^2\right )}{1-i \sqrt{3}+2 x}+\frac{\left (1+i \sqrt{3}\right ) \log \left (1+x+x^2\right )}{1+i \sqrt{3}+2 x}\right ) \, dx+4 \int \frac{x (1+2 x)}{1+x+x^2} \, dx\\ &=8 x-4 x \log \left (1+x+x^2\right )+x \log ^2\left (1+x+x^2\right )+4 \int \frac{-2-x}{1+x+x^2} \, dx+\left (2 \left (1-i \sqrt{3}\right )\right ) \int \frac{\log \left (1+x+x^2\right )}{1-i \sqrt{3}+2 x} \, dx+\left (2 \left (1+i \sqrt{3}\right )\right ) \int \frac{\log \left (1+x+x^2\right )}{1+i \sqrt{3}+2 x} \, dx\\ &=8 x-4 x \log \left (1+x+x^2\right )+\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+\left (1+i \sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+x \log ^2\left (1+x+x^2\right )-2 \int \frac{1+2 x}{1+x+x^2} \, dx-6 \int \frac{1}{1+x+x^2} \, dx+\left (-1-i \sqrt{3}\right ) \int \frac{(1+2 x) \log \left (1+i \sqrt{3}+2 x\right )}{1+x+x^2} \, dx+\left (-1+i \sqrt{3}\right ) \int \frac{(1+2 x) \log \left (1-i \sqrt{3}+2 x\right )}{1+x+x^2} \, dx\\ &=8 x-2 \log \left (1+x+x^2\right )-4 x \log \left (1+x+x^2\right )+\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+\left (1+i \sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+x \log ^2\left (1+x+x^2\right )+12 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x\right )+\left (-1-i \sqrt{3}\right ) \int \left (\frac{2 \log \left (1+i \sqrt{3}+2 x\right )}{1-i \sqrt{3}+2 x}+\frac{2 \log \left (1+i \sqrt{3}+2 x\right )}{1+i \sqrt{3}+2 x}\right ) \, dx+\left (-1+i \sqrt{3}\right ) \int \left (\frac{2 \log \left (1-i \sqrt{3}+2 x\right )}{1-i \sqrt{3}+2 x}+\frac{2 \log \left (1-i \sqrt{3}+2 x\right )}{1+i \sqrt{3}+2 x}\right ) \, dx\\ &=8 x-4 \sqrt{3} \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )-2 \log \left (1+x+x^2\right )-4 x \log \left (1+x+x^2\right )+\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+\left (1+i \sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+x \log ^2\left (1+x+x^2\right )-\left (2 \left (1-i \sqrt{3}\right )\right ) \int \frac{\log \left (1-i \sqrt{3}+2 x\right )}{1-i \sqrt{3}+2 x} \, dx-\left (2 \left (1-i \sqrt{3}\right )\right ) \int \frac{\log \left (1-i \sqrt{3}+2 x\right )}{1+i \sqrt{3}+2 x} \, dx-\left (2 \left (1+i \sqrt{3}\right )\right ) \int \frac{\log \left (1+i \sqrt{3}+2 x\right )}{1-i \sqrt{3}+2 x} \, dx-\left (2 \left (1+i \sqrt{3}\right )\right ) \int \frac{\log \left (1+i \sqrt{3}+2 x\right )}{1+i \sqrt{3}+2 x} \, dx\\ &=8 x-4 \sqrt{3} \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )-\left (1+i \sqrt{3}\right ) \log \left (\frac{i \left (1-i \sqrt{3}+2 x\right )}{2 \sqrt{3}}\right ) \log \left (1+i \sqrt{3}+2 x\right )-\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x\right ) \log \left (-\frac{i \left (1+i \sqrt{3}+2 x\right )}{2 \sqrt{3}}\right )-2 \log \left (1+x+x^2\right )-4 x \log \left (1+x+x^2\right )+\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+\left (1+i \sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+x \log ^2\left (1+x+x^2\right )-\left (1-i \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-i \sqrt{3}+2 x\right )+\left (2 \left (1-i \sqrt{3}\right )\right ) \int \frac{\log \left (\frac{2 \left (1+i \sqrt{3}+2 x\right )}{-2 \left (1-i \sqrt{3}\right )+2 \left (1+i \sqrt{3}\right )}\right )}{1-i \sqrt{3}+2 x} \, dx-\left (1+i \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+i \sqrt{3}+2 x\right )+\left (2 \left (1+i \sqrt{3}\right )\right ) \int \frac{\log \left (\frac{2 \left (1-i \sqrt{3}+2 x\right )}{2 \left (1-i \sqrt{3}\right )-2 \left (1+i \sqrt{3}\right )}\right )}{1+i \sqrt{3}+2 x} \, dx\\ &=8 x-4 \sqrt{3} \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )-\frac{1}{2} \left (1-i \sqrt{3}\right ) \log ^2\left (1-i \sqrt{3}+2 x\right )-\left (1+i \sqrt{3}\right ) \log \left (\frac{i \left (1-i \sqrt{3}+2 x\right )}{2 \sqrt{3}}\right ) \log \left (1+i \sqrt{3}+2 x\right )-\frac{1}{2} \left (1+i \sqrt{3}\right ) \log ^2\left (1+i \sqrt{3}+2 x\right )-\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x\right ) \log \left (-\frac{i \left (1+i \sqrt{3}+2 x\right )}{2 \sqrt{3}}\right )-2 \log \left (1+x+x^2\right )-4 x \log \left (1+x+x^2\right )+\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+\left (1+i \sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+x \log ^2\left (1+x+x^2\right )+\left (1-i \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{-2 \left (1-i \sqrt{3}\right )+2 \left (1+i \sqrt{3}\right )}\right )}{x} \, dx,x,1-i \sqrt{3}+2 x\right )+\left (1+i \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{2 \left (1-i \sqrt{3}\right )-2 \left (1+i \sqrt{3}\right )}\right )}{x} \, dx,x,1+i \sqrt{3}+2 x\right )\\ &=8 x-4 \sqrt{3} \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )-\frac{1}{2} \left (1-i \sqrt{3}\right ) \log ^2\left (1-i \sqrt{3}+2 x\right )-\left (1+i \sqrt{3}\right ) \log \left (\frac{i \left (1-i \sqrt{3}+2 x\right )}{2 \sqrt{3}}\right ) \log \left (1+i \sqrt{3}+2 x\right )-\frac{1}{2} \left (1+i \sqrt{3}\right ) \log ^2\left (1+i \sqrt{3}+2 x\right )-\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x\right ) \log \left (-\frac{i \left (1+i \sqrt{3}+2 x\right )}{2 \sqrt{3}}\right )-2 \log \left (1+x+x^2\right )-4 x \log \left (1+x+x^2\right )+\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+\left (1+i \sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+x \log ^2\left (1+x+x^2\right )-\left (1+i \sqrt{3}\right ) \text{Li}_2\left (-\frac{i-\sqrt{3}+2 i x}{2 \sqrt{3}}\right )-\left (1-i \sqrt{3}\right ) \text{Li}_2\left (\frac{i+\sqrt{3}+2 i x}{2 \sqrt{3}}\right )\\ \end{align*}
Mathematica [A] time = 0.146165, size = 323, normalized size = 0.87 \[ -\frac{1}{2} i \left (\sqrt{3}-i\right ) \left (2 \text{PolyLog}\left (2,\frac{-2 i x+\sqrt{3}-i}{2 \sqrt{3}}\right )+\log \left (2 x+i \sqrt{3}+1\right ) \left (2 \log \left (\frac{2 i x+\sqrt{3}+i}{2 \sqrt{3}}\right )+\log \left (2 x+i \sqrt{3}+1\right )\right )\right )+\frac{1}{2} i \left (\sqrt{3}+i\right ) \left (2 \text{PolyLog}\left (2,\frac{2 i x+\sqrt{3}+i}{2 \sqrt{3}}\right )+\log \left (2 x-i \sqrt{3}+1\right ) \left (2 \log \left (\frac{-2 i x+\sqrt{3}-i}{2 \sqrt{3}}\right )+\log \left (2 x-i \sqrt{3}+1\right )\right )\right )+x \log ^2\left (x^2+x+1\right )-4 x \log \left (x^2+x+1\right )+\left (1-i \sqrt{3}\right ) \log \left (2 x-i \sqrt{3}+1\right ) \log \left (x^2+x+1\right )+\left (1+i \sqrt{3}\right ) \log \left (2 x+i \sqrt{3}+1\right ) \log \left (x^2+x+1\right )-2 \log \left (x^2+x+1\right )+8 x-4 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.027, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ({x}^{2}+x+1 \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} x \log \left (x^{2} + x + 1\right )^{2} - \int \frac{2 \,{\left (2 \, x^{2} + x\right )} \log \left (x^{2} + x + 1\right )}{x^{2} + x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\log \left (x^{2} + x + 1\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RecursionError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \log \left (x^{2} + x + 1\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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