Optimal. Leaf size=33 \[ -\frac{1}{2} \log \left (x^2+x+1\right )+\log (x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0186396, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {705, 29, 634, 618, 204, 628} \[ -\frac{1}{2} \log \left (x^2+x+1\right )+\log (x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 705
Rule 29
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x \left (1+x+x^2\right )} \, dx &=\int \frac{1}{x} \, dx+\int \frac{-1-x}{1+x+x^2} \, dx\\ &=\log (x)-\frac{1}{2} \int \frac{1}{1+x+x^2} \, dx-\frac{1}{2} \int \frac{1+2 x}{1+x+x^2} \, dx\\ &=\log (x)-\frac{1}{2} \log \left (1+x+x^2\right )+\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=-\frac{\tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )}{\sqrt{3}}+\log (x)-\frac{1}{2} \log \left (1+x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0072856, size = 33, normalized size = 1. \[ -\frac{1}{2} \log \left (x^2+x+1\right )+\log (x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 29, normalized size = 0.9 \begin{align*} \ln \left ( x \right ) -{\frac{\ln \left ({x}^{2}+x+1 \right ) }{2}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46229, size = 38, normalized size = 1.15 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{2} \, \log \left (x^{2} + x + 1\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.26684, size = 103, normalized size = 3.12 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{2} \, \log \left (x^{2} + x + 1\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.125574, size = 37, normalized size = 1.12 \begin{align*} \log{\left (x \right )} - \frac{\log{\left (x^{2} + x + 1 \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12207, size = 39, normalized size = 1.18 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{2} \, \log \left (x^{2} + x + 1\right ) + \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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