Optimal. Leaf size=22 \[ \frac{2}{3} \log (1-x)-\frac{1}{3} \log \left (x^2+x+1\right ) \]
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Rubi [A] time = 0.0106187, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {1861, 31, 628} \[ \frac{2}{3} \log (1-x)-\frac{1}{3} \log \left (x^2+x+1\right ) \]
Antiderivative was successfully verified.
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Rule 1861
Rule 31
Rule 628
Rubi steps
\begin{align*} \int \frac{1+x}{-1+x^3} \, dx &=\frac{1}{3} \int \frac{-1-2 x}{1+x+x^2} \, dx-\frac{2}{3} \int \frac{1}{1-x} \, dx\\ &=\frac{2}{3} \log (1-x)-\frac{1}{3} \log \left (1+x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0026044, size = 22, normalized size = 1. \[ \frac{2}{3} \log (1-x)-\frac{1}{3} \log \left (x^2+x+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 17, normalized size = 0.8 \begin{align*}{\frac{2\,\ln \left ( -1+x \right ) }{3}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42956, size = 22, normalized size = 1. \begin{align*} -\frac{1}{3} \, \log \left (x^{2} + x + 1\right ) + \frac{2}{3} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.973768, size = 54, normalized size = 2.45 \begin{align*} -\frac{1}{3} \, \log \left (x^{2} + x + 1\right ) + \frac{2}{3} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.086469, size = 17, normalized size = 0.77 \begin{align*} \frac{2 \log{\left (x - 1 \right )}}{3} - \frac{\log{\left (x^{2} + x + 1 \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11627, size = 23, normalized size = 1.05 \begin{align*} -\frac{1}{3} \, \log \left (x^{2} + x + 1\right ) + \frac{2}{3} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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