Optimal. Leaf size=22 \[ \frac{1}{3} \log \left (x^2+x+1\right )-\frac{2}{3} \log (1-x) \]
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Rubi [A] time = 0.0121661, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1861, 31, 628} \[ \frac{1}{3} \log \left (x^2+x+1\right )-\frac{2}{3} \log (1-x) \]
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 &=-\left (\frac{1}{3} \int \frac{-1-2 x}{1+x+x^2} \, dx\right )+\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 = 0.003773, size = 22, normalized size = 1. \[ \frac{1}{3} \log \left (x^2+x+1\right )-\frac{2}{3} \log (1-x) \]
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 = 1.43893, 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 = 0.917206, size = 53, normalized size = 2.41 \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 = 0.09598, 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 = 1.19189, 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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