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 = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 31
Rule 628
Rule 1861
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*}
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Mathematica [A] time = 0.00, size = 22, normalized size = 1.00 \[ \frac {2}{3} \log (1-x)-\frac {1}{3} \log \left (x^2+x+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 16, normalized size = 0.73 \[ -\frac {1}{3} \, \log \left (x^{2} + x + 1\right ) + \frac {2}{3} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 17, normalized size = 0.77 \[ -\frac {1}{3} \, \log \left (x^{2} + x + 1\right ) + \frac {2}{3} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 17, normalized size = 0.77 \[ \frac {2 \ln \left (x -1\right )}{3}-\frac {\ln \left (x^{2}+x +1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 16, normalized size = 0.73 \[ -\frac {1}{3} \, \log \left (x^{2} + x + 1\right ) + \frac {2}{3} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 16, normalized size = 0.73 \[ \frac {2\,\ln \left (x-1\right )}{3}-\frac {\ln \left (x^2+x+1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 17, normalized size = 0.77 \[ \frac {2 \log {\left (x - 1 \right )}}{3} - \frac {\log {\left (x^{2} + x + 1 \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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