Optimal. Leaf size=16 \[ \log \left (x^2+x+1\right )+2 \log (1-x) \]
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Rubi [A] time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1875, 31, 628} \[ \log \left (x^2+x+1\right )+2 \log (1-x) \]
Antiderivative was successfully verified.
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Rule 31
Rule 628
Rule 1875
Rubi steps
\begin {align*} \int \frac {1+x+4 x^2}{-1+x^3} \, dx &=-\left (\frac {1}{3} \int \frac {-3-6 x}{1+x+x^2} \, dx\right )-2 \int \frac {1}{1-x} \, dx\\ &=2 \log (1-x)+\log \left (1+x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 1.00 \[ \log \left (x^2+x+1\right )+2 \log (1-x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 14, normalized size = 0.88 \[ \log \left (x^{2} + x + 1\right ) + 2 \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 15, normalized size = 0.94 \[ \log \left (x^{2} + x + 1\right ) + 2 \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 15, normalized size = 0.94 \[ 2 \ln \left (x -1\right )+\ln \left (x^{2}+x +1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 14, normalized size = 0.88 \[ \log \left (x^{2} + x + 1\right ) + 2 \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 14, normalized size = 0.88 \[ \ln \left (x^2+x+1\right )+2\,\ln \left (x-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 14, normalized size = 0.88 \[ 2 \log {\left (x - 1 \right )} + \log {\left (x^{2} + x + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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