Optimal. Leaf size=18 \[ -\log \left (x^2+x+1\right )-2 \log (1-x) \]
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Rubi [A] time = 0.0188959, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, 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 1875
Rule 31
Rule 628
Rubi steps
\begin{align*} \int \frac{1+x+4 x^2}{1-x^3} \, dx &=\frac{1}{3} \int \frac{-3-6 x}{1+x+x^2} \, dx+2 \int \frac{1}{1-x} \, dx\\ &=-2 \log (1-x)-\log \left (1+x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0050965, size = 18, normalized size = 1. \[ -\log \left (x^2+x+1\right )-2 \log (1-x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 17, normalized size = 0.9 \begin{align*} -2\,\ln \left ( -1+x \right ) -\ln \left ({x}^{2}+x+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44261, size = 22, normalized size = 1.22 \begin{align*} -\log \left (x^{2} + x + 1\right ) - 2 \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4008, size = 46, normalized size = 2.56 \begin{align*} -\log \left (x^{2} + x + 1\right ) - 2 \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.112162, size = 15, normalized size = 0.83 \begin{align*} - 2 \log{\left (x - 1 \right )} - \log{\left (x^{2} + x + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07339, size = 23, normalized size = 1.28 \begin{align*} -\log \left (x^{2} + x + 1\right ) - 2 \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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