Optimal. Leaf size=30 \[ -\frac{x^2}{2}+\frac{1}{3} \log \left (x^2-x+1\right )+x-\frac{2}{3} \log (x+1) \]
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Rubi [A] time = 0.0400098, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1887, 1860, 31, 628} \[ -\frac{x^2}{2}+\frac{1}{3} \log \left (x^2-x+1\right )+x-\frac{2}{3} \log (x+1) \]
Antiderivative was successfully verified.
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Rule 1887
Rule 1860
Rule 31
Rule 628
Rubi steps
\begin{align*} \int \frac{(1-x) x^3}{1+x^3} \, dx &=\int \left (1-x-\frac{1-x}{1+x^3}\right ) \, dx\\ &=x-\frac{x^2}{2}-\int \frac{1-x}{1+x^3} \, dx\\ &=x-\frac{x^2}{2}-\frac{1}{3} \int \frac{1-2 x}{1-x+x^2} \, dx-\frac{2}{3} \int \frac{1}{1+x} \, dx\\ &=x-\frac{x^2}{2}-\frac{2}{3} \log (1+x)+\frac{1}{3} \log \left (1-x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0043203, size = 30, normalized size = 1. \[ -\frac{x^2}{2}+\frac{1}{3} \log \left (x^2-x+1\right )+x-\frac{2}{3} \log (x+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 25, normalized size = 0.8 \begin{align*} x-{\frac{{x}^{2}}{2}}-{\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 = 1.42281, size = 32, normalized size = 1.07 \begin{align*} -\frac{1}{2} \, x^{2} + x + \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 = 1.51494, size = 73, normalized size = 2.43 \begin{align*} -\frac{1}{2} \, x^{2} + x + \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 = 0.089936, size = 24, normalized size = 0.8 \begin{align*} - \frac{x^{2}}{2} + x - \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 = 1.07455, size = 34, normalized size = 1.13 \begin{align*} -\frac{1}{2} \, x^{2} + x + \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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