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 = 0.04, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 31
Rule 628
Rule 1860
Rule 1887
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*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 1.00 \[ -\frac {x^2}{2}+\frac {1}{3} \log \left (x^2-x+1\right )+x-\frac {2}{3} \log (x+1) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 24, normalized size = 0.80 \[ -\frac {1}{2} \, x^{2} + x + \frac {1}{3} \, \log \left (x^{2} - x + 1\right ) - \frac {2}{3} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 25, normalized size = 0.83 \[ -\frac {1}{2} \, x^{2} + x + \frac {1}{3} \, \log \left (x^{2} - x + 1\right ) - \frac {2}{3} \, \log \left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 25, normalized size = 0.83 \[ -\frac {x^{2}}{2}+x -\frac {2 \ln \left (x +1\right )}{3}+\frac {\ln \left (x^{2}-x +1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.01, size = 24, normalized size = 0.80 \[ -\frac {1}{2} \, x^{2} + x + \frac {1}{3} \, \log \left (x^{2} - x + 1\right ) - \frac {2}{3} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 24, normalized size = 0.80 \[ x-\frac {2\,\ln \left (x+1\right )}{3}+\frac {\ln \left (x^2-x+1\right )}{3}-\frac {x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 24, normalized size = 0.80 \[ - \frac {x^{2}}{2} + x - \frac {2 \log {\left (x + 1 \right )}}{3} + \frac {\log {\left (x^{2} - x + 1 \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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