Optimal. Leaf size=54 \[ -\frac{x^3}{3}+\frac{x^2}{2}+\frac{1}{6} \log \left (x^2-x+1\right )+\frac{2}{3} \log (x+1)+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0719371, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {1887, 1874, 31, 634, 618, 204, 628} \[ -\frac{x^3}{3}+\frac{x^2}{2}+\frac{1}{6} \log \left (x^2-x+1\right )+\frac{2}{3} \log (x+1)+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1887
Rule 1874
Rule 31
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{(1-x) x^4}{1+x^3} \, dx &=\int \left (x-x^2+\frac{(-1+x) x}{1+x^3}\right ) \, dx\\ &=\frac{x^2}{2}-\frac{x^3}{3}+\int \frac{(-1+x) x}{1+x^3} \, dx\\ &=\frac{x^2}{2}-\frac{x^3}{3}+\frac{1}{3} \int \frac{-2+x}{1-x+x^2} \, dx+\frac{2}{3} \int \frac{1}{1+x} \, dx\\ &=\frac{x^2}{2}-\frac{x^3}{3}+\frac{2}{3} \log (1+x)+\frac{1}{6} \int \frac{-1+2 x}{1-x+x^2} \, dx-\frac{1}{2} \int \frac{1}{1-x+x^2} \, dx\\ &=\frac{x^2}{2}-\frac{x^3}{3}+\frac{2}{3} \log (1+x)+\frac{1}{6} \log \left (1-x+x^2\right )+\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )\\ &=\frac{x^2}{2}-\frac{x^3}{3}-\frac{\tan ^{-1}\left (\frac{-1+2 x}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{2}{3} \log (1+x)+\frac{1}{6} \log \left (1-x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0139, size = 59, normalized size = 1.09 \[ \frac{1}{6} \left (-2 x^3+3 x^2-\log \left (x^2-x+1\right )+2 \log \left (x^3+1\right )+2 \log (x+1)-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 45, normalized size = 0.8 \begin{align*} -{\frac{{x}^{3}}{3}}+{\frac{{x}^{2}}{2}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{6}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{2\,\ln \left ( 1+x \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4146, size = 59, normalized size = 1.09 \begin{align*} -\frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} - \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{6} \, \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 = 1.55536, size = 140, normalized size = 2.59 \begin{align*} -\frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} - \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{6} \, \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 = 0.1289, size = 53, normalized size = 0.98 \begin{align*} - \frac{x^{3}}{3} + \frac{x^{2}}{2} + \frac{2 \log{\left (x + 1 \right )}}{3} + \frac{\log{\left (x^{2} - x + 1 \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05003, size = 61, normalized size = 1.13 \begin{align*} -\frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} - \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{6} \, \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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