Optimal. Leaf size=54 \[ -\frac{x^3}{3}+\frac{x^2}{2}+\frac{1}{6} \log \left (x^2-x+1\right )+\frac{2}{3} \log (x+1)+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]
[Out]
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Rubi [A] time = 0.136182, 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 \[ -\frac{x^3}{3}+\frac{x^2}{2}+\frac{1}{6} \log \left (x^2-x+1\right )+\frac{2}{3} \log (x+1)+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[((1 - x)*x^4)/(1 + x^3),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{x^{3}}{3} + \frac{2 \log{\left (x + 1 \right )}}{3} + \frac{\log{\left (x^{2} - x + 1 \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{3} + \int x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-x)*x**4/(x**3+1),x)
[Out]
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Mathematica [A] time = 0.0256297, size = 59, normalized size = 1.09 \[ \frac{1}{6} \left (-2 x^3+2 \log \left (x^3+1\right )+3 x^2-\log \left (x^2-x+1\right )+2 \log (x+1)-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((1 - x)*x^4)/(1 + x^3),x]
[Out]
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Maple [A] time = 0.009, size = 45, normalized size = 0.8 \[ -{\frac{{x}^{3}}{3}}+{\frac{{x}^{2}}{2}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{6}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{2\,\ln \left ( 1+x \right ) }{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-x)*x^4/(x^3+1),x)
[Out]
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Maxima [A] time = 1.52153, size = 59, normalized size = 1.09 \[ -\frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} - \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{6} \, \log \left (x^{2} - x + 1\right ) + \frac{2}{3} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x - 1)*x^4/(x^3 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.215939, size = 77, normalized size = 1.43 \[ -\frac{1}{18} \, \sqrt{3}{\left (\sqrt{3}{\left (2 \, x^{3} - 3 \, x^{2}\right )} - \sqrt{3} \log \left (x^{2} - x + 1\right ) - 4 \, \sqrt{3} \log \left (x + 1\right ) + 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x - 1)*x^4/(x^3 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.158589, size = 53, normalized size = 0.98 \[ - \frac{x^{3}}{3} + \frac{x^{2}}{2} + \frac{2 \log{\left (x + 1 \right )}}{3} + \frac{\log{\left (x^{2} - x + 1 \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-x)*x**4/(x**3+1),x)
[Out]
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GIAC/XCAS [A] time = 0.212363, size = 61, normalized size = 1.13 \[ -\frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} - \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{6} \,{\rm ln}\left (x^{2} - x + 1\right ) + \frac{2}{3} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x - 1)*x^4/(x^3 + 1),x, algorithm="giac")
[Out]