3.304 \(\int \frac{(1-x) x^4}{1+x^3} \, dx\)

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]

x^2/2 - x^3/3 + ArcTan[(1 - 2*x)/Sqrt[3]]/Sqrt[3] + (2*Log[1 + x])/3 + Log[1 - x
+ x^2]/6

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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]

x^2/2 - x^3/3 + ArcTan[(1 - 2*x)/Sqrt[3]]/Sqrt[3] + (2*Log[1 + x])/3 + Log[1 - x
+ x^2]/6

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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]

-x**3/3 + 2*log(x + 1)/3 + log(x**2 - x + 1)/6 - sqrt(3)*atan(sqrt(3)*(2*x/3 - 1
/3))/3 + Integral(x, x)

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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]

(3*x^2 - 2*x^3 - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 2*Log[1 + x] - Log[1 - x
+ x^2] + 2*Log[1 + x^3])/6

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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]

-1/3*x^3+1/2*x^2+1/6*ln(x^2-x+1)-1/3*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))+2/3*ln(
1+x)

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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]

-1/3*x^3 + 1/2*x^2 - 1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/6*log(x^2 - x
+ 1) + 2/3*log(x + 1)

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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]

-1/18*sqrt(3)*(sqrt(3)*(2*x^3 - 3*x^2) - sqrt(3)*log(x^2 - x + 1) - 4*sqrt(3)*lo
g(x + 1) + 6*arctan(1/3*sqrt(3)*(2*x - 1)))

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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]

-x**3/3 + x**2/2 + 2*log(x + 1)/3 + log(x**2 - x + 1)/6 - sqrt(3)*atan(2*sqrt(3)
*x/3 - sqrt(3)/3)/3

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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]

-1/3*x^3 + 1/2*x^2 - 1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/6*ln(x^2 - x
+ 1) + 2/3*ln(abs(x + 1))