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

Optimal. Leaf size=41 \[ -\frac{1}{6} \log \left (x^2-x+1\right )-\frac{2}{3} \log (x+1)-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

-(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 = 0.0826407, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ -\frac{1}{6} \log \left (x^2-x+1\right )-\frac{2}{3} \log (x+1)-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - x)*x)/(1 + x^3),x]

[Out]

-(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 [A]  time = 12.45, size = 39, normalized size = 0.95 \[ - \frac{2 \log{\left (x + 1 \right )}}{3} - \frac{\log{\left (x^{2} - x + 1 \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)*x/(x**3+1),x)

[Out]

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

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Mathematica [A]  time = 0.0132038, size = 50, normalized size = 1.22 \[ -\frac{1}{3} \log \left (x^3+1\right )+\frac{1}{6} \log \left (x^2-x+1\right )-\frac{1}{3} \log (x+1)+\frac{\tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - x)*x)/(1 + x^3),x]

[Out]

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

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Maple [A]  time = 0.007, size = 35, normalized size = 0.9 \[ -{\frac{\ln \left ({x}^{2}-x+1 \right ) }{6}}+{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{2\,\ln \left ( 1+x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)*x/(x^3+1),x)

[Out]

-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 = 1.51995, size = 46, normalized size = 1.12 \[ \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{6} \, \log \left (x^{2} - x + 1\right ) - \frac{2}{3} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 1)*x/(x^3 + 1),x, algorithm="maxima")

[Out]

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 = 0.214494, size = 55, normalized size = 1.34 \[ -\frac{1}{18} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{2} - x + 1\right ) + 4 \, \sqrt{3} \log \left (x + 1\right ) - 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 1)*x/(x^3 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.166668, size = 42, normalized size = 1.02 \[ - \frac{2 \log{\left (x + 1 \right )}}{3} - \frac{\log{\left (x^{2} - x + 1 \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-x)*x/(x**3+1),x)

[Out]

-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 = 0.21205, size = 47, normalized size = 1.15 \[ \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{6} \,{\rm ln}\left (x^{2} - x + 1\right ) - \frac{2}{3} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 1)*x/(x^3 + 1),x, algorithm="giac")

[Out]

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))