3.20 \(\int \frac{x^8 \left (1-x^3\right )}{1-x^3+x^6} \, dx\)

Optimal. Leaf size=46 \[ -\frac{x^6}{6}-\frac{\tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{1}{6} \log \left (x^6-x^3+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.12434, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ -\frac{x^6}{6}-\frac{\tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{1}{6} \log \left (x^6-x^3+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x^8*(1 - x^3))/(1 - x^3 + x^6),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\log{\left (x^{6} - x^{3} + 1 \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{3}}{3} - \frac{1}{3}\right ) \right )}}{9} - \frac{\int ^{x^{3}} x\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(-x**3+1)/(x**6-x**3+1),x)

[Out]

log(x**6 - x**3 + 1)/6 + sqrt(3)*atan(sqrt(3)*(2*x**3/3 - 1/3))/9 - Integral(x,
(x, x**3))/3

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Mathematica [A]  time = 0.0245158, size = 46, normalized size = 1. \[ -\frac{x^6}{6}+\frac{\tan ^{-1}\left (\frac{2 x^3-1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{1}{6} \log \left (x^6-x^3+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(x^8*(1 - x^3))/(1 - x^3 + x^6),x]

[Out]

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

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Maple [A]  time = 0.004, size = 38, normalized size = 0.8 \[ -{\frac{{x}^{6}}{6}}+{\frac{\ln \left ({x}^{6}-{x}^{3}+1 \right ) }{6}}+{\frac{\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 2\,{x}^{3}-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(-x^3+1)/(x^6-x^3+1),x)

[Out]

-1/6*x^6+1/6*ln(x^6-x^3+1)+1/9*3^(1/2)*arctan(1/3*(2*x^3-1)*3^(1/2))

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Maxima [A]  time = 0.817217, size = 50, normalized size = 1.09 \[ -\frac{1}{6} \, x^{6} + \frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) + \frac{1}{6} \, \log \left (x^{6} - x^{3} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^3 - 1)*x^8/(x^6 - x^3 + 1),x, algorithm="maxima")

[Out]

-1/6*x^6 + 1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^3 - 1)) + 1/6*log(x^6 - x^3 + 1)

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Fricas [A]  time = 0.256763, size = 59, normalized size = 1.28 \[ -\frac{1}{18} \, \sqrt{3}{\left (\sqrt{3} x^{6} - \sqrt{3} \log \left (x^{6} - x^{3} + 1\right ) - 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^3 - 1)*x^8/(x^6 - x^3 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.286037, size = 42, normalized size = 0.91 \[ - \frac{x^{6}}{6} + \frac{\log{\left (x^{6} - x^{3} + 1 \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{3}}{3} - \frac{\sqrt{3}}{3} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(-x**3+1)/(x**6-x**3+1),x)

[Out]

-x**6/6 + log(x**6 - x**3 + 1)/6 + sqrt(3)*atan(2*sqrt(3)*x**3/3 - sqrt(3)/3)/9

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GIAC/XCAS [A]  time = 0.269074, size = 50, normalized size = 1.09 \[ -\frac{1}{6} \, x^{6} + \frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) + \frac{1}{6} \,{\rm ln}\left (x^{6} - x^{3} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^3 - 1)*x^8/(x^6 - x^3 + 1),x, algorithm="giac")

[Out]

-1/6*x^6 + 1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^3 - 1)) + 1/6*ln(x^6 - x^3 + 1)