3.21 \(\int \frac{x^5 \left (1-x^3\right )}{1-x^3+x^6} \, dx\)

Optimal. Leaf size=31 \[ -\frac{x^3}{3}-\frac{2 \tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0831293, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{x^3}{3}-\frac{2 \tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(x^5*(1 - x^3))/(1 - x^3 + x^6),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.2477, size = 29, normalized size = 0.94 \[ - \frac{x^{3}}{3} + \frac{2 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{3}}{3} - \frac{1}{3}\right ) \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(-x**3+1)/(x**6-x**3+1),x)

[Out]

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

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Mathematica [A]  time = 0.0123437, size = 31, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{2 x^3-1}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{x^3}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^5*(1 - x^3))/(1 - x^3 + x^6),x]

[Out]

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

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Maple [A]  time = 0.004, size = 25, normalized size = 0.8 \[ -{\frac{{x}^{3}}{3}}+{\frac{2\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 2\,{x}^{3}-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(-x^3+1)/(x^6-x^3+1),x)

[Out]

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

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Maxima [A]  time = 0.816813, size = 32, normalized size = 1.03 \[ -\frac{1}{3} \, x^{3} + \frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^3 - 1)*x^5/(x^6 - x^3 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.272018, size = 38, normalized size = 1.23 \[ -\frac{1}{9} \, \sqrt{3}{\left (\sqrt{3} x^{3} - 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^3 - 1)*x^5/(x^6 - x^3 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.269313, size = 32, normalized size = 1.03 \[ - \frac{x^{3}}{3} + \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{3}}{3} - \frac{\sqrt{3}}{3} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(-x**3+1)/(x**6-x**3+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.271258, size = 32, normalized size = 1.03 \[ -\frac{1}{3} \, x^{3} + \frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^3 - 1)*x^5/(x^6 - x^3 + 1),x, algorithm="giac")

[Out]

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