3.24 \(\int \frac{1-x^3}{x^4 \left (1-x^3+x^6\right )} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.102452, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{2 \tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^3)/(x^4*(1 - x^3 + x^6)),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.5165, size = 32, normalized size = 1.03 \[ - \frac{2 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{3}}{3} - \frac{1}{3}\right ) \right )}}{9} - \frac{1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**3+1)/x**4/(x**6-x**3+1),x)

[Out]

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

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Mathematica [C]  time = 0.0202159, size = 45, normalized size = 1.45 \[ -\frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\&\right ]-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x^3)/(x^4*(1 - x^3 + x^6)),x]

[Out]

-1/(3*x^3) - RootSum[1 - #1^3 + #1^6 & , Log[x - #1]/(-1 + 2*#1^3) & ]/3

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Maple [A]  time = 0.006, size = 25, normalized size = 0.8 \[ -{\frac{1}{3\,{x}^{3}}}-{\frac{2\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 2\,{x}^{3}-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^3+1)/x^4/(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 = 0.816764, size = 32, normalized size = 1.03 \[ -\frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) - \frac{1}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^3 - 1)/((x^6 - x^3 + 1)*x^4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.253693, size = 41, normalized size = 1.32 \[ -\frac{\sqrt{3}{\left (2 \, x^{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) + \sqrt{3}\right )}}{9 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^3 - 1)/((x^6 - x^3 + 1)*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.375926, size = 36, normalized size = 1.16 \[ - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{3}}{3} - \frac{\sqrt{3}}{3} \right )}}{9} - \frac{1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**3+1)/x**4/(x**6-x**3+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.27478, size = 32, normalized size = 1.03 \[ -\frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) - \frac{1}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^3 - 1)/((x^6 - x^3 + 1)*x^4),x, algorithm="giac")

[Out]

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