3.352 \(\int \frac{x^3}{1-x^4+x^8} \, dx\)

Optimal. Leaf size=23 \[ -\frac{\tan ^{-1}\left (\frac{1-2 x^4}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

[Out]

-ArcTan[(1 - 2*x^4)/Sqrt[3]]/(2*Sqrt[3])

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Rubi [A]  time = 0.045978, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{\tan ^{-1}\left (\frac{1-2 x^4}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(1 - x^4 + x^8),x]

[Out]

-ArcTan[(1 - 2*x^4)/Sqrt[3]]/(2*Sqrt[3])

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Rubi in Sympy [A]  time = 5.98158, size = 22, normalized size = 0.96 \[ \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{4}}{3} - \frac{1}{3}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(x**8-x**4+1),x)

[Out]

sqrt(3)*atan(sqrt(3)*(2*x**4/3 - 1/3))/6

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Mathematica [A]  time = 0.0109242, size = 23, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{2 x^4-1}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(1 - x^4 + x^8),x]

[Out]

ArcTan[(-1 + 2*x^4)/Sqrt[3]]/(2*Sqrt[3])

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Maple [A]  time = 0.001, size = 19, normalized size = 0.8 \[{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,{x}^{4}-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(x^8-x^4+1),x)

[Out]

1/6*3^(1/2)*arctan(1/3*(2*x^4-1)*3^(1/2))

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Maxima [A]  time = 0.824623, size = 24, normalized size = 1.04 \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(x^8 - x^4 + 1),x, algorithm="maxima")

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 - 1))

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Fricas [A]  time = 0.256167, size = 24, normalized size = 1.04 \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(x^8 - x^4 + 1),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 - 1))

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Sympy [A]  time = 0.292467, size = 26, normalized size = 1.13 \[ \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{4}}{3} - \frac{\sqrt{3}}{3} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(x**8-x**4+1),x)

[Out]

sqrt(3)*atan(2*sqrt(3)*x**4/3 - sqrt(3)/3)/6

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GIAC/XCAS [A]  time = 0.274977, size = 24, normalized size = 1.04 \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(x^8 - x^4 + 1),x, algorithm="giac")

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 - 1))