3.45 \(\int \frac{x^2}{13-6 x^3+x^6} \, dx\)

Optimal. Leaf size=14 \[ \frac{1}{6} \tan ^{-1}\left (\frac{1}{2} \left (x^3-3\right )\right ) \]

[Out]

ArcTan[(-3 + x^3)/2]/6

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Rubi [A]  time = 0.0337604, antiderivative size = 14, 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 \[ \frac{1}{6} \tan ^{-1}\left (\frac{1}{2} \left (x^3-3\right )\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2/(13 - 6*x^3 + x^6),x]

[Out]

ArcTan[(-3 + x^3)/2]/6

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Rubi in Sympy [A]  time = 2.50087, size = 10, normalized size = 0.71 \[ \frac{\operatorname{atan}{\left (\frac{x^{3}}{2} - \frac{3}{2} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(x**6-6*x**3+13),x)

[Out]

atan(x**3/2 - 3/2)/6

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Mathematica [A]  time = 0.0075148, size = 14, normalized size = 1. \[ \frac{1}{6} \tan ^{-1}\left (\frac{1}{2} \left (x^3-3\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(13 - 6*x^3 + x^6),x]

[Out]

ArcTan[(-3 + x^3)/2]/6

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Maple [A]  time = 0.004, size = 11, normalized size = 0.8 \[{\frac{1}{6}\arctan \left ({\frac{{x}^{3}}{2}}-{\frac{3}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^6-6*x^3+13),x)

[Out]

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

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Maxima [A]  time = 1.47843, size = 14, normalized size = 1. \[ \frac{1}{6} \, \arctan \left (\frac{1}{2} \, x^{3} - \frac{3}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^6 - 6*x^3 + 13),x, algorithm="maxima")

[Out]

1/6*arctan(1/2*x^3 - 3/2)

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Fricas [A]  time = 0.206583, size = 14, normalized size = 1. \[ \frac{1}{6} \, \arctan \left (\frac{1}{2} \, x^{3} - \frac{3}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^6 - 6*x^3 + 13),x, algorithm="fricas")

[Out]

1/6*arctan(1/2*x^3 - 3/2)

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Sympy [A]  time = 0.118414, size = 10, normalized size = 0.71 \[ \frac{\operatorname{atan}{\left (\frac{x^{3}}{2} - \frac{3}{2} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(x**6-6*x**3+13),x)

[Out]

atan(x**3/2 - 3/2)/6

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GIAC/XCAS [A]  time = 0.209145, size = 14, normalized size = 1. \[ \frac{1}{6} \, \arctan \left (\frac{1}{2} \, x^{3} - \frac{3}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^6 - 6*x^3 + 13),x, algorithm="giac")

[Out]

1/6*arctan(1/2*x^3 - 3/2)