3.16 \(\int \frac{2-3 x^2}{4-9 x^4} \, dx\)

Optimal. Leaf size=16 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{6}} \]

[Out]

ArcTan[Sqrt[3/2]*x]/Sqrt[6]

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Rubi [A]  time = 0.00853843, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{6}} \]

Antiderivative was successfully verified.

[In]  Int[(2 - 3*x^2)/(4 - 9*x^4),x]

[Out]

ArcTan[Sqrt[3/2]*x]/Sqrt[6]

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Rubi in Sympy [A]  time = 2.58155, size = 15, normalized size = 0.94 \[ \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x}{2} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-3*x**2+2)/(-9*x**4+4),x)

[Out]

sqrt(6)*atan(sqrt(6)*x/2)/6

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Mathematica [A]  time = 0.00814293, size = 16, normalized size = 1. \[ \frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{6}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 - 3*x^2)/(4 - 9*x^4),x]

[Out]

ArcTan[Sqrt[3/2]*x]/Sqrt[6]

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Maple [A]  time = 0.004, size = 13, normalized size = 0.8 \[{\frac{\sqrt{6}}{6}\arctan \left ({\frac{x\sqrt{6}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-3*x^2+2)/(-9*x^4+4),x)

[Out]

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

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Maxima [A]  time = 0.862497, size = 16, normalized size = 1. \[ \frac{1}{6} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 - 2)/(9*x^4 - 4),x, algorithm="maxima")

[Out]

1/6*sqrt(6)*arctan(1/2*sqrt(6)*x)

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Fricas [A]  time = 0.279951, size = 16, normalized size = 1. \[ \frac{1}{6} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 - 2)/(9*x^4 - 4),x, algorithm="fricas")

[Out]

1/6*sqrt(6)*arctan(1/2*sqrt(6)*x)

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Sympy [A]  time = 0.177656, size = 15, normalized size = 0.94 \[ \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x}{2} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*x**2+2)/(-9*x**4+4),x)

[Out]

sqrt(6)*atan(sqrt(6)*x/2)/6

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GIAC/XCAS [A]  time = 0.267811, size = 16, normalized size = 1. \[ \frac{1}{6} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 - 2)/(9*x^4 - 4),x, algorithm="giac")

[Out]

1/6*sqrt(6)*arctan(1/2*sqrt(6)*x)