3.63 \(\int \frac{1-2 x^2}{1+6 x^2+4 x^4} \, dx\)

Optimal. Leaf size=46 \[ \frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3-\sqrt{5}}}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{5}}}\right )}{\sqrt{2}} \]

[Out]

ArcTan[(2*x)/Sqrt[3 - Sqrt[5]]]/Sqrt[2] - ArcTan[(2*x)/Sqrt[3 + Sqrt[5]]]/Sqrt[2
]

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Rubi [A]  time = 0.0741989, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3-\sqrt{5}}}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{5}}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x^2)/(1 + 6*x^2 + 4*x^4),x]

[Out]

ArcTan[(2*x)/Sqrt[3 - Sqrt[5]]]/Sqrt[2] - ArcTan[(2*x)/Sqrt[3 + Sqrt[5]]]/Sqrt[2
]

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Rubi in Sympy [A]  time = 9.42177, size = 68, normalized size = 1.48 \[ - \frac{\left (- \sqrt{5} + 1\right ) \operatorname{atan}{\left (\frac{2 x}{\sqrt{- \sqrt{5} + 3}} \right )}}{2 \sqrt{- \sqrt{5} + 3}} - \frac{\left (1 + \sqrt{5}\right ) \operatorname{atan}{\left (\frac{2 x}{\sqrt{\sqrt{5} + 3}} \right )}}{2 \sqrt{\sqrt{5} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-2*x**2+1)/(4*x**4+6*x**2+1),x)

[Out]

-(-sqrt(5) + 1)*atan(2*x/sqrt(-sqrt(5) + 3))/(2*sqrt(-sqrt(5) + 3)) - (1 + sqrt(
5))*atan(2*x/sqrt(sqrt(5) + 3))/(2*sqrt(sqrt(5) + 3))

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Mathematica [A]  time = 0.122399, size = 84, normalized size = 1.83 \[ \frac{-\left (\sqrt{5}-5\right ) \sqrt{3+\sqrt{5}} \tan ^{-1}\left (\frac{2 x}{\sqrt{3-\sqrt{5}}}\right )-\sqrt{3-\sqrt{5}} \left (5+\sqrt{5}\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{5}}}\right )}{4 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x^2)/(1 + 6*x^2 + 4*x^4),x]

[Out]

(-((-5 + Sqrt[5])*Sqrt[3 + Sqrt[5]]*ArcTan[(2*x)/Sqrt[3 - Sqrt[5]]]) - Sqrt[3 -
Sqrt[5]]*(5 + Sqrt[5])*ArcTan[(2*x)/Sqrt[3 + Sqrt[5]]])/(4*Sqrt[5])

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Maple [B]  time = 0.019, size = 136, normalized size = 3. \[ 2\,{\frac{\sqrt{5}}{2\,\sqrt{10}-2\,\sqrt{2}}\arctan \left ( 8\,{\frac{x}{2\,\sqrt{10}-2\,\sqrt{2}}} \right ) }-2\,{\frac{1}{2\,\sqrt{10}-2\,\sqrt{2}}\arctan \left ( 8\,{\frac{x}{2\,\sqrt{10}-2\,\sqrt{2}}} \right ) }-2\,{\frac{\sqrt{5}}{2\,\sqrt{10}+2\,\sqrt{2}}\arctan \left ( 8\,{\frac{x}{2\,\sqrt{10}+2\,\sqrt{2}}} \right ) }-2\,{\frac{1}{2\,\sqrt{10}+2\,\sqrt{2}}\arctan \left ( 8\,{\frac{x}{2\,\sqrt{10}+2\,\sqrt{2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-2*x^2+1)/(4*x^4+6*x^2+1),x)

[Out]

2*5^(1/2)/(2*10^(1/2)-2*2^(1/2))*arctan(8*x/(2*10^(1/2)-2*2^(1/2)))-2/(2*10^(1/2
)-2*2^(1/2))*arctan(8*x/(2*10^(1/2)-2*2^(1/2)))-2*5^(1/2)/(2*10^(1/2)+2*2^(1/2))
*arctan(8*x/(2*10^(1/2)+2*2^(1/2)))-2/(2*10^(1/2)+2*2^(1/2))*arctan(8*x/(2*10^(1
/2)+2*2^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{2 \, x^{2} - 1}{4 \, x^{4} + 6 \, x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x^2 - 1)/(4*x^4 + 6*x^2 + 1),x, algorithm="maxima")

[Out]

-integrate((2*x^2 - 1)/(4*x^4 + 6*x^2 + 1), x)

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Fricas [A]  time = 0.275214, size = 34, normalized size = 0.74 \[ \frac{1}{2} \, \sqrt{2}{\left (\arctan \left (2 \, \sqrt{2}{\left (x^{3} + x\right )}\right ) - \arctan \left (\sqrt{2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x^2 - 1)/(4*x^4 + 6*x^2 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.246948, size = 39, normalized size = 0.85 \[ - \frac{\sqrt{2} \left (2 \operatorname{atan}{\left (\sqrt{2} x \right )} - 2 \operatorname{atan}{\left (2 \sqrt{2} x^{3} + 2 \sqrt{2} x \right )}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x**2+1)/(4*x**4+6*x**2+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.27799, size = 53, normalized size = 1.15 \[ -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{4 \, x}{\sqrt{10} + \sqrt{2}}\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{4 \, x}{\sqrt{10} - \sqrt{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x^2 - 1)/(4*x^4 + 6*x^2 + 1),x, algorithm="giac")

[Out]

-1/2*sqrt(2)*arctan(4*x/(sqrt(10) + sqrt(2))) + 1/2*sqrt(2)*arctan(4*x/(sqrt(10)
- sqrt(2)))