3.89 \(\int \frac{1-x^2}{1+5 x^2+x^4} \, dx\)

Optimal. Leaf size=50 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{1}{2} \left (5+\sqrt{21}\right )} x\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right )}{\sqrt{3}} \]

[Out]

-(ArcTan[Sqrt[2/(5 + Sqrt[21])]*x]/Sqrt[3]) + ArcTan[Sqrt[(5 + Sqrt[21])/2]*x]/S
qrt[3]

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Rubi [A]  time = 0.128619, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{1}{2} \left (5+\sqrt{21}\right )} x\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^2)/(1 + 5*x^2 + x^4),x]

[Out]

-(ArcTan[Sqrt[2/(5 + Sqrt[21])]*x]/Sqrt[3]) + ArcTan[Sqrt[(5 + Sqrt[21])/2]*x]/S
qrt[3]

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Rubi in Sympy [A]  time = 9.36661, size = 88, normalized size = 1.76 \[ - \frac{\sqrt{2} \left (- \frac{\sqrt{21}}{6} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{- \sqrt{21} + 5}} \right )}}{\sqrt{- \sqrt{21} + 5}} - \frac{\sqrt{2} \left (\frac{1}{2} + \frac{\sqrt{21}}{6}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{21} + 5}} \right )}}{\sqrt{\sqrt{21} + 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**2+1)/(x**4+5*x**2+1),x)

[Out]

-sqrt(2)*(-sqrt(21)/6 + 1/2)*atan(sqrt(2)*x/sqrt(-sqrt(21) + 5))/sqrt(-sqrt(21)
+ 5) - sqrt(2)*(1/2 + sqrt(21)/6)*atan(sqrt(2)*x/sqrt(sqrt(21) + 5))/sqrt(sqrt(2
1) + 5)

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Mathematica [A]  time = 0.216824, size = 87, normalized size = 1.74 \[ \frac{\left (7-\sqrt{21}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{5-\sqrt{21}}} x\right )}{\sqrt{42 \left (5-\sqrt{21}\right )}}+\frac{\left (-7-\sqrt{21}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right )}{\sqrt{42 \left (5+\sqrt{21}\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x^2)/(1 + 5*x^2 + x^4),x]

[Out]

((7 - Sqrt[21])*ArcTan[Sqrt[2/(5 - Sqrt[21])]*x])/Sqrt[42*(5 - Sqrt[21])] + ((-7
- Sqrt[21])*ArcTan[Sqrt[2/(5 + Sqrt[21])]*x])/Sqrt[42*(5 + Sqrt[21])]

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Maple [B]  time = 0.017, size = 136, normalized size = 2.7 \[ -{\frac{2\,\sqrt{21}}{6\,\sqrt{7}+6\,\sqrt{3}}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{7}+2\,\sqrt{3}}} \right ) }-2\,{\frac{1}{2\,\sqrt{7}+2\,\sqrt{3}}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{7}+2\,\sqrt{3}}} \right ) }+{\frac{2\,\sqrt{21}}{6\,\sqrt{7}-6\,\sqrt{3}}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{7}-2\,\sqrt{3}}} \right ) }-2\,{\frac{1}{2\,\sqrt{7}-2\,\sqrt{3}}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{7}-2\,\sqrt{3}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+1)/(x^4+5*x^2+1),x)

[Out]

-2/3*21^(1/2)/(2*7^(1/2)+2*3^(1/2))*arctan(4*x/(2*7^(1/2)+2*3^(1/2)))-2/(2*7^(1/
2)+2*3^(1/2))*arctan(4*x/(2*7^(1/2)+2*3^(1/2)))+2/3*21^(1/2)/(2*7^(1/2)-2*3^(1/2
))*arctan(4*x/(2*7^(1/2)-2*3^(1/2)))-2/(2*7^(1/2)-2*3^(1/2))*arctan(4*x/(2*7^(1/
2)-2*3^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{x^{2} - 1}{x^{4} + 5 \, x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 1)/(x^4 + 5*x^2 + 1),x, algorithm="maxima")

[Out]

-integrate((x^2 - 1)/(x^4 + 5*x^2 + 1), x)

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Fricas [A]  time = 0.273551, size = 38, normalized size = 0.76 \[ \frac{1}{3} \, \sqrt{3}{\left (\arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x^{3} + 4 \, x\right )}\right ) - \arctan \left (\frac{1}{3} \, \sqrt{3} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 1)/(x^4 + 5*x^2 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.254755, size = 42, normalized size = 0.84 \[ - \frac{\sqrt{3} \left (2 \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} \right )} - 2 \operatorname{atan}{\left (\frac{\sqrt{3} x^{3}}{3} + \frac{4 \sqrt{3} x}{3} \right )}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+1)/(x**4+5*x**2+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.277505, size = 35, normalized size = 0.7 \[ \frac{1}{6} \, \sqrt{3}{\left (\pi{\rm sign}\left (x\right ) - 2 \, \arctan \left (\frac{\sqrt{3}{\left (x^{2} + 1\right )}}{3 \, x}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 1)/(x^4 + 5*x^2 + 1),x, algorithm="giac")

[Out]

1/6*sqrt(3)*(pi*sign(x) - 2*arctan(1/3*sqrt(3)*(x^2 + 1)/x))