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

Optimal. Leaf size=49 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right )}{\sqrt{7}}+\frac{\tan ^{-1}\left (\sqrt{\frac{1}{2} \left (5+\sqrt{21}\right )} x\right )}{\sqrt{7}} \]

[Out]

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

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Rubi [A]  time = 0.154938, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right )}{\sqrt{7}}+\frac{\tan ^{-1}\left (\sqrt{\frac{1}{2} \left (5+\sqrt{21}\right )} x\right )}{\sqrt{7}} \]

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

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Rubi in Sympy [A]  time = 8.99053, size = 87, normalized size = 1.78 \[ \frac{\sqrt{2} \left (- \frac{\sqrt{21}}{14} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{- \sqrt{21} + 5}} \right )}}{\sqrt{- \sqrt{21} + 5}} + \frac{\sqrt{2} \left (\frac{\sqrt{21}}{14} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{21} + 5}} \right )}}{\sqrt{\sqrt{21} + 5}} \]

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)/14 + 1/2)*atan(sqrt(2)*x/sqrt(-sqrt(21) + 5))/sqrt(-sqrt(21)
+ 5) + sqrt(2)*(sqrt(21)/14 + 1/2)*atan(sqrt(2)*x/sqrt(sqrt(21) + 5))/sqrt(sqrt(
21) + 5)

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Mathematica [A]  time = 0.231082, size = 83, normalized size = 1.69 \[ \frac{\left (\sqrt{21}-3\right ) \tan ^{-1}\left (\sqrt{\frac{2}{5-\sqrt{21}}} x\right )}{\sqrt{42 \left (5-\sqrt{21}\right )}}+\frac{\left (3+\sqrt{21}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right )}{\sqrt{42 \left (5+\sqrt{21}\right )}} \]

Antiderivative was successfully verified.

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

[Out]

((-3 + Sqrt[21])*ArcTan[Sqrt[2/(5 - Sqrt[21])]*x])/Sqrt[42*(5 - Sqrt[21])] + ((3
+ Sqrt[21])*ArcTan[Sqrt[2/(5 + Sqrt[21])]*x])/Sqrt[42*(5 + Sqrt[21])]

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Maple [B]  time = 0.058, size = 136, normalized size = 2.8 \[{\frac{2\,\sqrt{21}}{14\,\sqrt{7}+14\,\sqrt{3}}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{7}+2\,\sqrt{3}}} \right ) }+2\,{\frac{1}{2\,\sqrt{7}+2\,\sqrt{3}}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{7}+2\,\sqrt{3}}} \right ) }-{\frac{2\,\sqrt{21}}{14\,\sqrt{7}-14\,\sqrt{3}}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{7}-2\,\sqrt{3}}} \right ) }+2\,{\frac{1}{2\,\sqrt{7}-2\,\sqrt{3}}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{7}-2\,\sqrt{3}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+1)/(x^4+5*x^2+1),x)

[Out]

2/7*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)))-2/7*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 = 0.264699, size = 35, normalized size = 0.71 \[ \frac{1}{7} \, \sqrt{7}{\left (\arctan \left (\frac{1}{7} \, \sqrt{7}{\left (x^{3} + 6 \, x\right )}\right ) + \arctan \left (\frac{1}{7} \, \sqrt{7} x\right )\right )} \]

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]

1/7*sqrt(7)*(arctan(1/7*sqrt(7)*(x^3 + 6*x)) + arctan(1/7*sqrt(7)*x))

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Sympy [A]  time = 0.236008, size = 41, normalized size = 0.84 \[ \frac{\sqrt{7} \left (2 \operatorname{atan}{\left (\frac{\sqrt{7} x}{7} \right )} + 2 \operatorname{atan}{\left (\frac{\sqrt{7} x^{3}}{7} + \frac{6 \sqrt{7} x}{7} \right )}\right )}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(7)*(2*atan(sqrt(7)*x/7) + 2*atan(sqrt(7)*x**3/7 + 6*sqrt(7)*x/7))/14

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GIAC/XCAS [A]  time = 0.277339, size = 35, normalized size = 0.71 \[ \frac{1}{14} \, \sqrt{7}{\left (\pi{\rm sign}\left (x\right ) + 2 \, \arctan \left (\frac{\sqrt{7}{\left (x^{2} - 1\right )}}{7 \, x}\right )\right )} \]

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]

1/14*sqrt(7)*(pi*sign(x) + 2*arctan(1/7*sqrt(7)*(x^2 - 1)/x))