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

Optimal. Leaf size=67 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{3 \left (2+\sqrt{3}\right )}} \]

[Out]

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

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Rubi [A]  time = 0.0296746, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{3 \left (2+\sqrt{3}\right )}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + 4*x^2 + x^4)^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 1.10556, size = 60, normalized size = 0.9 \[ \frac{\sqrt{3} \operatorname{atan}{\left (\frac{x}{\sqrt{- \sqrt{3} + 2}} \right )}}{6 \sqrt{- \sqrt{3} + 2}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{x}{\sqrt{\sqrt{3} + 2}} \right )}}{6 \sqrt{\sqrt{3} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0271374, size = 67, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{3 \left (2+\sqrt{3}\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.019, size = 60, normalized size = 0.9 \[ -{\frac{\sqrt{3}}{3\,\sqrt{6}+3\,\sqrt{2}}\arctan \left ( 2\,{\frac{x}{\sqrt{6}+\sqrt{2}}} \right ) }+{\frac{\sqrt{3}}{3\,\sqrt{6}-3\,\sqrt{2}}\arctan \left ( 2\,{\frac{x}{\sqrt{6}-\sqrt{2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^4+4*x^2+1),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(x^4 + 4*x^2 + 1), x)

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Fricas [A]  time = 0.222937, size = 182, normalized size = 2.72 \[ \frac{1}{3} \, \sqrt{\sqrt{3}{\left (2 \, \sqrt{3} - 3\right )}} \arctan \left (\frac{\sqrt{\sqrt{3}{\left (2 \, \sqrt{3} - 3\right )}}{\left (\sqrt{3} + 2\right )}}{\sqrt{3} \sqrt{\frac{1}{3}} \sqrt{\sqrt{3}{\left (\sqrt{3}{\left (x^{2} + 2\right )} + 3\right )}} + \sqrt{3} x}\right ) + \frac{1}{3} \, \sqrt{\sqrt{3}{\left (2 \, \sqrt{3} + 3\right )}} \arctan \left (\frac{\sqrt{\sqrt{3}{\left (2 \, \sqrt{3} + 3\right )}}{\left (\sqrt{3} - 2\right )}}{\sqrt{3} \sqrt{\frac{1}{3}} \sqrt{\sqrt{3}{\left (\sqrt{3}{\left (x^{2} + 2\right )} - 3\right )}} + \sqrt{3} x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.291233, size = 92, normalized size = 1.37 \[ - 2 \sqrt{- \frac{\sqrt{3}}{48} + \frac{1}{24}} \operatorname{atan}{\left (\frac{x}{\sqrt{3} \sqrt{- \sqrt{3} + 2} + 2 \sqrt{- \sqrt{3} + 2}} \right )} - 2 \sqrt{\frac{\sqrt{3}}{48} + \frac{1}{24}} \operatorname{atan}{\left (\frac{x}{- 2 \sqrt{\sqrt{3} + 2} + \sqrt{3} \sqrt{\sqrt{3} + 2}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**4+4*x**2+1),x)

[Out]

-2*sqrt(-sqrt(3)/48 + 1/24)*atan(x/(sqrt(3)*sqrt(-sqrt(3) + 2) + 2*sqrt(-sqrt(3)
+ 2))) - 2*sqrt(sqrt(3)/48 + 1/24)*atan(x/(-2*sqrt(sqrt(3) + 2) + sqrt(3)*sqrt(
sqrt(3) + 2)))

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GIAC/XCAS [A]  time = 0.203876, size = 69, normalized size = 1.03 \[ \frac{1}{12} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{2 \, x}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{12} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{2 \, x}{\sqrt{6} - \sqrt{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(sqrt(6) - 3*sqrt(2))*arctan(2*x/(sqrt(6) + sqrt(2))) + 1/12*(sqrt(6) + 3*s
qrt(2))*arctan(2*x/(sqrt(6) - sqrt(2)))