3.12 \(\int \frac{1}{3+4 x^2+x^4} \, dx\)

Optimal. Leaf size=24 \[ \frac{1}{2} \tan ^{-1}(x)-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0193395, antiderivative size = 24, 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 \[ \frac{1}{2} \tan ^{-1}(x)-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(3 + 4*x^2 + x^4)^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.8243, size = 20, normalized size = 0.83 \[ \frac{\operatorname{atan}{\left (x \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**4+4*x**2+3),x)

[Out]

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

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Mathematica [A]  time = 0.0182419, size = 24, normalized size = 1. \[ \frac{1}{2} \tan ^{-1}(x)-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 4*x^2 + x^4)^(-1),x]

[Out]

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

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Maple [A]  time = 0.01, size = 18, normalized size = 0.8 \[{\frac{\arctan \left ( x \right ) }{2}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{x\sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^4+4*x^2+3),x)

[Out]

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

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Maxima [A]  time = 0.763653, size = 23, normalized size = 0.96 \[ -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) + \frac{1}{2} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^4 + 4*x^2 + 3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.260195, size = 28, normalized size = 1.17 \[ \frac{1}{6} \, \sqrt{3}{\left (\sqrt{3} \arctan \left (x\right ) - \arctan \left (\frac{1}{3} \, \sqrt{3} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^4 + 4*x^2 + 3),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*(sqrt(3)*arctan(x) - arctan(1/3*sqrt(3)*x))

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Sympy [A]  time = 0.394925, size = 20, normalized size = 0.83 \[ \frac{\operatorname{atan}{\left (x \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**4+4*x**2+3),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.268327, size = 23, normalized size = 0.96 \[ -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) + \frac{1}{2} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^4 + 4*x^2 + 3),x, algorithm="giac")

[Out]

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