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

Optimal. Leaf size=18 \[ x-\frac{8}{3} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{3} \tan ^{-1}(x) \]

[Out]

x - (8*ArcTan[x/2])/3 + ArcTan[x]/3

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Rubi [A]  time = 0.0332514, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ x-\frac{8}{3} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{3} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[x^4/(4 + 5*x^2 + x^4),x]

[Out]

x - (8*ArcTan[x/2])/3 + ArcTan[x]/3

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Rubi in Sympy [A]  time = 5.02038, size = 14, normalized size = 0.78 \[ x - \frac{8 \operatorname{atan}{\left (\frac{x}{2} \right )}}{3} + \frac{\operatorname{atan}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - 8*atan(x/2)/3 + atan(x)/3

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Mathematica [A]  time = 0.0110295, size = 18, normalized size = 1. \[ x+\frac{8}{3} \tan ^{-1}\left (\frac{2}{x}\right )+\frac{1}{3} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

x + (8*ArcTan[2/x])/3 + ArcTan[x]/3

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Maple [A]  time = 0.011, size = 13, normalized size = 0.7 \[ x-{\frac{8}{3}\arctan \left ({\frac{x}{2}} \right ) }+{\frac{\arctan \left ( x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(x^4+5*x^2+4),x)

[Out]

x-8/3*arctan(1/2*x)+1/3*arctan(x)

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Maxima [A]  time = 1.51905, size = 16, normalized size = 0.89 \[ x - \frac{8}{3} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{3} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - 8/3*arctan(1/2*x) + 1/3*arctan(x)

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Fricas [A]  time = 0.195418, size = 16, normalized size = 0.89 \[ x - \frac{8}{3} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{3} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - 8/3*arctan(1/2*x) + 1/3*arctan(x)

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Sympy [A]  time = 0.192233, size = 14, normalized size = 0.78 \[ x - \frac{8 \operatorname{atan}{\left (\frac{x}{2} \right )}}{3} + \frac{\operatorname{atan}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - 8*atan(x/2)/3 + atan(x)/3

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GIAC/XCAS [A]  time = 0.215385, size = 16, normalized size = 0.89 \[ x - \frac{8}{3} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{3} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - 8/3*arctan(1/2*x) + 1/3*arctan(x)