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