Optimal. Leaf size=23 \[ \tan ^{-1}\left (2 x+\sqrt{3}\right )-\tan ^{-1}\left (\sqrt{3}-2 x\right ) \]
[Out]
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Rubi [A] time = 0.036119, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \tan ^{-1}\left (2 x+\sqrt{3}\right )-\tan ^{-1}\left (\sqrt{3}-2 x\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 + x^2)/(1 - x^2 + x^4),x]
[Out]
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Rubi in Sympy [A] time = 7.65397, size = 19, normalized size = 0.83 \[ \operatorname{atan}{\left (2 x - \sqrt{3} \right )} + \operatorname{atan}{\left (2 x + \sqrt{3} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**2+1)/(x**4-x**2+1),x)
[Out]
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Mathematica [A] time = 0.0107325, size = 12, normalized size = 0.52 \[ -\tan ^{-1}\left (\frac{x}{x^2-1}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 + x^2)/(1 - x^2 + x^4),x]
[Out]
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Maple [A] time = 0.018, size = 20, normalized size = 0.9 \[ \arctan \left ( 2\,x-\sqrt{3} \right ) +\arctan \left ( 2\,x+\sqrt{3} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^2+1)/(x^4-x^2+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} + 1}{x^{4} - x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 1)/(x^4 - x^2 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.284387, size = 9, normalized size = 0.39 \[ \arctan \left (x^{3}\right ) + \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 1)/(x^4 - x^2 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.191684, size = 7, normalized size = 0.3 \[ \operatorname{atan}{\left (x \right )} + \operatorname{atan}{\left (x^{3} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**2+1)/(x**4-x**2+1),x)
[Out]
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GIAC/XCAS [A] time = 0.273567, size = 41, normalized size = 1.78 \[ \frac{1}{4} \, \pi{\rm sign}\left (x\right ) + \frac{1}{2} \, \arctan \left (\frac{x^{4} - 3 \, x^{2} + 1}{2 \,{\left (x^{3} - x\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 1)/(x^4 - x^2 + 1),x, algorithm="giac")
[Out]