Optimal. Leaf size=39 \[ \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )-\tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right ) \]
[Out]
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Rubi [A] time = 0.0922117, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )-\tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 - x^2)/(1 + 3*x^2 + x^4),x]
[Out]
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Rubi in Sympy [A] time = 9.2912, size = 88, normalized size = 2.26 \[ - \frac{\sqrt{2} \left (- \frac{\sqrt{5}}{2} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{- \sqrt{5} + 3}} \right )}}{\sqrt{- \sqrt{5} + 3}} - \frac{\sqrt{2} \left (\frac{1}{2} + \frac{\sqrt{5}}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{5} + 3}} \right )}}{\sqrt{\sqrt{5} + 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**2+1)/(x**4+3*x**2+1),x)
[Out]
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Mathematica [A] time = 0.0100228, size = 10, normalized size = 0.26 \[ \tan ^{-1}\left (\frac{x}{x^2+1}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 - x^2)/(1 + 3*x^2 + x^4),x]
[Out]
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Maple [B] time = 0.017, size = 104, normalized size = 2.7 \[ -2\,{\frac{\sqrt{5}}{2\,\sqrt{5}+2}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{5}+2}} \right ) }-2\,{\frac{1}{2\,\sqrt{5}+2}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{5}+2}} \right ) }+2\,{\frac{\sqrt{5}}{-2+2\,\sqrt{5}}\arctan \left ( 4\,{\frac{x}{-2+2\,\sqrt{5}}} \right ) }-2\,{\frac{1}{-2+2\,\sqrt{5}}\arctan \left ( 4\,{\frac{x}{-2+2\,\sqrt{5}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^2+1)/(x^4+3*x^2+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{x^{2} - 1}{x^{4} + 3 \, x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^2 - 1)/(x^4 + 3*x^2 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.264358, size = 18, normalized size = 0.46 \[ \arctan \left (x^{3} + 2 \, x\right ) - \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^2 - 1)/(x^4 + 3*x^2 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.201657, size = 10, normalized size = 0.26 \[ - \operatorname{atan}{\left (x \right )} + \operatorname{atan}{\left (x^{3} + 2 x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x**2+1)/(x**4+3*x**2+1),x)
[Out]
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GIAC/XCAS [A] time = 0.274356, size = 35, normalized size = 0.9 \[ \frac{1}{4} \, \pi{\rm sign}\left (x\right ) - \frac{1}{2} \, \arctan \left (\frac{x^{4} + x^{2} + 1}{2 \,{\left (x^{3} + x\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^2 - 1)/(x^4 + 3*x^2 + 1),x, algorithm="giac")
[Out]