Optimal. Leaf size=46 \[ \frac{\log \left (x^2+\sqrt{3} x+1\right )}{2 \sqrt{3}}-\frac{\log \left (x^2-\sqrt{3} x+1\right )}{2 \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.0400427, antiderivative size = 46, 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 \[ \frac{\log \left (x^2+\sqrt{3} x+1\right )}{2 \sqrt{3}}-\frac{\log \left (x^2-\sqrt{3} x+1\right )}{2 \sqrt{3}} \]
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 = 12.8177, size = 39, normalized size = 0.85 \[ - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{6} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{6} \]
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 = 0.02027, size = 40, normalized size = 0.87 \[ \frac{\log \left (x^2+\sqrt{3} x+1\right )-\log \left (-x^2+\sqrt{3} x-1\right )}{2 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - x^2)/(1 - x^2 + x^4),x]
[Out]
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Maple [A] time = 0.014, size = 35, normalized size = 0.8 \[ -{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{6}}+{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{6}} \]
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 = 0.290388, size = 57, normalized size = 1.24 \[ \frac{1}{6} \, \sqrt{3} \log \left (\frac{6 \, x^{3} + \sqrt{3}{\left (x^{4} + 5 \, x^{2} + 1\right )} + 6 \, x}{x^{4} - x^{2} + 1}\right ) \]
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 = 0.197104, size = 39, normalized size = 0.85 \[ - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{6} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{6} \]
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 = 0.27022, size = 53, normalized size = 1.15 \[ -\frac{1}{6} \, \sqrt{3}{\rm ln}\left (\frac{{\left | 2 \, x - 2 \, \sqrt{3} + \frac{2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt{3} + \frac{2}{x} \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^2 - 1)/(x^4 - x^2 + 1),x, algorithm="giac")
[Out]