Optimal. Leaf size=46 \[ \frac{\log \left (x^2+\sqrt{2} x+1\right )}{2 \sqrt{2}}-\frac{\log \left (x^2-\sqrt{2} x+1\right )}{2 \sqrt{2}} \]
[Out]
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Rubi [A] time = 0.0373196, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\log \left (x^2+\sqrt{2} x+1\right )}{2 \sqrt{2}}-\frac{\log \left (x^2-\sqrt{2} x+1\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[(1 - x^2)/(1 + x^4),x]
[Out]
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Rubi in Sympy [A] time = 9.39688, size = 39, normalized size = 0.85 \[ - \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{4} + \frac{\sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**2+1)/(x**4+1),x)
[Out]
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Mathematica [A] time = 0.0172064, size = 40, normalized size = 0.87 \[ \frac{\log \left (x^2+\sqrt{2} x+1\right )-\log \left (-x^2+\sqrt{2} x-1\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - x^2)/(1 + x^4),x]
[Out]
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Maple [A] time = 0.003, size = 62, normalized size = 1.4 \[{\frac{\sqrt{2}}{8}\ln \left ({\frac{1+{x}^{2}+\sqrt{2}x}{1+{x}^{2}-\sqrt{2}x}} \right ) }-{\frac{\sqrt{2}}{8}\ln \left ({\frac{1+{x}^{2}-\sqrt{2}x}{1+{x}^{2}+\sqrt{2}x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^2+1)/(x^4+1),x)
[Out]
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Maxima [A] time = 0.811743, size = 46, normalized size = 1. \[ \frac{1}{4} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{4} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^2 - 1)/(x^4 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.278512, size = 50, normalized size = 1.09 \[ \frac{1}{4} \, \sqrt{2} \log \left (\frac{4 \, x^{3} + \sqrt{2}{\left (x^{4} + 4 \, x^{2} + 1\right )} + 4 \, x}{x^{4} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^2 - 1)/(x^4 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.182918, size = 39, normalized size = 0.85 \[ - \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{4} + \frac{\sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x**2+1)/(x**4+1),x)
[Out]
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GIAC/XCAS [A] time = 0.273028, size = 46, normalized size = 1. \[ \frac{1}{4} \, \sqrt{2}{\rm ln}\left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{4} \, \sqrt{2}{\rm ln}\left (x^{2} - \sqrt{2} x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^2 - 1)/(x^4 + 1),x, algorithm="giac")
[Out]