Optimal. Leaf size=50 \[ \frac{\log \left (2 x^2+\sqrt{5} x+1\right )}{2 \sqrt{5}}-\frac{\log \left (2 x^2-\sqrt{5} x+1\right )}{2 \sqrt{5}} \]
[Out]
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Rubi [A] time = 0.0476567, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\log \left (2 x^2+\sqrt{5} x+1\right )}{2 \sqrt{5}}-\frac{\log \left (2 x^2-\sqrt{5} x+1\right )}{2 \sqrt{5}} \]
Antiderivative was successfully verified.
[In] Int[(1 - 2*x^2)/(1 - x^2 + 4*x^4),x]
[Out]
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Rubi in Sympy [A] time = 13.8308, size = 46, normalized size = 0.92 \[ - \frac{\sqrt{5} \log{\left (x^{2} - \frac{\sqrt{5} x}{2} + \frac{1}{2} \right )}}{10} + \frac{\sqrt{5} \log{\left (x^{2} + \frac{\sqrt{5} x}{2} + \frac{1}{2} \right )}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-2*x**2+1)/(4*x**4-x**2+1),x)
[Out]
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Mathematica [A] time = 0.0236739, size = 42, normalized size = 0.84 \[ \frac{\log \left (2 x^2+\sqrt{5} x+1\right )-\log \left (-2 x^2+\sqrt{5} x-1\right )}{2 \sqrt{5}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - 2*x^2)/(1 - x^2 + 4*x^4),x]
[Out]
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Maple [A] time = 0.015, size = 39, normalized size = 0.8 \[ -{\frac{\ln \left ( 1+2\,{x}^{2}-x\sqrt{5} \right ) \sqrt{5}}{10}}+{\frac{\ln \left ( 1+2\,{x}^{2}+x\sqrt{5} \right ) \sqrt{5}}{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-2*x^2+1)/(4*x^4-x^2+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{2 \, x^{2} - 1}{4 \, x^{4} - x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*x^2 - 1)/(4*x^4 - x^2 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.270817, size = 62, normalized size = 1.24 \[ \frac{1}{10} \, \sqrt{5} \log \left (\frac{20 \, x^{3} + \sqrt{5}{\left (4 \, x^{4} + 9 \, x^{2} + 1\right )} + 10 \, x}{4 \, x^{4} - x^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*x^2 - 1)/(4*x^4 - x^2 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.196494, size = 46, normalized size = 0.92 \[ - \frac{\sqrt{5} \log{\left (x^{2} - \frac{\sqrt{5} x}{2} + \frac{1}{2} \right )}}{10} + \frac{\sqrt{5} \log{\left (x^{2} + \frac{\sqrt{5} x}{2} + \frac{1}{2} \right )}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x**2+1)/(4*x**4-x**2+1),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{2 \, x^{2} - 1}{4 \, x^{4} - x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*x^2 - 1)/(4*x^4 - x^2 + 1),x, algorithm="giac")
[Out]