3.70 \(\int \frac{1-2 x^2}{1-x^2+4 x^4} \, dx\)

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]

-Log[1 - Sqrt[5]*x + 2*x^2]/(2*Sqrt[5]) + Log[1 + Sqrt[5]*x + 2*x^2]/(2*Sqrt[5])

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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]

-Log[1 - Sqrt[5]*x + 2*x^2]/(2*Sqrt[5]) + Log[1 + Sqrt[5]*x + 2*x^2]/(2*Sqrt[5])

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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]

-sqrt(5)*log(x**2 - sqrt(5)*x/2 + 1/2)/10 + sqrt(5)*log(x**2 + sqrt(5)*x/2 + 1/2
)/10

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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]

(-Log[-1 + Sqrt[5]*x - 2*x^2] + Log[1 + Sqrt[5]*x + 2*x^2])/(2*Sqrt[5])

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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]

-1/10*ln(1+2*x^2-x*5^(1/2))*5^(1/2)+1/10*ln(1+2*x^2+x*5^(1/2))*5^(1/2)

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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]

-integrate((2*x^2 - 1)/(4*x^4 - x^2 + 1), x)

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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]

1/10*sqrt(5)*log((20*x^3 + sqrt(5)*(4*x^4 + 9*x^2 + 1) + 10*x)/(4*x^4 - x^2 + 1)
)

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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]

-sqrt(5)*log(x**2 - sqrt(5)*x/2 + 1/2)/10 + sqrt(5)*log(x**2 + sqrt(5)*x/2 + 1/2
)/10

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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]

integrate(-(2*x^2 - 1)/(4*x^4 - x^2 + 1), x)