3.66 \(\int \frac{1-2 x^2}{1+3 x^2+4 x^4} \, dx\)

Optimal. Leaf size=29 \[ \frac{1}{2} \log \left (2 x^2+x+1\right )-\frac{1}{2} \log \left (2 x^2-x+1\right ) \]

[Out]

-Log[1 - x + 2*x^2]/2 + Log[1 + x + 2*x^2]/2

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Rubi [A]  time = 0.0326194, antiderivative size = 29, 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 \[ \frac{1}{2} \log \left (2 x^2+x+1\right )-\frac{1}{2} \log \left (2 x^2-x+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x^2)/(1 + 3*x^2 + 4*x^4),x]

[Out]

-Log[1 - x + 2*x^2]/2 + Log[1 + x + 2*x^2]/2

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Rubi in Sympy [A]  time = 11.4104, size = 26, normalized size = 0.9 \[ - \frac{\log{\left (x^{2} - \frac{x}{2} + \frac{1}{2} \right )}}{2} + \frac{\log{\left (x^{2} + \frac{x}{2} + \frac{1}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-2*x**2+1)/(4*x**4+3*x**2+1),x)

[Out]

-log(x**2 - x/2 + 1/2)/2 + log(x**2 + x/2 + 1/2)/2

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Mathematica [A]  time = 0.00986284, size = 29, normalized size = 1. \[ \frac{1}{2} \log \left (2 x^2+x+1\right )-\frac{1}{2} \log \left (2 x^2-x+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x^2)/(1 + 3*x^2 + 4*x^4),x]

[Out]

-Log[1 - x + 2*x^2]/2 + Log[1 + x + 2*x^2]/2

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Maple [A]  time = 0.005, size = 26, normalized size = 0.9 \[ -{\frac{\ln \left ( 2\,{x}^{2}-x+1 \right ) }{2}}+{\frac{\ln \left ( 2\,{x}^{2}+x+1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-2*x^2+1)/(4*x^4+3*x^2+1),x)

[Out]

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

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Maxima [A]  time = 0.744593, size = 34, normalized size = 1.17 \[ \frac{1}{2} \, \log \left (2 \, x^{2} + x + 1\right ) - \frac{1}{2} \, \log \left (2 \, x^{2} - x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x^2 - 1)/(4*x^4 + 3*x^2 + 1),x, algorithm="maxima")

[Out]

1/2*log(2*x^2 + x + 1) - 1/2*log(2*x^2 - x + 1)

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Fricas [A]  time = 0.282175, size = 34, normalized size = 1.17 \[ \frac{1}{2} \, \log \left (2 \, x^{2} + x + 1\right ) - \frac{1}{2} \, \log \left (2 \, x^{2} - x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x^2 - 1)/(4*x^4 + 3*x^2 + 1),x, algorithm="fricas")

[Out]

1/2*log(2*x^2 + x + 1) - 1/2*log(2*x^2 - x + 1)

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Sympy [A]  time = 0.19189, size = 26, normalized size = 0.9 \[ - \frac{\log{\left (x^{2} - \frac{x}{2} + \frac{1}{2} \right )}}{2} + \frac{\log{\left (x^{2} + \frac{x}{2} + \frac{1}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x**2+1)/(4*x**4+3*x**2+1),x)

[Out]

-log(x**2 - x/2 + 1/2)/2 + log(x**2 + x/2 + 1/2)/2

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GIAC/XCAS [A]  time = 0.271278, size = 34, normalized size = 1.17 \[ \frac{1}{2} \,{\rm ln}\left (2 \, x^{2} + x + 1\right ) - \frac{1}{2} \,{\rm ln}\left (2 \, x^{2} - x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x^2 - 1)/(4*x^4 + 3*x^2 + 1),x, algorithm="giac")

[Out]

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