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

Optimal. Leaf size=11 \[ \frac{x}{2 x^2+1} \]

[Out]

x/(1 + 2*x^2)

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Rubi [A]  time = 0.0103355, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x}{2 x^2+1} \]

Antiderivative was successfully verified.

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

[Out]

x/(1 + 2*x^2)

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Rubi in Sympy [A]  time = 6.09194, size = 8, normalized size = 0.73 \[ \frac{2 x}{4 x^{2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x/(4*x**2 + 2)

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Mathematica [A]  time = 0.00683356, size = 11, normalized size = 1. \[ \frac{x}{2 x^2+1} \]

Antiderivative was successfully verified.

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

[Out]

x/(1 + 2*x^2)

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Maple [A]  time = 0.008, size = 11, normalized size = 1. \[{\frac{x}{2} \left ({x}^{2}+{\frac{1}{2}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.73177, size = 15, normalized size = 1.36 \[ \frac{x}{2 \, x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(2*x^2 + 1)

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Fricas [A]  time = 0.274797, size = 15, normalized size = 1.36 \[ \frac{x}{2 \, x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(2*x^2 + 1)

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Sympy [A]  time = 0.154271, size = 7, normalized size = 0.64 \[ \frac{x}{2 x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(2*x**2 + 1)

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GIAC/XCAS [A]  time = 0.27068, size = 15, normalized size = 1.36 \[ \frac{x}{2 \, x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(2*x^2 + 1)