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

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

[Out]

x/(1 - 2*x^2)

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Rubi [A]  time = 0.00988683, 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 \[ \frac{x}{1-2 x^2} \]

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 = 6.04925, 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 = 0.00833844, size = 12, normalized size = 1.09 \[ -\frac{x}{2 x^2-1} \]

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 = 0.009, 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 = 0.732339, size = 16, normalized size = 1.45 \[ -\frac{x}{2 \, x^{2} - 1} \]

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 = 0.291718, size = 16, normalized size = 1.45 \[ -\frac{x}{2 \, x^{2} - 1} \]

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 = 0.15684, size = 8, normalized size = 0.73 \[ - \frac{x}{2 x^{2} - 1} \]

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 = 0.269831, size = 16, normalized size = 1.45 \[ -\frac{x}{2 \, x^{2} - 1} \]

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)