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

Optimal. Leaf size=11 \[ -\frac{1}{2 \left (x^2+1\right )} \]

[Out]

-1/(2*(1 + x^2))

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Rubi [A]  time = 0.00759608, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{1}{2 \left (x^2+1\right )} \]

Antiderivative was successfully verified.

[In]  Int[x/(1 + 2*x^2 + x^4),x]

[Out]

-1/(2*(1 + x^2))

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Rubi in Sympy [A]  time = 2.74399, size = 8, normalized size = 0.73 \[ - \frac{1}{2 \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*(x**2 + 1))

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Mathematica [A]  time = 0.00225236, size = 11, normalized size = 1. \[ -\frac{1}{2 \left (x^2+1\right )} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*(1 + x^2))

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Maple [A]  time = 0.004, size = 10, normalized size = 0.9 \[ -{\frac{1}{2\,{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^4+2*x^2+1),x)

[Out]

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

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Maxima [A]  time = 0.697025, size = 12, normalized size = 1.09 \[ -\frac{1}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(x^2 + 1)

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Fricas [A]  time = 0.248907, size = 12, normalized size = 1.09 \[ -\frac{1}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(x^2 + 1)

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Sympy [A]  time = 0.140513, size = 8, normalized size = 0.73 \[ - \frac{1}{2 x^{2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**4+2*x**2+1),x)

[Out]

-1/(2*x**2 + 2)

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GIAC/XCAS [A]  time = 0.26804, size = 12, normalized size = 1.09 \[ -\frac{1}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(x^2 + 1)