3.110 \(\int \frac{1-x^2}{\left (1+x^2\right )^2} \, dx\)

Optimal. Leaf size=9 \[ \frac{x}{x^2+1} \]

[Out]

x/(1 + x^2)

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Rubi [A]  time = 0.0071113, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{x}{x^2+1} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^2)/(1 + x^2)^2,x]

[Out]

x/(1 + x^2)

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Rubi in Sympy [A]  time = 4.16044, size = 5, normalized size = 0.56 \[ \frac{x}{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(x**2 + 1)

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Mathematica [A]  time = 0.00642078, size = 9, normalized size = 1. \[ \frac{x}{x^2+1} \]

Antiderivative was successfully verified.

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

[Out]

x/(1 + x^2)

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Maple [A]  time = 0.009, size = 10, normalized size = 1.1 \[{\frac{x}{{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+1)/(x^2+1)^2,x)

[Out]

x/(x^2+1)

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Maxima [A]  time = 1.33772, size = 12, normalized size = 1.33 \[ \frac{x}{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(x^2 + 1)

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Fricas [A]  time = 0.218094, size = 12, normalized size = 1.33 \[ \frac{x}{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(x^2 + 1)

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Sympy [A]  time = 0.175413, size = 5, normalized size = 0.56 \[ \frac{x}{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(x**2 + 1)

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GIAC/XCAS [A]  time = 0.240044, size = 9, normalized size = 1. \[ \frac{1}{x + \frac{1}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(x + 1/x)