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

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

[Out]

x/(1 - x^2)

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Rubi [A]  time = 0.00728537, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{x}{1-x^2} \]

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 = 3.67251, size = 5, normalized size = 0.45 \[ \frac{x}{- x^{2} + 1} \]

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 = 0.00643358, size = 10, normalized size = 0.91 \[ -\frac{x}{x^2-1} \]

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 = 0.008, size = 16, normalized size = 1.5 \[ -{\frac{1}{-2+2\,x}}-{\frac{1}{2+2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+1)/(x^2-1)^2,x)

[Out]

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

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Maxima [A]  time = 1.33919, size = 14, normalized size = 1.27 \[ -\frac{x}{x^{2} - 1} \]

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 = 0.230401, size = 14, normalized size = 1.27 \[ -\frac{x}{x^{2} - 1} \]

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 = 0.166185, size = 7, normalized size = 0.64 \[ - \frac{x}{x^{2} - 1} \]

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 = 0.218714, size = 15, normalized size = 1.36 \[ -\frac{1}{x - \frac{1}{x}} \]

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)