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

Optimal. Leaf size=8 \[ \frac{1}{x}-\tanh ^{-1}(x) \]

[Out]

x^(-1) - ArcTanh[x]

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Rubi [A]  time = 0.0119814, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{1}{x}-\tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(-x^2 + x^4)^(-1),x]

[Out]

x^(-1) - ArcTanh[x]

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Rubi in Sympy [A]  time = 1.70363, size = 5, normalized size = 0.62 \[ - \operatorname{atanh}{\left (x \right )} + \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**4-x**2),x)

[Out]

-atanh(x) + 1/x

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Mathematica [B]  time = 0.00399979, size = 22, normalized size = 2.75 \[ \frac{1}{x}+\frac{1}{2} \log (1-x)-\frac{1}{2} \log (x+1) \]

Antiderivative was successfully verified.

[In]  Integrate[(-x^2 + x^4)^(-1),x]

[Out]

x^(-1) + Log[1 - x]/2 - Log[1 + x]/2

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Maple [A]  time = 0.008, size = 17, normalized size = 2.1 \[ -{\frac{\ln \left ( 1+x \right ) }{2}}+{x}^{-1}+{\frac{\ln \left ( -1+x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^4-x^2),x)

[Out]

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

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Maxima [A]  time = 1.36111, size = 22, normalized size = 2.75 \[ \frac{1}{x} - \frac{1}{2} \, \log \left (x + 1\right ) + \frac{1}{2} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/x - 1/2*log(x + 1) + 1/2*log(x - 1)

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Fricas [A]  time = 0.195611, size = 27, normalized size = 3.38 \[ -\frac{x \log \left (x + 1\right ) - x \log \left (x - 1\right ) - 2}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.099833, size = 15, normalized size = 1.88 \[ \frac{\log{\left (x - 1 \right )}}{2} - \frac{\log{\left (x + 1 \right )}}{2} + \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**4-x**2),x)

[Out]

log(x - 1)/2 - log(x + 1)/2 + 1/x

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GIAC/XCAS [A]  time = 0.204851, size = 24, normalized size = 3. \[ \frac{1}{x} - \frac{1}{2} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/x - 1/2*ln(abs(x + 1)) + 1/2*ln(abs(x - 1))