3.31 \(\int \frac{1}{x-x^3} \, dx\)

Optimal. Leaf size=15 \[ \log (x)-\frac{1}{2} \log \left (1-x^2\right ) \]

[Out]

Log[x] - Log[1 - x^2]/2

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Rubi [A]  time = 0.0208824, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556 \[ \log (x)-\frac{1}{2} \log \left (1-x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x - x^3)^(-1),x]

[Out]

Log[x] - Log[1 - x^2]/2

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-x**3+x),x)

[Out]

Exception raised: TypeError

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Mathematica [A]  time = 0.00394251, size = 15, normalized size = 1. \[ \log (x)-\frac{1}{2} \log \left (1-x^2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(x - x^3)^(-1),x]

[Out]

Log[x] - Log[1 - x^2]/2

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Maple [A]  time = 0.009, size = 16, normalized size = 1.1 \[ \ln \left ( x \right ) -{\frac{\ln \left ( -1+x \right ) }{2}}-{\frac{\ln \left ( 1+x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^3+x),x)

[Out]

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

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Maxima [A]  time = 1.36284, size = 20, normalized size = 1.33 \[ -\frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(x^3 - x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.201793, size = 15, normalized size = 1. \[ -\frac{1}{2} \, \log \left (x^{2} - 1\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(x^3 - x),x, algorithm="fricas")

[Out]

-1/2*log(x^2 - 1) + log(x)

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Sympy [A]  time = 0.175263, size = 10, normalized size = 0.67 \[ \log{\left (x \right )} - \frac{\log{\left (x^{2} - 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-x**3+x),x)

[Out]

log(x) - log(x**2 - 1)/2

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GIAC/XCAS [A]  time = 0.221381, size = 22, normalized size = 1.47 \[ \frac{1}{2} \,{\rm ln}\left (x^{2}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | x^{2} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(x^3 - x),x, algorithm="giac")

[Out]

1/2*ln(x^2) - 1/2*ln(abs(x^2 - 1))