3.433 \(\int \frac{-1+x^2}{-2 x+x^3} \, dx\)

Optimal. Leaf size=19 \[ \frac{1}{4} \log \left (2-x^2\right )+\frac{\log (x)}{2} \]

[Out]

Log[x]/2 + Log[2 - x^2]/4

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Rubi [A]  time = 0.0496399, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{4} \log \left (2-x^2\right )+\frac{\log (x)}{2} \]

Antiderivative was successfully verified.

[In]  Int[(-1 + x^2)/(-2*x + x^3),x]

[Out]

Log[x]/2 + Log[2 - x^2]/4

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Rubi in Sympy [A]  time = 8.20481, size = 14, normalized size = 0.74 \[ \frac{\log{\left (x^{2} \right )}}{4} + \frac{\log{\left (- x^{2} + 2 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2-1)/(x**3-2*x),x)

[Out]

log(x**2)/4 + log(-x**2 + 2)/4

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Mathematica [A]  time = 0.00559458, size = 19, normalized size = 1. \[ \frac{1}{4} \log \left (2-x^2\right )+\frac{\log (x)}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[(-1 + x^2)/(-2*x + x^3),x]

[Out]

Log[x]/2 + Log[2 - x^2]/4

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Maple [A]  time = 0.007, size = 14, normalized size = 0.7 \[{\frac{\ln \left ( x \right ) }{2}}+{\frac{\ln \left ({x}^{2}-2 \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2-1)/(x^3-2*x),x)

[Out]

1/2*ln(x)+1/4*ln(x^2-2)

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Maxima [A]  time = 0.78842, size = 18, normalized size = 0.95 \[ \frac{1}{4} \, \log \left (x^{2} - 2\right ) + \frac{1}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 1)/(x^3 - 2*x),x, algorithm="maxima")

[Out]

1/4*log(x^2 - 2) + 1/2*log(x)

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Fricas [A]  time = 0.258814, size = 18, normalized size = 0.95 \[ \frac{1}{4} \, \log \left (x^{2} - 2\right ) + \frac{1}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 1)/(x^3 - 2*x),x, algorithm="fricas")

[Out]

1/4*log(x^2 - 2) + 1/2*log(x)

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Sympy [A]  time = 0.178408, size = 12, normalized size = 0.63 \[ \frac{\log{\left (x \right )}}{2} + \frac{\log{\left (x^{2} - 2 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2-1)/(x**3-2*x),x)

[Out]

log(x)/2 + log(x**2 - 2)/4

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GIAC/XCAS [A]  time = 0.259422, size = 22, normalized size = 1.16 \[ \frac{1}{4} \,{\rm ln}\left (x^{2}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | x^{2} - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 1)/(x^3 - 2*x),x, algorithm="giac")

[Out]

1/4*ln(x^2) + 1/4*ln(abs(x^2 - 2))