3.436 \(\int \frac{-2+4 x}{-x+x^3} \, dx\)

Optimal. Leaf size=17 \[ \log (1-x)+2 \log (x)-3 \log (x+1) \]

[Out]

Log[1 - x] + 2*Log[x] - 3*Log[1 + x]

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Rubi [A]  time = 0.0436409, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \log (1-x)+2 \log (x)-3 \log (x+1) \]

Antiderivative was successfully verified.

[In]  Int[(-2 + 4*x)/(-x + x^3),x]

[Out]

Log[1 - x] + 2*Log[x] - 3*Log[1 + x]

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Rubi in Sympy [A]  time = 6.5731, size = 15, normalized size = 0.88 \[ 2 \log{\left (x \right )} + \log{\left (- x + 1 \right )} - 3 \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-2+4*x)/(x**3-x),x)

[Out]

2*log(x) + log(-x + 1) - 3*log(x + 1)

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Mathematica [A]  time = 0.00896208, size = 17, normalized size = 1. \[ \log (1-x)+2 \log (x)-3 \log (x+1) \]

Antiderivative was successfully verified.

[In]  Integrate[(-2 + 4*x)/(-x + x^3),x]

[Out]

Log[1 - x] + 2*Log[x] - 3*Log[1 + x]

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Maple [A]  time = 0.01, size = 16, normalized size = 0.9 \[ \ln \left ( -1+x \right ) -3\,\ln \left ( 1+x \right ) +2\,\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-2+4*x)/(x^3-x),x)

[Out]

ln(-1+x)-3*ln(1+x)+2*ln(x)

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Maxima [A]  time = 0.794937, size = 20, normalized size = 1.18 \[ -3 \, \log \left (x + 1\right ) + \log \left (x - 1\right ) + 2 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*log(x + 1) + log(x - 1) + 2*log(x)

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Fricas [A]  time = 0.264144, size = 20, normalized size = 1.18 \[ -3 \, \log \left (x + 1\right ) + \log \left (x - 1\right ) + 2 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*log(x + 1) + log(x - 1) + 2*log(x)

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Sympy [A]  time = 0.252031, size = 15, normalized size = 0.88 \[ 2 \log{\left (x \right )} + \log{\left (x - 1 \right )} - 3 \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2+4*x)/(x**3-x),x)

[Out]

2*log(x) + log(x - 1) - 3*log(x + 1)

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GIAC/XCAS [A]  time = 0.260208, size = 24, normalized size = 1.41 \[ -3 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) +{\rm ln}\left ({\left | x - 1 \right |}\right ) + 2 \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*ln(abs(x + 1)) + ln(abs(x - 1)) + 2*ln(abs(x))