3.100 \(\int \frac{1}{(-4+x) (-3+x) (-2+x) (-1+x)} \, dx\)

Optimal. Leaf size=41 \[ -\frac{1}{6} \log (1-x)+\frac{1}{2} \log (2-x)-\frac{1}{2} \log (3-x)+\frac{1}{6} \log (4-x) \]

[Out]

-Log[1 - x]/6 + Log[2 - x]/2 - Log[3 - x]/2 + Log[4 - x]/6

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Rubi [A]  time = 0.0666397, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{1}{6} \log (1-x)+\frac{1}{2} \log (2-x)-\frac{1}{2} \log (3-x)+\frac{1}{6} \log (4-x) \]

Antiderivative was successfully verified.

[In]  Int[1/((-4 + x)*(-3 + x)*(-2 + x)*(-1 + x)),x]

[Out]

-Log[1 - x]/6 + Log[2 - x]/2 - Log[3 - x]/2 + Log[4 - x]/6

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Rubi in Sympy [A]  time = 3.77353, size = 26, normalized size = 0.63 \[ - \frac{\log{\left (- x + 1 \right )}}{6} + \frac{\log{\left (- x + 2 \right )}}{2} - \frac{\log{\left (- x + 3 \right )}}{2} + \frac{\log{\left (- x + 4 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-4+x)/(-3+x)/(-2+x)/(-1+x),x)

[Out]

-log(-x + 1)/6 + log(-x + 2)/2 - log(-x + 3)/2 + log(-x + 4)/6

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Mathematica [A]  time = 0.0118567, size = 41, normalized size = 1. \[ -\frac{1}{6} \log (1-x)+\frac{1}{2} \log (2-x)-\frac{1}{2} \log (3-x)+\frac{1}{6} \log (4-x) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((-4 + x)*(-3 + x)*(-2 + x)*(-1 + x)),x]

[Out]

-Log[1 - x]/6 + Log[2 - x]/2 - Log[3 - x]/2 + Log[4 - x]/6

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Maple [A]  time = 0.013, size = 26, normalized size = 0.6 \[ -{\frac{\ln \left ( -3+x \right ) }{2}}+{\frac{\ln \left ( x-4 \right ) }{6}}-{\frac{\ln \left ( -1+x \right ) }{6}}+{\frac{\ln \left ( -2+x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x-4)/(-3+x)/(-2+x)/(-1+x),x)

[Out]

-1/2*ln(-3+x)+1/6*ln(x-4)-1/6*ln(-1+x)+1/2*ln(-2+x)

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Maxima [A]  time = 1.37337, size = 34, normalized size = 0.83 \[ -\frac{1}{6} \, \log \left (x - 1\right ) + \frac{1}{2} \, \log \left (x - 2\right ) - \frac{1}{2} \, \log \left (x - 3\right ) + \frac{1}{6} \, \log \left (x - 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x - 1)*(x - 2)*(x - 3)*(x - 4)),x, algorithm="maxima")

[Out]

-1/6*log(x - 1) + 1/2*log(x - 2) - 1/2*log(x - 3) + 1/6*log(x - 4)

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Fricas [A]  time = 0.207978, size = 34, normalized size = 0.83 \[ -\frac{1}{6} \, \log \left (x - 1\right ) + \frac{1}{2} \, \log \left (x - 2\right ) - \frac{1}{2} \, \log \left (x - 3\right ) + \frac{1}{6} \, \log \left (x - 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x - 1)*(x - 2)*(x - 3)*(x - 4)),x, algorithm="fricas")

[Out]

-1/6*log(x - 1) + 1/2*log(x - 2) - 1/2*log(x - 3) + 1/6*log(x - 4)

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Sympy [A]  time = 0.228868, size = 26, normalized size = 0.63 \[ \frac{\log{\left (x - 4 \right )}}{6} - \frac{\log{\left (x - 3 \right )}}{2} + \frac{\log{\left (x - 2 \right )}}{2} - \frac{\log{\left (x - 1 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-4+x)/(-3+x)/(-2+x)/(-1+x),x)

[Out]

log(x - 4)/6 - log(x - 3)/2 + log(x - 2)/2 - log(x - 1)/6

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GIAC/XCAS [A]  time = 0.201241, size = 39, normalized size = 0.95 \[ -\frac{1}{6} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | x - 2 \right |}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | x - 3 \right |}\right ) + \frac{1}{6} \,{\rm ln}\left ({\left | x - 4 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x - 1)*(x - 2)*(x - 3)*(x - 4)),x, algorithm="giac")

[Out]

-1/6*ln(abs(x - 1)) + 1/2*ln(abs(x - 2)) - 1/2*ln(abs(x - 3)) + 1/6*ln(abs(x - 4
))