3.44 \(\int \frac{1}{5-6 x+x^2} \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{4} \log (5-x)-\frac{1}{4} \log (1-x) \]

[Out]

-Log[1 - x]/4 + Log[5 - x]/4

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Rubi [A]  time = 0.0111936, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{4} \log (5-x)-\frac{1}{4} \log (1-x) \]

Antiderivative was successfully verified.

[In]  Int[(5 - 6*x + x^2)^(-1),x]

[Out]

-Log[1 - x]/4 + Log[5 - x]/4

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Rubi in Sympy [A]  time = 0.739674, size = 12, normalized size = 0.57 \[ - \frac{\log{\left (- x + 1 \right )}}{4} + \frac{\log{\left (- x + 5 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**2-6*x+5),x)

[Out]

-log(-x + 1)/4 + log(-x + 5)/4

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Mathematica [A]  time = 0.00445992, size = 21, normalized size = 1. \[ \frac{1}{4} \log (5-x)-\frac{1}{4} \log (1-x) \]

Antiderivative was successfully verified.

[In]  Integrate[(5 - 6*x + x^2)^(-1),x]

[Out]

-Log[1 - x]/4 + Log[5 - x]/4

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Maple [A]  time = 0.007, size = 14, normalized size = 0.7 \[{\frac{\ln \left ( -5+x \right ) }{4}}-{\frac{\ln \left ( -1+x \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2-6*x+5),x)

[Out]

1/4*ln(-5+x)-1/4*ln(-1+x)

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Maxima [A]  time = 1.33546, size = 18, normalized size = 0.86 \[ -\frac{1}{4} \, \log \left (x - 1\right ) + \frac{1}{4} \, \log \left (x - 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^2 - 6*x + 5),x, algorithm="maxima")

[Out]

-1/4*log(x - 1) + 1/4*log(x - 5)

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Fricas [A]  time = 0.208437, size = 18, normalized size = 0.86 \[ -\frac{1}{4} \, \log \left (x - 1\right ) + \frac{1}{4} \, \log \left (x - 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^2 - 6*x + 5),x, algorithm="fricas")

[Out]

-1/4*log(x - 1) + 1/4*log(x - 5)

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Sympy [A]  time = 0.087385, size = 12, normalized size = 0.57 \[ \frac{\log{\left (x - 5 \right )}}{4} - \frac{\log{\left (x - 1 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**2-6*x+5),x)

[Out]

log(x - 5)/4 - log(x - 1)/4

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GIAC/XCAS [A]  time = 0.203373, size = 20, normalized size = 0.95 \[ -\frac{1}{4} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | x - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^2 - 6*x + 5),x, algorithm="giac")

[Out]

-1/4*ln(abs(x - 1)) + 1/4*ln(abs(x - 5))