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

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

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Rubi [A]  time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {616, 31} \[ \frac {1}{4} \log (5-x)-\frac {1}{4} \log (1-x) \]

Antiderivative was successfully verified.

[In]

Int[(5 - 6*x + x^2)^(-1),x]

[Out]

-Log[1 - x]/4 + Log[5 - x]/4

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {1}{5-6 x+x^2} \, dx &=\frac {1}{4} \int \frac {1}{-5+x} \, dx-\frac {1}{4} \int \frac {1}{-1+x} \, dx\\ &=-\frac {1}{4} \log (1-x)+\frac {1}{4} \log (5-x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \[ \frac {1}{4} \log (5-x)-\frac {1}{4} \log (1-x) \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - 6*x + x^2)^(-1),x]

[Out]

-1/4*Log[1 - x] + Log[5 - x]/4

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IntegrateAlgebraic [A]  time = 0.01, size = 14, normalized size = 0.67 \[ \frac {1}{2} \tanh ^{-1}\left (\frac {3}{2}-\frac {x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[3/2 - x/2]/2

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fricas [A]  time = 0.70, size = 13, normalized size = 0.62 \[ -\frac {1}{4} \, \log \left (x - 1\right ) + \frac {1}{4} \, \log \left (x - 5\right ) \]

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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giac [A]  time = 0.85, size = 15, normalized size = 0.71 \[ -\frac {1}{4} \, \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | x - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.31, size = 14, normalized size = 0.67

method	result	size
default	\(\frac {\ln \left (-5+x \right )}{4}-\frac {\ln \left (-1+x \right )}{4}\)	\(14\)
norman	\(\frac {\ln \left (-5+x \right )}{4}-\frac {\ln \left (-1+x \right )}{4}\)	\(14\)
risch	\(\frac {\ln \left (-5+x \right )}{4}-\frac {\ln \left (-1+x \right )}{4}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2-6*x+5),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.42, size = 13, normalized size = 0.62 \[ -\frac {1}{4} \, \log \left (x - 1\right ) + \frac {1}{4} \, \log \left (x - 5\right ) \]

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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mupad [B]  time = 0.26, size = 8, normalized size = 0.38 \[ -\frac {\mathrm {atanh}\left (\frac {x}{2}-\frac {3}{2}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2 - 6*x + 5),x)

[Out]

-atanh(x/2 - 3/2)/2

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sympy [A]  time = 0.10, 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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