3.2.14 \(\int \frac {2}{6-11 x+6 x^2-x^3} \, dx\) [114]

Optimal. Leaf size=25 \[ -\log (1-x)+2 \log (2-x)-\log (3-x) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 2083} \begin {gather*} -\log (1-x)+2 \log (2-x)-\log (3-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2/(6 - 11*x + 6*x^2 - x^3),x]

[Out]

-Log[1 - x] + 2*Log[2 - x] - Log[3 - x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2083

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[ExpandIntegrand[u^p, x], x] /;  !SumQ[NonfreeFactors[u,
x]]] /; PolyQ[P, x] && ILtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=2 \int \frac {1}{6-11 x+6 x^2-x^3} \, dx\\ &=2 \int \left (-\frac {1}{2 (-3+x)}+\frac {1}{-2+x}-\frac {1}{2 (-1+x)}\right ) \, dx\\ &=-\log (1-x)+2 \log (2-x)-\log (3-x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 24, normalized size = 0.96 \begin {gather*} -2 \left (-\log (2-x)+\frac {1}{2} \log \left (3-4 x+x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2/(6 - 11*x + 6*x^2 - x^3),x]

[Out]

-2*(-Log[2 - x] + Log[3 - 4*x + x^2]/2)

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Maple [A]
time = 0.02, size = 20, normalized size = 0.80

method	result	size
risch	\(2 \ln \left (x -2\right )-\ln \left (x^{2}-4 x +3\right )\)	\(19\)
default	\(2 \ln \left (x -2\right )-\ln \left (-3+x \right )-\ln \left (x -1\right )\)	\(20\)
norman	\(2 \ln \left (x -2\right )-\ln \left (-3+x \right )-\ln \left (x -1\right )\)	\(20\)
parallelrisch	\(2 \ln \left (x -2\right )-\ln \left (-3+x \right )-\ln \left (x -1\right )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/(-x^3+6*x^2-11*x+6),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.34, size = 19, normalized size = 0.76 \begin {gather*} -\log \left (x - 1\right ) + 2 \, \log \left (x - 2\right ) - \log \left (x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(-x^3+6*x^2-11*x+6),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.56, size = 18, normalized size = 0.72 \begin {gather*} -\log \left (x^{2} - 4 \, x + 3\right ) + 2 \, \log \left (x - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(-x^3+6*x^2-11*x+6),x, algorithm="fricas")

[Out]

-log(x^2 - 4*x + 3) + 2*log(x - 2)

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Sympy [A]
time = 0.05, size = 15, normalized size = 0.60 \begin {gather*} 2 \log {\left (x - 2 \right )} - \log {\left (x^{2} - 4 x + 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(-x**3+6*x**2-11*x+6),x)

[Out]

2*log(x - 2) - log(x**2 - 4*x + 3)

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Giac [A]
time = 0.43, size = 22, normalized size = 0.88 \begin {gather*} -\log \left ({\left | x - 1 \right |}\right ) + 2 \, \log \left ({\left | x - 2 \right |}\right ) - \log \left ({\left | x - 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(-x^3+6*x^2-11*x+6),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.06, size = 18, normalized size = 0.72 \begin {gather*} 2\,\ln \left (x-2\right )-\ln \left (x^2-4\,x+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-2/(11*x - 6*x^2 + x^3 - 6),x)

[Out]

2*log(x - 2) - log(x^2 - 4*x + 3)

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Chatgpt [F] Failed to verify
time = 1.00, size = 21, normalized size = 0.84 \begin {gather*} \frac {\ln \left (x -1\right )}{8}-\frac {\ln \left (x -2\right )}{8}+\frac {\arctan \left (2 x -11\right )}{4} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(2/(-x^3+6*x^2-11*x+6),x)

[Out]

1/8*ln(x-1)-1/8*ln(x-2)+1/4*arctan(2*x-11)

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