3.76.87 \(\int -\frac {12}{-12 x+x^2} \, dx\)

Optimal. Leaf size=11 \[ \log \left (\frac {4 x}{12-x}\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 615} \begin {gather*} \log (x)-\log (12-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-12/(-12*x + x^2),x]

[Out]

-Log[12 - x] + Log[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 615

Int[((b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[Log[x]/b, x] - Simp[Log[RemoveContent[b + c*x, x]]/b,
x] /; FreeQ[{b, c}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (12 \int \frac {1}{-12 x+x^2} \, dx\right )\\ &=-\log (12-x)+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.73 \begin {gather*} -12 \left (\frac {1}{12} \log (12-x)-\frac {\log (x)}{12}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-12/(-12*x + x^2),x]

[Out]

-12*(Log[12 - x]/12 - Log[x]/12)

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fricas [A]  time = 0.75, size = 9, normalized size = 0.82 \begin {gather*} -\log \left (x - 12\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-12/(x^2-12*x),x, algorithm="fricas")

[Out]

-log(x - 12) + log(x)

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giac [A]  time = 0.14, size = 11, normalized size = 1.00 \begin {gather*} -\log \left ({\left | x - 12 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-12/(x^2-12*x),x, algorithm="giac")

[Out]

-log(abs(x - 12)) + log(abs(x))

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maple [A]  time = 0.22, size = 10, normalized size = 0.91

method	result	size
default	\(\ln \relax (x )-\ln \left (x -12\right )\)	\(10\)
norman	\(\ln \relax (x )-\ln \left (x -12\right )\)	\(10\)
risch	\(\ln \relax (x )-\ln \left (x -12\right )\)	\(10\)
meijerg	\(\ln \relax (x )-2 \ln \relax (2)-\ln \relax (3)+i \pi -\ln \left (1-\frac {x}{12}\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-12/(x^2-12*x),x,method=_RETURNVERBOSE)

[Out]

ln(x)-ln(x-12)

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maxima [A]  time = 0.42, size = 9, normalized size = 0.82 \begin {gather*} -\log \left (x - 12\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-12/(x^2-12*x),x, algorithm="maxima")

[Out]

-log(x - 12) + log(x)

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mupad [B]  time = 4.89, size = 8, normalized size = 0.73 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {x}{6}-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(12/(12*x - x^2),x)

[Out]

2*atanh(x/6 - 1)

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sympy [A]  time = 0.09, size = 7, normalized size = 0.64 \begin {gather*} \log {\relax (x )} - \log {\left (x - 12 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-12/(x**2-12*x),x)

[Out]

log(x) - log(x - 12)

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