\(\int -\frac {12}{-12 x+x^2} \, dx\) [7587]

Optimal result

Integrand size = 11, antiderivative size = 11 \[ \int -\frac {12}{-12 x+x^2} \, dx=\log \left (\frac {4 x}{12-x}\right ) \]

[Out]

ln(4/(12-x)*x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 629} \[ \int -\frac {12}{-12 x+x^2} \, dx=\log (x)-\log (12-x) \]

[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 629

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.73 \[ \int -\frac {12}{-12 x+x^2} \, dx=-12 \left (\frac {1}{12} \log (12-x)-\frac {\log (x)}{12}\right ) \]

[In]

Integrate[-12/(-12*x + x^2),x]

[Out]

-12*(Log[12 - x]/12 - Log[x]/12)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

ln(x)-ln(x-12)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int -\frac {12}{-12 x+x^2} \, dx=-\log \left (x - 12\right ) + \log \left (x\right ) \]

[In]

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

[Out]

-log(x - 12) + log(x)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.64 \[ \int -\frac {12}{-12 x+x^2} \, dx=\log {\left (x \right )} - \log {\left (x - 12 \right )} \]

[In]

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

[Out]

log(x) - log(x - 12)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int -\frac {12}{-12 x+x^2} \, dx=-\log \left (x - 12\right ) + \log \left (x\right ) \]

[In]

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

[Out]

-log(x - 12) + log(x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int -\frac {12}{-12 x+x^2} \, dx=-\log \left ({\left | x - 12 \right |}\right ) + \log \left ({\left | x \right |}\right ) \]

[In]

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

[Out]

-log(abs(x - 12)) + log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int -\frac {12}{-12 x+x^2} \, dx=2\,\mathrm {atanh}\left (\frac {x}{6}-1\right ) \]

[In]

int(12/(12*x - x^2),x)

[Out]

2*atanh(x/6 - 1)