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

Optimal. Leaf size=13 \[ \log \left (\frac {1}{4} (2-x) x \log (4)\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 0.77, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {628} \begin {gather*} \log \left (2 x-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 + 2*x)/(-2*x + x^2),x]

[Out]

Log[2*x - x^2]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (2 x-x^2\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 19, normalized size = 1.46 \begin {gather*} 2 \left (\frac {1}{2} \log (2-x)+\frac {\log (x)}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 + 2*x)/(-2*x + x^2),x]

[Out]

2*(Log[2 - x]/2 + Log[x]/2)

________________________________________________________________________________________

fricas [A]  time = 0.64, size = 8, normalized size = 0.62 \begin {gather*} \log \left (x^{2} - 2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 - 2*x)

________________________________________________________________________________________

giac [A]  time = 0.21, size = 13, normalized size = 1.00 \begin {gather*} \log \left (2 \, {\left | \frac {1}{2} \, x^{2} - x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2*abs(1/2*x^2 - x))

________________________________________________________________________________________

maple [A]  time = 0.41, size = 7, normalized size = 0.54

method	result	size
default	\(\ln \left (\left (x -2\right ) x \right )\)	\(7\)
norman	\(\ln \relax (x )+\ln \left (x -2\right )\)	\(8\)
derivativedivides	\(\ln \left (x^{2}-2 x \right )\)	\(9\)
risch	\(\ln \left (x^{2}-2 x \right )\)	\(9\)
meijerg	\(\ln \left (1-\frac {x}{2}\right )+\ln \relax (x )-\ln \relax (2)+i \pi \)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln((x-2)*x)

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 8, normalized size = 0.62 \begin {gather*} \log \left (x^{2} - 2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 - 2*x)

________________________________________________________________________________________

mupad [B]  time = 0.06, size = 6, normalized size = 0.46 \begin {gather*} \ln \left (x\,\left (x-2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 2)/(2*x - x^2),x)

[Out]

log(x*(x - 2))

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 7, normalized size = 0.54 \begin {gather*} \log {\left (x^{2} - 2 x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x**2 - 2*x)

________________________________________________________________________________________