3.39.81 \(\int -\frac {1}{-x+x^2} \, dx\)

Optimal. Leaf size=13 \[ \log \left (\frac {x}{6 (1-x)}\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 0.85, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {615} \begin {gather*} \log (x)-\log (1-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-(-x + x^2)^(-1),x]

[Out]

-Log[1 - x] + Log[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} &=-\log (1-x)+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 0.85 \begin {gather*} -\log (1-x)+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-(-x + x^2)^(-1),x]

[Out]

-Log[1 - x] + Log[x]

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fricas [A]  time = 1.14, size = 9, normalized size = 0.69 \begin {gather*} -\log \left (x - 1\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/(x^2-x),x, algorithm="fricas")

[Out]

-log(x - 1) + log(x)

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giac [A]  time = 0.12, size = 11, normalized size = 0.85 \begin {gather*} -\log \left ({\left | x - 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/(x^2-x),x, algorithm="giac")

[Out]

-log(abs(x - 1)) + log(abs(x))

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maple [A]  time = 0.16, size = 10, normalized size = 0.77

method	result	size
default	\(\ln \relax (x )-\ln \left (x -1\right )\)	\(10\)
norman	\(\ln \relax (x )-\ln \left (x -1\right )\)	\(10\)
risch	\(\ln \relax (x )-\ln \left (x -1\right )\)	\(10\)
meijerg	\(-\ln \left (1-x \right )+\ln \relax (x )+i \pi \)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(x^2-x),x,method=_RETURNVERBOSE)

[Out]

ln(x)-ln(x-1)

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maxima [A]  time = 0.34, size = 9, normalized size = 0.69 \begin {gather*} -\log \left (x - 1\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/(x^2-x),x, algorithm="maxima")

[Out]

-log(x - 1) + log(x)

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mupad [B]  time = 2.26, size = 8, normalized size = 0.62 \begin {gather*} 2\,\mathrm {atanh}\left (2\,x-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x - x^2),x)

[Out]

2*atanh(2*x - 1)

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sympy [A]  time = 0.09, size = 7, normalized size = 0.54 \begin {gather*} \log {\relax (x )} - \log {\left (x - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/(x**2-x),x)

[Out]

log(x) - log(x - 1)

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