3.9.64 \(\int \frac {-6-2 x}{-36-16 x+x^2+(2+x) \log (-2-x)} \, dx\)

Optimal. Leaf size=14 \[ \log \left (\frac {16}{(-18+x+\log (-2-x))^2}\right ) \]

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Rubi [A]  time = 0.15, antiderivative size = 16, normalized size of antiderivative = 1.14, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6741, 6684} \begin {gather*} -2 \log (-x-\log (-x-2)+18) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6 - 2*x)/(-36 - 16*x + x^2 + (2 + x)*Log[-2 - x]),x]

[Out]

-2*Log[18 - x - Log[-2 - x]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6+2 x}{(2+x) (18-x-\log (-2-x))} \, dx\\ &=-2 \log (18-x-\log (-2-x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 12, normalized size = 0.86 \begin {gather*} -2 \log (-18+x+\log (-2-x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 - 2*x)/(-36 - 16*x + x^2 + (2 + x)*Log[-2 - x]),x]

[Out]

-2*Log[-18 + x + Log[-2 - x]]

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fricas [A]  time = 0.65, size = 12, normalized size = 0.86 \begin {gather*} -2 \, \log \left (x + \log \left (-x - 2\right ) - 18\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-6)/((2+x)*log(-x-2)+x^2-16*x-36),x, algorithm="fricas")

[Out]

-2*log(x + log(-x - 2) - 18)

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giac [A]  time = 0.25, size = 12, normalized size = 0.86 \begin {gather*} -2 \, \log \left (x + \log \left (-x - 2\right ) - 18\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-6)/((2+x)*log(-x-2)+x^2-16*x-36),x, algorithm="giac")

[Out]

-2*log(x + log(-x - 2) - 18)

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maple [A]  time = 0.03, size = 13, normalized size = 0.93

method	result	size
norman	\(-2 \ln \left (\ln \left (-x -2\right )+x -18\right )\)	\(13\)
risch	\(-2 \ln \left (\ln \left (-x -2\right )+x -18\right )\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x-6)/((2+x)*ln(-x-2)+x^2-16*x-36),x,method=_RETURNVERBOSE)

[Out]

-2*ln(ln(-x-2)+x-18)

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maxima [A]  time = 0.55, size = 12, normalized size = 0.86 \begin {gather*} -2 \, \log \left (x + \log \left (-x - 2\right ) - 18\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-6)/((2+x)*log(-x-2)+x^2-16*x-36),x, algorithm="maxima")

[Out]

-2*log(x + log(-x - 2) - 18)

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mupad [B]  time = 0.24, size = 12, normalized size = 0.86 \begin {gather*} -2\,\ln \left (x+\ln \left (-x-2\right )-18\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 6)/(16*x - x^2 - log(- x - 2)*(x + 2) + 36),x)

[Out]

-2*log(x + log(- x - 2) - 18)

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sympy [A]  time = 0.15, size = 14, normalized size = 1.00 \begin {gather*} - 2 \log {\left (x + \log {\left (- x - 2 \right )} - 18 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-6)/((2+x)*ln(-x-2)+x**2-16*x-36),x)

[Out]

-2*log(x + log(-x - 2) - 18)

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