3.37.68 \(\int \frac {8-8 x}{(-2 x+x^2) \log (-4 x+2 x^2)} \, dx\)

Optimal. Leaf size=18 \[ \log \left (\frac {256}{9 e^{16} \log ^4(x (-4+2 x))}\right ) \]

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Rubi [A]  time = 0.10, antiderivative size = 13, normalized size of antiderivative = 0.72, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1593, 6684} \begin {gather*} -4 \log \left (\log \left (2 x^2-4 x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8 - 8*x)/((-2*x + x^2)*Log[-4*x + 2*x^2]),x]

[Out]

-4*Log[Log[-4*x + 2*x^2]]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6684

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8-8 x}{(-2+x) x \log \left (-4 x+2 x^2\right )} \, dx\\ &=-4 \log \left (\log \left (-4 x+2 x^2\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 10, normalized size = 0.56 \begin {gather*} -4 \log (\log (2 (-2+x) x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 - 8*x)/((-2*x + x^2)*Log[-4*x + 2*x^2]),x]

[Out]

-4*Log[Log[2*(-2 + x)*x]]

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fricas [A]  time = 0.90, size = 13, normalized size = 0.72 \begin {gather*} -4 \, \log \left (\log \left (2 \, x^{2} - 4 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x+8)/(x^2-2*x)/log(2*x^2-4*x),x, algorithm="fricas")

[Out]

-4*log(log(2*x^2 - 4*x))

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giac [A]  time = 0.15, size = 13, normalized size = 0.72 \begin {gather*} -4 \, \log \left (\log \left (2 \, x^{2} - 4 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x+8)/(x^2-2*x)/log(2*x^2-4*x),x, algorithm="giac")

[Out]

-4*log(log(2*x^2 - 4*x))

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maple [A]  time = 0.08, size = 14, normalized size = 0.78

method	result	size
norman	\(-4 \ln \left (\ln \left (2 x^{2}-4 x \right )\right )\)	\(14\)
risch	\(-4 \ln \left (\ln \left (2 x^{2}-4 x \right )\right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*x+8)/(x^2-2*x)/ln(2*x^2-4*x),x,method=_RETURNVERBOSE)

[Out]

-4*ln(ln(2*x^2-4*x))

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maxima [A]  time = 0.50, size = 12, normalized size = 0.67 \begin {gather*} -4 \, \log \left (\log \relax (2) + \log \left (x - 2\right ) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x+8)/(x^2-2*x)/log(2*x^2-4*x),x, algorithm="maxima")

[Out]

-4*log(log(2) + log(x - 2) + log(x))

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mupad [B]  time = 2.39, size = 13, normalized size = 0.72 \begin {gather*} -4\,\ln \left (\ln \left (2\,x^2-4\,x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x - 8)/(log(2*x^2 - 4*x)*(2*x - x^2)),x)

[Out]

-4*log(log(2*x^2 - 4*x))

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sympy [A]  time = 0.16, size = 14, normalized size = 0.78 \begin {gather*} - 4 \log {\left (\log {\left (2 x^{2} - 4 x \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x+8)/(x**2-2*x)/ln(2*x**2-4*x),x)

[Out]

-4*log(log(2*x**2 - 4*x))

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