3.49.74 \(\int \frac {4+4 x+2 x^2}{-2 x^2-x^3+(2 x+x^2) \log (\frac {4+2 x}{x})} \, dx\)

Optimal. Leaf size=21 \[ \log \left (\frac {3 \log (3)}{\left (-x+\log \left (\frac {4+2 x}{x}\right )\right )^2}\right ) \]

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Rubi [A]  time = 0.17, antiderivative size = 15, normalized size of antiderivative = 0.71, number of steps used = 2, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6741, 6684} \begin {gather*} -2 \log \left (x-\log \left (\frac {4}{x}+2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 + 4*x + 2*x^2)/(-2*x^2 - x^3 + (2*x + x^2)*Log[(4 + 2*x)/x]),x]

[Out]

-2*Log[x - Log[2 + 4/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 {-4-4 x-2 x^2}{x (2+x) \left (x-\log \left (2+\frac {4}{x}\right )\right )} \, dx\\ &=-2 \log \left (x-\log \left (2+\frac {4}{x}\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 5.01, size = 15, normalized size = 0.71 \begin {gather*} -2 \log \left (x-\log \left (2+\frac {4}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + 4*x + 2*x^2)/(-2*x^2 - x^3 + (2*x + x^2)*Log[(4 + 2*x)/x]),x]

[Out]

-2*Log[x - Log[2 + 4/x]]

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fricas [A]  time = 0.93, size = 16, normalized size = 0.76 \begin {gather*} -2 \, \log \left (-x + \log \left (\frac {2 \, {\left (x + 2\right )}}{x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+4*x+4)/((x^2+2*x)*log((2*x+4)/x)-x^3-2*x^2),x, algorithm="fricas")

[Out]

-2*log(-x + log(2*(x + 2)/x))

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giac [B]  time = 0.19, size = 47, normalized size = 2.24 \begin {gather*} -2 \, \log \left (\frac {2 \, {\left (x + 2\right )} \log \left (\frac {2 \, {\left (x + 2\right )}}{x}\right )}{x} - 2 \, \log \left (\frac {2 \, {\left (x + 2\right )}}{x}\right ) - 4\right ) + 2 \, \log \left (\frac {2 \, {\left (x + 2\right )}}{x} - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+4*x+4)/((x^2+2*x)*log((2*x+4)/x)-x^3-2*x^2),x, algorithm="giac")

[Out]

-2*log(2*(x + 2)*log(2*(x + 2)/x)/x - 2*log(2*(x + 2)/x) - 4) + 2*log(2*(x + 2)/x - 2)

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maple [A]  time = 0.12, size = 18, normalized size = 0.86

method	result	size
norman	\(-2 \ln \left (x -\ln \left (\frac {2 x +4}{x}\right )\right )\)	\(18\)
risch	\(-2 \ln \left (\ln \left (\frac {2 x +4}{x}\right )-x \right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2+4*x+4)/((x^2+2*x)*ln((2*x+4)/x)-x^3-2*x^2),x,method=_RETURNVERBOSE)

[Out]

-2*ln(x-ln((2*x+4)/x))

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maxima [A]  time = 0.49, size = 17, normalized size = 0.81 \begin {gather*} -2 \, \log \left (-x + \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((2*x^2+4*x+4)/((x^2+2*x)*log((2*x+4)/x)-x^3-2*x^2),x, algorithm="maxima")

[Out]

-2*log(-x + log(2) + log(x + 2) - log(x))

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mupad [B]  time = 3.92, size = 17, normalized size = 0.81 \begin {gather*} -2\,\ln \left (x-\ln \left (\frac {2\,x+4}{x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x + 2*x^2 + 4)/(2*x^2 - log((2*x + 4)/x)*(2*x + x^2) + x^3),x)

[Out]

-2*log(x - log((2*x + 4)/x))

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sympy [A]  time = 0.29, size = 14, normalized size = 0.67 \begin {gather*} - 2 \log {\left (- x + \log {\left (\frac {2 x + 4}{x} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2+4*x+4)/((x**2+2*x)*ln((2*x+4)/x)-x**3-2*x**2),x)

[Out]

-2*log(-x + log((2*x + 4)/x))

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