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

Optimal. Leaf size=14 \[ \log \left (\frac {1}{4} x \log \left (\frac {2}{2+x}\right )\right ) \]

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Rubi [A]  time = 0.12, antiderivative size = 11, normalized size of antiderivative = 0.79, number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1593, 6685} \begin {gather*} \log \left (x \log \left (\frac {2}{x+2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x*Log[2/(2 + x)]]

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 6685

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

Rubi steps

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

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

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] + Log[Log[2/(2 + x)]]

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fricas [A]  time = 0.74, size = 12, normalized size = 0.86 \begin {gather*} \log \relax (x) + \log \left (\log \left (\frac {2}{x + 2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) + log(log(2/(x + 2)))

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giac [B]  time = 0.17, size = 30, normalized size = 2.14 \begin {gather*} -\log \left (\frac {2}{x + 2}\right ) + \log \left (\frac {2}{x + 2} - 1\right ) + \log \left (\log \left (\frac {2}{x + 2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(2/(x + 2)) + log(2/(x + 2) - 1) + log(log(2/(x + 2)))

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maple [A]  time = 0.29, size = 13, normalized size = 0.93

method	result	size
norman	\(\ln \relax (x )+\ln \left (\ln \left (\frac {2}{2+x}\right )\right )\)	\(13\)
risch	\(\ln \relax (x )+\ln \left (\ln \left (\frac {2}{2+x}\right )\right )\)	\(13\)
derivativedivides	\(-\ln \left (\frac {2}{2+x}\right )+\ln \left (-1+\frac {2}{2+x}\right )+\ln \left (\ln \left (\frac {2}{2+x}\right )\right )\)	\(31\)
default	\(-\ln \left (\frac {2}{2+x}\right )+\ln \left (-1+\frac {2}{2+x}\right )+\ln \left (\ln \left (\frac {2}{2+x}\right )\right )\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x)+ln(ln(2/(2+x)))

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maxima [B]  time = 0.52, size = 57, normalized size = 4.07 \begin {gather*} {\left (\log \relax (2) - \log \left (x + 2\right )\right )} \log \left (-\log \relax (2) + \log \left (x + 2\right )\right ) - \log \left (\frac {2}{x + 2}\right ) \log \left (-\log \relax (2) + \log \left (x + 2\right )\right ) - \log \relax (2) + \log \relax (x) + \log \left (-\log \relax (2) + \log \left (x + 2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.25, size = 12, normalized size = 0.86 \begin {gather*} \ln \left (\ln \left (\frac {2}{x+2}\right )\right )+\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(2/(x + 2))) + log(x)

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sympy [A]  time = 0.15, size = 10, normalized size = 0.71 \begin {gather*} \log {\relax (x )} + \log {\left (\log {\left (\frac {2}{x + 2} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) + log(log(2/(x + 2)))

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