3.59.66 \(\int \frac {4 \log (x)}{-x+x \log (2)} \, dx\)

Optimal. Leaf size=14 \[ 1+\frac {2 \log ^2(x)}{-1+\log (2)} \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6, 12, 2301} \begin {gather*} -\frac {2 \log ^2(x)}{1-\log (2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*Log[x])/(-x + x*Log[2]),x]

[Out]

(-2*Log[x]^2)/(1 - Log[2])

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \log (x)}{x (-1+\log (2))} \, dx\\ &=-\frac {4 \int \frac {\log (x)}{x} \, dx}{1-\log (2)}\\ &=-\frac {2 \log ^2(x)}{1-\log (2)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -\frac {4 \log ^2(x)}{2-\log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Log[x]^2)/(2 - Log[4])

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fricas [A]  time = 0.65, size = 12, normalized size = 0.86 \begin {gather*} \frac {2 \, \log \relax (x)^{2}}{\log \relax (2) - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(x)^2/(log(2) - 1)

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giac [A]  time = 0.12, size = 12, normalized size = 0.86 \begin {gather*} \frac {2 \, \log \relax (x)^{2}}{\log \relax (2) - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(x)^2/(log(2) - 1)

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maple [A]  time = 0.04, size = 13, normalized size = 0.93

method	result	size
default	\(\frac {2 \ln \relax (x )^{2}}{\ln \relax (2)-1}\)	\(13\)
norman	\(\frac {2 \ln \relax (x )^{2}}{\ln \relax (2)-1}\)	\(13\)
risch	\(\frac {2 \ln \relax (x )^{2}}{\ln \relax (2)-1}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*ln(x)^2/(ln(2)-1)

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maxima [A]  time = 0.49, size = 12, normalized size = 0.86 \begin {gather*} \frac {2 \, \log \relax (x)^{2}}{\log \relax (2) - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(x)^2/(log(2) - 1)

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mupad [B]  time = 4.25, size = 12, normalized size = 0.86 \begin {gather*} \frac {2\,{\ln \relax (x)}^2}{\ln \relax (2)-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*log(x))/(x - x*log(2)),x)

[Out]

(2*log(x)^2)/(log(2) - 1)

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sympy [A]  time = 0.11, size = 10, normalized size = 0.71 \begin {gather*} \frac {2 \log {\relax (x )}^{2}}{-1 + \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(x)**2/(-1 + log(2))

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