3.40.88 \(\int \frac {6}{2 x+3 x \log (x)} \, dx\)

Optimal. Leaf size=13 \[ \log \left (\frac {(2+3 \log (x))^2}{e^2}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 9, normalized size of antiderivative = 0.69, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {12, 31} \begin {gather*} 2 \log (3 \log (x)+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[6/(2*x + 3*x*Log[x]),x]

[Out]

2*Log[2 + 3*Log[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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=6 \int \frac {1}{2 x+3 x \log (x)} \, dx\\ &=6 \operatorname {Subst}\left (\int \frac {1}{2+3 x} \, dx,x,\log (x)\right )\\ &=2 \log (2+3 \log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 9, normalized size = 0.69 \begin {gather*} 2 \log (2+3 \log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[6/(2*x + 3*x*Log[x]),x]

[Out]

2*Log[2 + 3*Log[x]]

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fricas [A]  time = 0.81, size = 9, normalized size = 0.69 \begin {gather*} 2 \, \log \left (3 \, \log \relax (x) + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6/(3*x*log(x)+2*x),x, algorithm="fricas")

[Out]

2*log(3*log(x) + 2)

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giac [A]  time = 0.24, size = 22, normalized size = 1.69 \begin {gather*} \log \left (\frac {9}{4} \, \pi ^{2} {\left (\mathrm {sgn}\relax (x) - 1\right )}^{2} + {\left (3 \, \log \left ({\left | x \right |}\right ) + 2\right )}^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6/(3*x*log(x)+2*x),x, algorithm="giac")

[Out]

log(9/4*pi^2*(sgn(x) - 1)^2 + (3*log(abs(x)) + 2)^2)

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maple [A]  time = 0.01, size = 8, normalized size = 0.62

method	result	size
risch	\(2 \ln \left (\ln \relax (x )+\frac {2}{3}\right )\)	\(8\)
default	\(2 \ln \left (3 \ln \relax (x )+2\right )\)	\(10\)
norman	\(2 \ln \left (3 \ln \relax (x )+2\right )\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(6/(3*x*ln(x)+2*x),x,method=_RETURNVERBOSE)

[Out]

2*ln(ln(x)+2/3)

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maxima [A]  time = 0.38, size = 7, normalized size = 0.54 \begin {gather*} 2 \, \log \left (\log \relax (x) + \frac {2}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6/(3*x*log(x)+2*x),x, algorithm="maxima")

[Out]

2*log(log(x) + 2/3)

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mupad [B]  time = 2.59, size = 9, normalized size = 0.69 \begin {gather*} 2\,\ln \left (3\,\ln \relax (x)+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(6/(2*x + 3*x*log(x)),x)

[Out]

2*log(3*log(x) + 2)

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sympy [A]  time = 0.09, size = 8, normalized size = 0.62 \begin {gather*} 2 \log {\left (\log {\relax (x )} + \frac {2}{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6/(3*x*ln(x)+2*x),x)

[Out]

2*log(log(x) + 2/3)

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