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

Optimal. Leaf size=10 \[ \log \left (2304 \log ^2(\log (4+x))\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 8, normalized size of antiderivative = 0.80, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 6684} \begin {gather*} 2 \log (\log (\log (x+4))) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*Log[Log[Log[4 + 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 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} &=2 \int \frac {1}{(4+x) \log (4+x) \log (\log (4+x))} \, dx\\ &=2 \log (\log (\log (4+x)))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 8, normalized size = 0.80 \begin {gather*} 2 \log (\log (\log (4+x))) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*Log[Log[Log[4 + x]]]

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fricas [A]  time = 0.97, size = 8, normalized size = 0.80 \begin {gather*} 2 \, \log \left (\log \left (\log \left (x + 4\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(log(log(x + 4)))

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giac [A]  time = 0.12, size = 8, normalized size = 0.80 \begin {gather*} 2 \, \log \left (\log \left (\log \left (x + 4\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(log(log(x + 4)))

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maple [A]  time = 0.04, size = 9, normalized size = 0.90

method	result	size
derivativedivides	\(2 \ln \left (\ln \left (\ln \left (4+x \right )\right )\right )\)	\(9\)
default	\(2 \ln \left (\ln \left (\ln \left (4+x \right )\right )\right )\)	\(9\)
norman	\(2 \ln \left (\ln \left (\ln \left (4+x \right )\right )\right )\)	\(9\)
risch	\(2 \ln \left (\ln \left (\ln \left (4+x \right )\right )\right )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.34, size = 8, normalized size = 0.80 \begin {gather*} 2 \, \log \left (\log \left (\log \left (x + 4\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(log(log(x + 4)))

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mupad [B]  time = 2.42, size = 8, normalized size = 0.80 \begin {gather*} 2\,\ln \left (\ln \left (\ln \left (x+4\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/(log(x + 4)*log(log(x + 4))*(x + 4)),x)

[Out]

2*log(log(log(x + 4)))

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sympy [A]  time = 0.29, size = 8, normalized size = 0.80 \begin {gather*} 2 \log {\left (\log {\left (\log {\left (x + 4 \right )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(4+x)/ln(4+x)/ln(ln(4+x)),x)

[Out]

2*log(log(log(x + 4)))

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