\(\int \frac {2}{(4+x) \log (4+x) \log (\log (4+x))} \, dx\) [3921]

Optimal result

Integrand size = 20, antiderivative size = 10 \[ \int \frac {2}{(4+x) \log (4+x) \log (\log (4+x))} \, dx=\log \left (2304 \log ^2(\log (4+x))\right ) \]

[Out]

ln(2304*ln(ln(4+x))^2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 6816} \[ \int \frac {2}{(4+x) \log (4+x) \log (\log (4+x))} \, dx=2 \log (\log (\log (x+4))) \]

[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 6816

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

Rubi steps \begin{align*} \text {integral}& = 2 \int \frac {1}{(4+x) \log (4+x) \log (\log (4+x))} \, dx \\ & = 2 \log (\log (\log (4+x))) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {2}{(4+x) \log (4+x) \log (\log (4+x))} \, dx=2 \log (\log (\log (4+x))) \]

[In]

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

[Out]

2*Log[Log[Log[4 + x]]]

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 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	\(2 \ln \left (\ln \left (\ln \left (4+x \right )\right )\right )\)	\(9\)
parallelrisch	\(2 \ln \left (\ln \left (\ln \left (4+x \right )\right )\right )\)	\(9\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {2}{(4+x) \log (4+x) \log (\log (4+x))} \, dx=2 \, \log \left (\log \left (\log \left (x + 4\right )\right )\right ) \]

[In]

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

[Out]

2*log(log(log(x + 4)))

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {2}{(4+x) \log (4+x) \log (\log (4+x))} \, dx=2 \log {\left (\log {\left (\log {\left (x + 4 \right )} \right )} \right )} \]

[In]

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

[Out]

2*log(log(log(x + 4)))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {2}{(4+x) \log (4+x) \log (\log (4+x))} \, dx=2 \, \log \left (\log \left (\log \left (x + 4\right )\right )\right ) \]

[In]

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

[Out]

2*log(log(log(x + 4)))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {2}{(4+x) \log (4+x) \log (\log (4+x))} \, dx=2 \, \log \left (\log \left (\log \left (x + 4\right )\right )\right ) \]

[In]

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

[Out]

2*log(log(log(x + 4)))

Mupad [B] (verification not implemented)

Time = 9.41 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {2}{(4+x) \log (4+x) \log (\log (4+x))} \, dx=2\,\ln \left (\ln \left (\ln \left (x+4\right )\right )\right ) \]

[In]

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

[Out]

2*log(log(log(x + 4)))