\(\int -\frac {2 \log (\log (4))}{x \log ^3(-x)} \, dx\) [2369]

Optimal result

Integrand size = 14, antiderivative size = 10 \[ \int -\frac {2 \log (\log (4))}{x \log ^3(-x)} \, dx=\frac {\log (\log (4))}{\log ^2(-x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {12, 2339, 30} \[ \int -\frac {2 \log (\log (4))}{x \log ^3(-x)} \, dx=\frac {\log (\log (4))}{\log ^2(-x)} \]

[In]

Int[(-2*Log[Log[4]])/(x*Log[-x]^3),x]

[Out]

Log[Log[4]]/Log[-x]^2

Rule 12

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rubi steps \begin{align*} \text {integral}& = -\left ((2 \log (\log (4))) \int \frac {1}{x \log ^3(-x)} \, dx\right ) \\ & = -\left ((2 \log (\log (4))) \text {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log (-x)\right )\right ) \\ & = \frac {\log (\log (4))}{\log ^2(-x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int -\frac {2 \log (\log (4))}{x \log ^3(-x)} \, dx=\frac {\log (\log (4))}{\log ^2(-x)} \]

[In]

Integrate[(-2*Log[Log[4]])/(x*Log[-x]^3),x]

[Out]

Log[Log[4]]/Log[-x]^2

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.30

method	result	size
derivativedivides	\(\frac {\ln \left (2 \ln \left (2\right )\right )}{\ln \left (-x \right )^{2}}\)	\(13\)
default	\(\frac {\ln \left (2 \ln \left (2\right )\right )}{\ln \left (-x \right )^{2}}\)	\(13\)
parallelrisch	\(\frac {\ln \left (2 \ln \left (2\right )\right )}{\ln \left (-x \right )^{2}}\)	\(13\)
norman	\(\frac {\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )}{\ln \left (-x \right )^{2}}\)	\(14\)
risch	\(\frac {\ln \left (2\right )}{\ln \left (-x \right )^{2}}+\frac {\ln \left (\ln \left (2\right )\right )}{\ln \left (-x \right )^{2}}\)	\(21\)

[In]

int(-2*ln(2*ln(2))/x/ln(-x)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int -\frac {2 \log (\log (4))}{x \log ^3(-x)} \, dx=\frac {\log \left (2 \, \log \left (2\right )\right )}{\log \left (-x\right )^{2}} \]

[In]

integrate(-2*log(2*log(2))/x/log(-x)^3,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.40 \[ \int -\frac {2 \log (\log (4))}{x \log ^3(-x)} \, dx=\frac {\log {\left (\log {\left (2 \right )} \right )} + \log {\left (2 \right )}}{\log {\left (- x \right )}^{2}} \]

[In]

integrate(-2*ln(2*ln(2))/x/ln(-x)**3,x)

[Out]

(log(log(2)) + log(2))/log(-x)**2

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int -\frac {2 \log (\log (4))}{x \log ^3(-x)} \, dx=\frac {\log \left (2 \, \log \left (2\right )\right )}{\log \left (-x\right )^{2}} \]

[In]

integrate(-2*log(2*log(2))/x/log(-x)^3,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int -\frac {2 \log (\log (4))}{x \log ^3(-x)} \, dx=\frac {\log \left (2 \, \log \left (2\right )\right )}{\log \left (-x\right )^{2}} \]

[In]

integrate(-2*log(2*log(2))/x/log(-x)^3,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.59 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int -\frac {2 \log (\log (4))}{x \log ^3(-x)} \, dx=\frac {\ln \left (\ln \left (4\right )\right )}{{\ln \left (-x\right )}^2} \]

[In]

int(-(2*log(2*log(2)))/(x*log(-x)^3),x)

[Out]

log(log(4))/log(-x)^2