\(\int \frac {\log ^3(\log (x))}{x} \, dx\) [630]

Optimal result

Integrand size = 9, antiderivative size = 29 \[ \int \frac {\log ^3(\log (x))}{x} \, dx=-6 \log (x)+6 \log (x) \log (\log (x))-3 \log (x) \log ^2(\log (x))+\log (x) \log ^3(\log (x)) \]

[Out]

-6*ln(x)+6*ln(x)*ln(ln(x))-3*ln(x)*ln(ln(x))^2+ln(x)*ln(ln(x))^3

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2333, 2332} \[ \int \frac {\log ^3(\log (x))}{x} \, dx=\log (x) \log ^3(\log (x))-3 \log (x) \log ^2(\log (x))+6 \log (x) \log (\log (x))-6 \log (x) \]

[In]

Int[Log[Log[x]]^3/x,x]

[Out]

-6*Log[x] + 6*Log[x]*Log[Log[x]] - 3*Log[x]*Log[Log[x]]^2 + Log[x]*Log[Log[x]]^3

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \log ^3(x) \, dx,x,\log (x)\right ) \\ & = \log (x) \log ^3(\log (x))-3 \text {Subst}\left (\int \log ^2(x) \, dx,x,\log (x)\right ) \\ & = -3 \log (x) \log ^2(\log (x))+\log (x) \log ^3(\log (x))+6 \text {Subst}(\int \log (x) \, dx,x,\log (x)) \\ & = -6 \log (x)+6 \log (x) \log (\log (x))-3 \log (x) \log ^2(\log (x))+\log (x) \log ^3(\log (x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^3(\log (x))}{x} \, dx=-6 \log (x)+6 \log (x) \log (\log (x))-3 \log (x) \log ^2(\log (x))+\log (x) \log ^3(\log (x)) \]

[In]

Integrate[Log[Log[x]]^3/x,x]

[Out]

-6*Log[x] + 6*Log[x]*Log[Log[x]] - 3*Log[x]*Log[Log[x]]^2 + Log[x]*Log[Log[x]]^3

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03

method	result	size
derivativedivides	\(-6 \ln \left (x \right )+6 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-3 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{3}\)	\(30\)
default	\(-6 \ln \left (x \right )+6 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-3 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{3}\)	\(30\)
norman	\(-6 \ln \left (x \right )+6 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-3 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{3}\)	\(30\)
risch	\(-6 \ln \left (x \right )+6 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-3 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{3}\)	\(30\)

[In]

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

[Out]

-6*ln(x)+6*ln(x)*ln(ln(x))-3*ln(x)*ln(ln(x))^2+ln(x)*ln(ln(x))^3

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^3(\log (x))}{x} \, dx=\log \left (x\right ) \log \left (\log \left (x\right )\right )^{3} - 3 \, \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} + 6 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) - 6 \, \log \left (x\right ) \]

[In]

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

[Out]

log(x)*log(log(x))^3 - 3*log(x)*log(log(x))^2 + 6*log(x)*log(log(x)) - 6*log(x)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {\log ^3(\log (x))}{x} \, dx=\log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )}^{3} - 3 \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )}^{2} + 6 \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )} - 6 \log {\left (x \right )} \]

[In]

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

[Out]

log(x)*log(log(x))**3 - 3*log(x)*log(log(x))**2 + 6*log(x)*log(log(x)) - 6*log(x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {\log ^3(\log (x))}{x} \, dx={\left (\log \left (\log \left (x\right )\right )^{3} - 3 \, \log \left (\log \left (x\right )\right )^{2} + 6 \, \log \left (\log \left (x\right )\right ) - 6\right )} \log \left (x\right ) \]

[In]

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

[Out]

(log(log(x))^3 - 3*log(log(x))^2 + 6*log(log(x)) - 6)*log(x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^3(\log (x))}{x} \, dx=\log \left (x\right ) \log \left (\log \left (x\right )\right )^{3} - 3 \, \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} + 6 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) - 6 \, \log \left (x\right ) \]

[In]

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

[Out]

log(x)*log(log(x))^3 - 3*log(x)*log(log(x))^2 + 6*log(x)*log(log(x)) - 6*log(x)

Mupad [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^3(\log (x))}{x} \, dx=\ln \left (x\right )\,{\ln \left (\ln \left (x\right )\right )}^3-3\,\ln \left (x\right )\,{\ln \left (\ln \left (x\right )\right )}^2+6\,\ln \left (x\right )\,\ln \left (\ln \left (x\right )\right )-6\,\ln \left (x\right ) \]

[In]

int(log(log(x))^3/x,x)

[Out]

6*log(log(x))*log(x) - 6*log(x) - 3*log(log(x))^2*log(x) + log(log(x))^3*log(x)