\(\int \frac {\log ^4(\log (x))}{x} \, dx\) [631]

Optimal result

Integrand size = 9, antiderivative size = 38 \[ \int \frac {\log ^4(\log (x))}{x} \, dx=24 \log (x)-24 \log (x) \log (\log (x))+12 \log (x) \log ^2(\log (x))-4 \log (x) \log ^3(\log (x))+\log (x) \log ^4(\log (x)) \]

[Out]

24*ln(x)-24*ln(x)*ln(ln(x))+12*ln(x)*ln(ln(x))^2-4*ln(x)*ln(ln(x))^3+ln(x)*ln(ln(x))^4

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2333, 2332} \[ \int \frac {\log ^4(\log (x))}{x} \, dx=\log (x) \log ^4(\log (x))-4 \log (x) \log ^3(\log (x))+12 \log (x) \log ^2(\log (x))-24 \log (x) \log (\log (x))+24 \log (x) \]

[In]

Int[Log[Log[x]]^4/x,x]

[Out]

24*Log[x] - 24*Log[x]*Log[Log[x]] + 12*Log[x]*Log[Log[x]]^2 - 4*Log[x]*Log[Log[x]]^3 + Log[x]*Log[Log[x]]^4

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 ^4(x) \, dx,x,\log (x)\right ) \\ & = \log (x) \log ^4(\log (x))-4 \text {Subst}\left (\int \log ^3(x) \, dx,x,\log (x)\right ) \\ & = -4 \log (x) \log ^3(\log (x))+\log (x) \log ^4(\log (x))+12 \text {Subst}\left (\int \log ^2(x) \, dx,x,\log (x)\right ) \\ & = 12 \log (x) \log ^2(\log (x))-4 \log (x) \log ^3(\log (x))+\log (x) \log ^4(\log (x))-24 \text {Subst}(\int \log (x) \, dx,x,\log (x)) \\ & = 24 \log (x)-24 \log (x) \log (\log (x))+12 \log (x) \log ^2(\log (x))-4 \log (x) \log ^3(\log (x))+\log (x) \log ^4(\log (x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^4(\log (x))}{x} \, dx=24 \log (x)-24 \log (x) \log (\log (x))+12 \log (x) \log ^2(\log (x))-4 \log (x) \log ^3(\log (x))+\log (x) \log ^4(\log (x)) \]

[In]

Integrate[Log[Log[x]]^4/x,x]

[Out]

24*Log[x] - 24*Log[x]*Log[Log[x]] + 12*Log[x]*Log[Log[x]]^2 - 4*Log[x]*Log[Log[x]]^3 + Log[x]*Log[Log[x]]^4

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.03

method	result	size
derivativedivides	\(24 \ln \left (x \right )-24 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+12 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}-4 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{3}+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{4}\)	\(39\)
default	\(24 \ln \left (x \right )-24 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+12 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}-4 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{3}+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{4}\)	\(39\)
norman	\(24 \ln \left (x \right )-24 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+12 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}-4 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{3}+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{4}\)	\(39\)
risch	\(24 \ln \left (x \right )-24 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+12 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}-4 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{3}+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{4}\)	\(39\)

[In]

int(ln(ln(x))^4/x,x,method=_RETURNVERBOSE)

[Out]

24*ln(x)-24*ln(x)*ln(ln(x))+12*ln(x)*ln(ln(x))^2-4*ln(x)*ln(ln(x))^3+ln(x)*ln(ln(x))^4

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^4(\log (x))}{x} \, dx=\log \left (x\right ) \log \left (\log \left (x\right )\right )^{4} - 4 \, \log \left (x\right ) \log \left (\log \left (x\right )\right )^{3} + 12 \, \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - 24 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 24 \, \log \left (x\right ) \]

[In]

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

[Out]

log(x)*log(log(x))^4 - 4*log(x)*log(log(x))^3 + 12*log(x)*log(log(x))^2 - 24*log(x)*log(log(x)) + 24*log(x)

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.26 \[ \int \frac {\log ^4(\log (x))}{x} \, dx=\log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )}^{4} - 4 \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )}^{3} + 12 \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )}^{2} - 24 \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )} + 24 \log {\left (x \right )} \]

[In]

integrate(ln(ln(x))**4/x,x)

[Out]

log(x)*log(log(x))**4 - 4*log(x)*log(log(x))**3 + 12*log(x)*log(log(x))**2 - 24*log(x)*log(log(x)) + 24*log(x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int \frac {\log ^4(\log (x))}{x} \, dx={\left (\log \left (\log \left (x\right )\right )^{4} - 4 \, \log \left (\log \left (x\right )\right )^{3} + 12 \, \log \left (\log \left (x\right )\right )^{2} - 24 \, \log \left (\log \left (x\right )\right ) + 24\right )} \log \left (x\right ) \]

[In]

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

[Out]

(log(log(x))^4 - 4*log(log(x))^3 + 12*log(log(x))^2 - 24*log(log(x)) + 24)*log(x)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^4(\log (x))}{x} \, dx=\log \left (x\right ) \log \left (\log \left (x\right )\right )^{4} - 4 \, \log \left (x\right ) \log \left (\log \left (x\right )\right )^{3} + 12 \, \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - 24 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 24 \, \log \left (x\right ) \]

[In]

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

[Out]

log(x)*log(log(x))^4 - 4*log(x)*log(log(x))^3 + 12*log(x)*log(log(x))^2 - 24*log(x)*log(log(x)) + 24*log(x)

Mupad [B] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^4(\log (x))}{x} \, dx=\ln \left (x\right )\,{\ln \left (\ln \left (x\right )\right )}^4-4\,\ln \left (x\right )\,{\ln \left (\ln \left (x\right )\right )}^3+12\,\ln \left (x\right )\,{\ln \left (\ln \left (x\right )\right )}^2-24\,\ln \left (x\right )\,\ln \left (\ln \left (x\right )\right )+24\,\ln \left (x\right ) \]

[In]

int(log(log(x))^4/x,x)

[Out]

24*log(x) - 24*log(log(x))*log(x) + 12*log(log(x))^2*log(x) - 4*log(log(x))^3*log(x) + log(log(x))^4*log(x)