3.631 \(\int \frac {\log ^4(\log (x))}{x} \, dx\)

Optimal. Leaf size=38 \[ \log (x) \log ^4(\log (x))-4 \log (x) \log ^3(\log (x))+12 \log (x) \log ^2(\log (x))-24 \log (x) \log (\log (x))+24 \log (x) \]

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Rubi [A]  time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2296, 2295} \[ \log (x) \log ^4(\log (x))-4 \log (x) \log ^3(\log (x))+12 \log (x) \log ^2(\log (x))-24 \log (x) \log (\log (x))+24 \log (x) \]

Antiderivative was successfully verified.

[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 2295

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

Rule 2296

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*} \int \frac {\log ^4(\log (x))}{x} \, dx &=\operatorname {Subst}\left (\int \log ^4(x) \, dx,x,\log (x)\right )\\ &=\log (x) \log ^4(\log (x))-4 \operatorname {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 \operatorname {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 \operatorname {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*}

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Mathematica [A]  time = 0.01, size = 38, normalized size = 1.00 \[ \log (x) \log ^4(\log (x))-4 \log (x) \log ^3(\log (x))+12 \log (x) \log ^2(\log (x))-24 \log (x) \log (\log (x))+24 \log (x) \]

Antiderivative was successfully verified.

[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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log ^4(\log (x))}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 0.93, size = 38, normalized size = 1.00 \[ \log \relax (x) \log \left (\log \relax (x)\right )^{4} - 4 \, \log \relax (x) \log \left (\log \relax (x)\right )^{3} + 12 \, \log \relax (x) \log \left (\log \relax (x)\right )^{2} - 24 \, \log \relax (x) \log \left (\log \relax (x)\right ) + 24 \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.95, size = 38, normalized size = 1.00 \[ \log \relax (x) \log \left (\log \relax (x)\right )^{4} - 4 \, \log \relax (x) \log \left (\log \relax (x)\right )^{3} + 12 \, \log \relax (x) \log \left (\log \relax (x)\right )^{2} - 24 \, \log \relax (x) \log \left (\log \relax (x)\right ) + 24 \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.04, size = 39, normalized size = 1.03

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

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [A]  time = 0.42, size = 29, normalized size = 0.76 \[ {\left (\log \left (\log \relax (x)\right )^{4} - 4 \, \log \left (\log \relax (x)\right )^{3} + 12 \, \log \left (\log \relax (x)\right )^{2} - 24 \, \log \left (\log \relax (x)\right ) + 24\right )} \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.37, size = 38, normalized size = 1.00 \[ \ln \relax (x)\,{\ln \left (\ln \relax (x)\right )}^4-4\,\ln \relax (x)\,{\ln \left (\ln \relax (x)\right )}^3+12\,\ln \relax (x)\,{\ln \left (\ln \relax (x)\right )}^2-24\,\ln \relax (x)\,\ln \left (\ln \relax (x)\right )+24\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.39, size = 48, normalized size = 1.26 \[ \log {\relax (x )} \log {\left (\log {\relax (x )} \right )}^{4} - 4 \log {\relax (x )} \log {\left (\log {\relax (x )} \right )}^{3} + 12 \log {\relax (x )} \log {\left (\log {\relax (x )} \right )}^{2} - 24 \log {\relax (x )} \log {\left (\log {\relax (x )} \right )} + 24 \log {\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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