3.630 \(\int \frac {\log ^3(\log (x))}{x} \, dx\)

Optimal. Leaf size=29 \[ \log (x) \log ^3(\log (x))-3 \log (x) \log ^2(\log (x))+6 \log (x) \log (\log (x))-6 \log (x) \]

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Rubi [A]  time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2296, 2295} \[ \log (x) \log ^3(\log (x))-3 \log (x) \log ^2(\log (x))+6 \log (x) \log (\log (x))-6 \log (x) \]

Antiderivative was successfully verified.

[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 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 ^3(\log (x))}{x} \, dx &=\operatorname {Subst}\left (\int \log ^3(x) \, dx,x,\log (x)\right )\\ &=\log (x) \log ^3(\log (x))-3 \operatorname {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 \operatorname {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*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 1.00 \[ \log (x) \log ^3(\log (x))-3 \log (x) \log ^2(\log (x))+6 \log (x) \log (\log (x))-6 \log (x) \]

Antiderivative was successfully verified.

[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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log ^3(\log (x))}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 1.03, size = 29, normalized size = 1.00 \[ \log \relax (x) \log \left (\log \relax (x)\right )^{3} - 3 \, \log \relax (x) \log \left (\log \relax (x)\right )^{2} + 6 \, \log \relax (x) \log \left (\log \relax (x)\right ) - 6 \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.79, size = 29, normalized size = 1.00 \[ \log \relax (x) \log \left (\log \relax (x)\right )^{3} - 3 \, \log \relax (x) \log \left (\log \relax (x)\right )^{2} + 6 \, \log \relax (x) \log \left (\log \relax (x)\right ) - 6 \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.04, size = 30, normalized size = 1.03

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

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [A]  time = 0.42, size = 22, normalized size = 0.76 \[ {\left (\log \left (\log \relax (x)\right )^{3} - 3 \, \log \left (\log \relax (x)\right )^{2} + 6 \, \log \left (\log \relax (x)\right ) - 6\right )} \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.36, size = 29, normalized size = 1.00 \[ \ln \relax (x)\,{\ln \left (\ln \relax (x)\right )}^3-3\,\ln \relax (x)\,{\ln \left (\ln \relax (x)\right )}^2+6\,\ln \relax (x)\,\ln \left (\ln \relax (x)\right )-6\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.35, size = 36, normalized size = 1.24 \[ \log {\relax (x )} \log {\left (\log {\relax (x )} \right )}^{3} - 3 \log {\relax (x )} \log {\left (\log {\relax (x )} \right )}^{2} + 6 \log {\relax (x )} \log {\left (\log {\relax (x )} \right )} - 6 \log {\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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