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

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

[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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Rubi [A]  time = 0.0215285, antiderivative size = 29, normalized size of antiderivative = 1., 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 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]

Rule 2295

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

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*}

Mathematica [A]  time = 0.0088646, size = 29, normalized size = 1. \[ \log (x) \log ^3(\log (x))-3 \log (x) \log ^2(\log (x))+6 \log (x) \log (\log (x))-6 \log (x) \]

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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Maple [A]  time = 0.002, size = 30, normalized size = 1. \begin{align*} -6\,\ln \left ( x \right ) +6\,\ln \left ( x \right ) \ln \left ( \ln \left ( x \right ) \right ) -3\,\ln \left ( x \right ) \left ( \ln \left ( \ln \left ( x \right ) \right ) \right ) ^{2}+\ln \left ( x \right ) \left ( \ln \left ( \ln \left ( x \right ) \right ) \right ) ^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 1.04443, size = 30, normalized size = 1.03 \begin{align*}{\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 ) \end{align*}

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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Fricas [A]  time = 2.32822, size = 109, normalized size = 3.76 \begin{align*} \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 ) \end{align*}

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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Sympy [A]  time = 0.356041, size = 36, normalized size = 1.24 \begin{align*} \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 )} \end{align*}

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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Giac [A]  time = 1.06163, size = 39, normalized size = 1.34 \begin{align*} \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 ) \end{align*}

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)