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

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

[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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Rubi [A]  time = 0.0256756, antiderivative size = 38, normalized size of antiderivative = 1., 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 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 ^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*}

Mathematica [A]  time = 0.0060759, size = 38, normalized size = 1. \[ \log (x) \log ^4(\log (x))-4 \log (x) \log ^3(\log (x))+12 \log (x) \log ^2(\log (x))-24 \log (x) \log (\log (x))+24 \log (x) \]

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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Maple [A]  time = 0.002, size = 39, normalized size = 1. \begin{align*} 24\,\ln \left ( x \right ) -24\,\ln \left ( x \right ) \ln \left ( \ln \left ( x \right ) \right ) +12\,\ln \left ( x \right ) \left ( \ln \left ( \ln \left ( x \right ) \right ) \right ) ^{2}-4\,\ln \left ( x \right ) \left ( \ln \left ( \ln \left ( x \right ) \right ) \right ) ^{3}+\ln \left ( x \right ) \left ( \ln \left ( \ln \left ( x \right ) \right ) \right ) ^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [A]  time = 0.981408, size = 39, normalized size = 1.03 \begin{align*}{\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 ) \end{align*}

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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Fricas [A]  time = 2.75544, size = 147, normalized size = 3.87 \begin{align*} \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 ) \end{align*}

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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Sympy [A]  time = 0.396785, size = 48, normalized size = 1.26 \begin{align*} \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 )} \end{align*}

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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Giac [A]  time = 1.06818, size = 51, normalized size = 1.34 \begin{align*} \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 ) \end{align*}

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)