3.629 \(\int \frac{\log ^2(\log (x))}{x} \, dx\)

Optimal. Leaf size=20 \[ \log (x) \log ^2(\log (x))-2 \log (x) \log (\log (x))+2 \log (x) \]

[Out]

2*Log[x] - 2*Log[x]*Log[Log[x]] + Log[x]*Log[Log[x]]^2

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Rubi [A]  time = 0.0177781, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2296, 2295} \[ \log (x) \log ^2(\log (x))-2 \log (x) \log (\log (x))+2 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[Log[Log[x]]^2/x,x]

[Out]

2*Log[x] - 2*Log[x]*Log[Log[x]] + Log[x]*Log[Log[x]]^2

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 ^2(\log (x))}{x} \, dx &=\operatorname{Subst}\left (\int \log ^2(x) \, dx,x,\log (x)\right )\\ &=\log (x) \log ^2(\log (x))-2 \operatorname{Subst}(\int \log (x) \, dx,x,\log (x))\\ &=2 \log (x)-2 \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))\\ \end{align*}

Mathematica [A]  time = 0.0059768, size = 20, normalized size = 1. \[ \log (x) \log ^2(\log (x))-2 \log (x) \log (\log (x))+2 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[Log[x]]^2/x,x]

[Out]

2*Log[x] - 2*Log[x]*Log[Log[x]] + Log[x]*Log[Log[x]]^2

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Maple [A]  time = 0.003, size = 21, normalized size = 1.1 \begin{align*} 2\,\ln \left ( x \right ) -2\,\ln \left ( x \right ) \ln \left ( \ln \left ( x \right ) \right ) +\ln \left ( x \right ) \left ( \ln \left ( \ln \left ( x \right ) \right ) \right ) ^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(ln(x))^2/x,x)

[Out]

2*ln(x)-2*ln(x)*ln(ln(x))+ln(x)*ln(ln(x))^2

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Maxima [A]  time = 0.958777, size = 20, normalized size = 1. \begin{align*}{\left (\log \left (\log \left (x\right )\right )^{2} - 2 \, \log \left (\log \left (x\right )\right ) + 2\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(log(x))^2/x,x, algorithm="maxima")

[Out]

(log(log(x))^2 - 2*log(log(x)) + 2)*log(x)

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Fricas [A]  time = 2.41174, size = 76, normalized size = 3.8 \begin{align*} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - 2 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 2 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(log(x))^2/x,x, algorithm="fricas")

[Out]

log(x)*log(log(x))^2 - 2*log(x)*log(log(x)) + 2*log(x)

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Sympy [A]  time = 0.316487, size = 24, normalized size = 1.2 \begin{align*} \log{\left (x \right )} \log{\left (\log{\left (x \right )} \right )}^{2} - 2 \log{\left (x \right )} \log{\left (\log{\left (x \right )} \right )} + 2 \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(ln(x))**2/x,x)

[Out]

log(x)*log(log(x))**2 - 2*log(x)*log(log(x)) + 2*log(x)

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Giac [A]  time = 1.06349, size = 27, normalized size = 1.35 \begin{align*} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - 2 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 2 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(log(x))^2/x,x, algorithm="giac")

[Out]

log(x)*log(log(x))^2 - 2*log(x)*log(log(x)) + 2*log(x)