3.7.29 \(\int \frac {\log ^2(\log (x))}{x} \, dx\) [629]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 = {2333, 2332} \begin {gather*} \log (x) \log ^2(\log (x))-2 \log (x) \log (\log (x))+2 \log (x) \end {gather*}

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 2332

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

Rule 2333

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 ^2(\log (x))}{x} \, dx &=\text {Subst}\left (\int \log ^2(x) \, dx,x,\log (x)\right )\\ &=\log (x) \log ^2(\log (x))-2 \text {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*}

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Mathematica [A]
time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} 2 \log (x)-2 \log (x) \log (\log (x))+\log (x) \log ^2(\log (x)) \end {gather*}

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 = 0.02, size = 21, normalized size = 1.05

method	result	size
derivativedivides	\(2 \ln \left (x \right )-2 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}\)	\(21\)
default	\(2 \ln \left (x \right )-2 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}\)	\(21\)
norman	\(2 \ln \left (x \right )-2 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}\)	\(21\)
risch	\(2 \ln \left (x \right )-2 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(ln(x))^2/x,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 5.29, size = 15, normalized size = 0.75 \begin {gather*} {\left (\log \left (\log \left (x\right )\right )^{2} - 2 \, \log \left (\log \left (x\right )\right ) + 2\right )} \log \left (x\right ) \end {gather*}

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 = 0.73, size = 20, normalized size = 1.00 \begin {gather*} \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 {gather*}

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 = 0.09, size = 24, normalized size = 1.20 \begin {gather*} \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 {gather*}

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 = 1.53, size = 20, normalized size = 1.00 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 0.38, size = 15, normalized size = 0.75 \begin {gather*} \ln \left (x\right )\,\left ({\ln \left (\ln \left (x\right )\right )}^2-2\,\ln \left (\ln \left (x\right )\right )+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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