### 3.138 $$\int \frac{\log (\log (6 x))}{x \log (6 x)} \, dx$$

Optimal. Leaf size=11 $\frac{1}{2} \log ^2(\log (6 x))$

[Out]

Log[Log[6*x]]^2/2

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Rubi [A]  time = 0.0283252, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.067, Rules used = {2301} $\frac{1}{2} \log ^2(\log (6 x))$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[Log[6*x]]/(x*Log[6*x]),x]

[Out]

Log[Log[6*x]]^2/2

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin{align*} \int \frac{\log (\log (6 x))}{x \log (6 x)} \, dx &=\operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,\log (6 x)\right )\\ &=\frac{1}{2} \log ^2(\log (6 x))\\ \end{align*}

Mathematica [A]  time = 0.0039504, size = 11, normalized size = 1. $\frac{1}{2} \log ^2(\log (6 x))$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[Log[6*x]]/(x*Log[6*x]),x]

[Out]

Log[Log[6*x]]^2/2

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Maple [A]  time = 0.004, size = 10, normalized size = 0.9 \begin{align*}{\frac{ \left ( \ln \left ( \ln \left ( 6\,x \right ) \right ) \right ) ^{2}}{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(ln(ln(6*x))/x/ln(6*x),x)

[Out]

1/2*ln(ln(6*x))^2

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Maxima [A]  time = 1.09572, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{2} \, \log \left (\log \left (6 \, x\right )\right )^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(log(6*x))/x/log(6*x),x, algorithm="maxima")

[Out]

1/2*log(log(6*x))^2

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Fricas [A]  time = 2.08825, size = 28, normalized size = 2.55 \begin{align*} \frac{1}{2} \, \log \left (\log \left (6 \, x\right )\right )^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(log(6*x))/x/log(6*x),x, algorithm="fricas")

[Out]

1/2*log(log(6*x))^2

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Sympy [A]  time = 0.290454, size = 8, normalized size = 0.73 \begin{align*} \frac{\log{\left (\log{\left (6 x \right )} \right )}^{2}}{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(ln(6*x))/x/ln(6*x),x)

[Out]

log(log(6*x))**2/2

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Giac [A]  time = 1.30393, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{2} \, \log \left (\log \left (6 \, x\right )\right )^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(log(6*x))/x/log(6*x),x, algorithm="giac")

[Out]

1/2*log(log(6*x))^2