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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 = 0.00, size = 11, normalized size = 1.00 \[ \frac {1}{2} \log ^2(\log (6 x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.56, size = 9, normalized size = 0.82 \[ \frac {1}{2} \, \log \left (\log \left (6 \, x\right )\right )^{2} \]

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

________________________________________________________________________________________

giac [A]  time = 0.19, size = 9, normalized size = 0.82 \[ \frac {1}{2} \, \log \left (\log \left (6 \, x\right )\right )^{2} \]

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

________________________________________________________________________________________

maple [A]  time = 0.06, size = 10, normalized size = 0.91 \[ \frac {\ln \left (\ln \left (6 x \right )\right )^{2}}{2} \]

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

________________________________________________________________________________________

maxima [A]  time = 0.60, size = 9, normalized size = 0.82 \[ \frac {1}{2} \, \log \left (\log \left (6 \, x\right )\right )^{2} \]

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

________________________________________________________________________________________

mupad [B]  time = 0.52, size = 9, normalized size = 0.82 \[ \frac {{\ln \left (\ln \left (6\,x\right )\right )}^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.25, size = 8, normalized size = 0.73 \[ \frac {\log {\left (\log {\left (6 x \right )} \right )}^{2}}{2} \]

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

________________________________________________________________________________________