3.8.44 \(\int \frac {-6+2 \log (\log (x))}{x \log (x)} \, dx\)

Optimal. Leaf size=26 \[ -5+\left (-1-x-\frac {2 x-x^2}{x}+\log (\log (x))\right )^2 \]

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Rubi [A]  time = 0.04, antiderivative size = 9, normalized size of antiderivative = 0.35, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {12, 2301} \begin {gather*} (3-\log (\log (x)))^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6 + 2*Log[Log[x]])/(x*Log[x]),x]

[Out]

(3 - Log[Log[x]])^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {2 (-3+\log (x))}{x} \, dx,x,\log (x)\right )\\ &=2 \operatorname {Subst}\left (\int \frac {-3+\log (x)}{x} \, dx,x,\log (x)\right )\\ &=(3-\log (\log (x)))^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 11, normalized size = 0.42 \begin {gather*} -6 \log (\log (x))+\log ^2(\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 + 2*Log[Log[x]])/(x*Log[x]),x]

[Out]

-6*Log[Log[x]] + Log[Log[x]]^2

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fricas [A]  time = 0.74, size = 11, normalized size = 0.42 \begin {gather*} \log \left (\log \relax (x)\right )^{2} - 6 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(log(x))-6)/x/log(x),x, algorithm="fricas")

[Out]

log(log(x))^2 - 6*log(log(x))

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giac [A]  time = 0.42, size = 11, normalized size = 0.42 \begin {gather*} \log \left (\log \relax (x)\right )^{2} - 6 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(log(x))-6)/x/log(x),x, algorithm="giac")

[Out]

log(log(x))^2 - 6*log(log(x))

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maple [A]  time = 0.03, size = 12, normalized size = 0.46

method	result	size
derivativedivides	\(\ln \left (\ln \relax (x )\right )^{2}-6 \ln \left (\ln \relax (x )\right )\)	\(12\)
default	\(\ln \left (\ln \relax (x )\right )^{2}-6 \ln \left (\ln \relax (x )\right )\)	\(12\)
norman	\(\ln \left (\ln \relax (x )\right )^{2}-6 \ln \left (\ln \relax (x )\right )\)	\(12\)
risch	\(\ln \left (\ln \relax (x )\right )^{2}-6 \ln \left (\ln \relax (x )\right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*ln(ln(x))-6)/x/ln(x),x,method=_RETURNVERBOSE)

[Out]

ln(ln(x))^2-6*ln(ln(x))

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maxima [A]  time = 0.49, size = 7, normalized size = 0.27 \begin {gather*} {\left (\log \left (\log \relax (x)\right ) - 3\right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(log(x))-6)/x/log(x),x, algorithm="maxima")

[Out]

(log(log(x)) - 3)^2

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mupad [B]  time = 0.57, size = 9, normalized size = 0.35 \begin {gather*} \ln \left (\ln \relax (x)\right )\,\left (\ln \left (\ln \relax (x)\right )-6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(log(x)) - 6)/(x*log(x)),x)

[Out]

log(log(x))*(log(log(x)) - 6)

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sympy [A]  time = 0.26, size = 12, normalized size = 0.46 \begin {gather*} \log {\left (\log {\relax (x )} \right )}^{2} - 6 \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*ln(ln(x))-6)/x/ln(x),x)

[Out]

log(log(x))**2 - 6*log(log(x))

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