3.83.41 \(\int \frac {1}{x \log (2 x)} \, dx\)

Optimal. Leaf size=7 \[ \log (-\log (2 x)) \]

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Rubi [A]  time = 0.01, antiderivative size = 5, normalized size of antiderivative = 0.71, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2302, 29} \begin {gather*} \log (\log (2 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Log[2*x]),x]

[Out]

Log[Log[2*x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (2 x)\right )\\ &=\log (\log (2 x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 5, normalized size = 0.71 \begin {gather*} \log (\log (2 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Log[2*x]),x]

[Out]

Log[Log[2*x]]

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fricas [A]  time = 0.58, size = 5, normalized size = 0.71 \begin {gather*} \log \left (\log \left (2 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(2*x),x, algorithm="fricas")

[Out]

log(log(2*x))

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giac [A]  time = 0.12, size = 5, normalized size = 0.71 \begin {gather*} \log \left (\log \left (2 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(2*x),x, algorithm="giac")

[Out]

log(log(2*x))

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maple [A]  time = 0.02, size = 6, normalized size = 0.86

method	result	size
derivativedivides	\(\ln \left (\ln \left (2 x \right )\right )\)	\(6\)
default	\(\ln \left (\ln \left (2 x \right )\right )\)	\(6\)
norman	\(\ln \left (\ln \left (2 x \right )\right )\)	\(6\)
risch	\(\ln \left (\ln \left (2 x \right )\right )\)	\(6\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/ln(2*x),x,method=_RETURNVERBOSE)

[Out]

ln(ln(2*x))

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maxima [A]  time = 0.37, size = 5, normalized size = 0.71 \begin {gather*} \log \left (\log \left (2 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(2*x),x, algorithm="maxima")

[Out]

log(log(2*x))

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mupad [B]  time = 4.83, size = 5, normalized size = 0.71 \begin {gather*} \ln \left (\ln \left (2\,x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*log(2*x)),x)

[Out]

log(log(2*x))

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sympy [A]  time = 0.09, size = 5, normalized size = 0.71 \begin {gather*} \log {\left (\log {\left (2 x \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/ln(2*x),x)

[Out]

log(log(2*x))

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