3.3.14 \(\int \frac {\log (2 x)}{x \log (x)} \, dx\) [214]

Optimal. Leaf size=9 \[ \log (x)+\log (2) \log (\log (x)) \]

[Out]

ln(x)+ln(2)*ln(ln(x))

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Rubi [A]
time = 0.02, antiderivative size = 18, normalized size of antiderivative = 2.00, number of steps used = 2, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2339, 29, 2413, 2601} \begin {gather*} -\log (\log (x)) \log (x)+\log (x)+\log (2 x) \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] - Log[x]*Log[Log[x]] + Log[2*x]*Log[Log[x]]

Rule 29

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

Rule 2339

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]

Rule 2413

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rule 2601

Int[((a_.) + Log[Log[(d_.)*(x_)^(n_.)]^(p_.)*(c_.)]*(b_.))/(x_), x_Symbol] :> Simp[Log[d*x^n]*((a + b*Log[c*Lo
g[d*x^n]^p])/n), x] - Simp[b*p*Log[x], x] /; FreeQ[{a, b, c, d, n, p}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\log (2 x) \log (\log (x))-\int \frac {\log (\log (x))}{x} \, dx\\ &=\log (x)-\log (x) \log (\log (x))+\log (2 x) \log (\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 9, normalized size = 1.00 \begin {gather*} \log (x)+\log (2) \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] + Log[2]*Log[Log[x]]

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Maple [A]
time = 0.03, size = 10, normalized size = 1.11

method	result	size
default	\(\ln \left (x \right )+\ln \left (2\right ) \ln \left (\ln \left (x \right )\right )\)	\(10\)
risch	\(\ln \left (x \right )+\ln \left (2\right ) \ln \left (\ln \left (x \right )\right )\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x)+ln(2)*ln(ln(x))

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Maxima [A]
time = 0.34, size = 18, normalized size = 2.00 \begin {gather*} \log \left (2 \, x\right ) \log \left (\log \left (x\right )\right ) - \log \left (x\right ) \log \left (\log \left (x\right )\right ) + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.59, size = 9, normalized size = 1.00 \begin {gather*} \log \left (2\right ) \log \left (\log \left (x\right )\right ) + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2)*log(log(x)) + log(x)

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Sympy [A]
time = 0.04, size = 10, normalized size = 1.11 \begin {gather*} \log {\left (x \right )} + \log {\left (2 \right )} \log {\left (\log {\left (x \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) + log(2)*log(log(x))

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Giac [A]
time = 0.46, size = 10, normalized size = 1.11 \begin {gather*} \log \left (2\right ) \log \left ({\left | \log \left (x\right ) \right |}\right ) + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2)*log(abs(log(x))) + log(x)

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Mupad [B]
time = 0.14, size = 9, normalized size = 1.00 \begin {gather*} \ln \left (x\right )+\ln \left (\ln \left (x\right )\right )\,\ln \left (2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) + log(log(x))*log(2)

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Chatgpt [F] Failed to verify
time = 1.00, size = 9, normalized size = 1.00 \begin {gather*} \hyperbolicCosineIntegral \left (2 x \right )-\hyperbolicCosineIntegral \left (x \right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(ln(2*x)/x/ln(x),x)

[Out]

Li(2*x)-Li(x)

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