3.23 \(\int \frac {1}{x (1+\log (\frac {x}{a}))} \, dx\)

Optimal. Leaf size=9 \[ \log \left (\log \left (\frac {x}{a}\right )+1\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2302, 29} \[ \log \left (\log \left (\frac {x}{a}\right )+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(1 + Log[x/a])),x]

[Out]

Log[1 + Log[x/a]]

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 {align*} \int \frac {1}{x \left (1+\log \left (\frac {x}{a}\right )\right )} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,1+\log \left (\frac {x}{a}\right )\right )\\ &=\log \left (1+\log \left (\frac {x}{a}\right )\right )\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 9, normalized size = 1.00 \[ \log \left (\log \left (\frac {x}{a}\right )+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 + Log[x/a])),x]

[Out]

Log[1 + Log[x/a]]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (1+\log \left (\frac {x}{a}\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[1/(x*(1 + Log[x/a])),x]

[Out]

Could not integrate

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fricas [A]  time = 1.38, size = 9, normalized size = 1.00 \[ \log \left (\log \left (\frac {x}{a}\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(1+log(x/a)),x, algorithm="fricas")

[Out]

log(log(x/a) + 1)

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giac [A]  time = 0.88, size = 9, normalized size = 1.00 \[ \log \left (\log \left (\frac {x}{a}\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(1+log(x/a)),x, algorithm="giac")

[Out]

log(log(x/a) + 1)

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maple [A]  time = 0.02, size = 10, normalized size = 1.11

method	result	size
derivativedivides	\(\ln \left (1+\ln \left (\frac {x}{a}\right )\right )\)	\(10\)
default	\(\ln \left (1+\ln \left (\frac {x}{a}\right )\right )\)	\(10\)
norman	\(\ln \left (1+\ln \left (\frac {x}{a}\right )\right )\)	\(10\)
risch	\(\ln \left (1+\ln \left (\frac {x}{a}\right )\right )\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(1+ln(x/a)),x,method=_RETURNVERBOSE)

[Out]

ln(1+ln(x/a))

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maxima [A]  time = 0.42, size = 9, normalized size = 1.00 \[ \log \left (\log \left (\frac {x}{a}\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(1+log(x/a)),x, algorithm="maxima")

[Out]

log(log(x/a) + 1)

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mupad [B]  time = 0.21, size = 9, normalized size = 1.00 \[ \ln \left (\ln \left (\frac {x}{a}\right )+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(log(x/a) + 1)),x)

[Out]

log(log(x/a) + 1)

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sympy [A]  time = 0.11, size = 7, normalized size = 0.78 \[ \log {\left (\log {\left (\frac {x}{a} \right )} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(1+ln(x/a)),x)

[Out]

log(log(x/a) + 1)

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