3.101.78 \(\int \frac {-1+\log (3 x)}{x \log (3 x)} \, dx\)

Optimal. Leaf size=9 \[ \log \left (\frac {x}{\log (3 x)}\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 10, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2365, 43} \begin {gather*} \log (x)-\log (\log (3 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + Log[3*x])/(x*Log[3*x]),x]

[Out]

Log[x] - Log[Log[3*x]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2365

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {-1+x}{x} \, dx,x,\log (3 x)\right )\\ &=\operatorname {Subst}\left (\int \left (1-\frac {1}{x}\right ) \, dx,x,\log (3 x)\right )\\ &=\log (x)-\log (\log (3 x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 10, normalized size = 1.11 \begin {gather*} \log (x)-\log (\log (3 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + Log[3*x])/(x*Log[3*x]),x]

[Out]

Log[x] - Log[Log[3*x]]

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fricas [A]  time = 1.00, size = 12, normalized size = 1.33 \begin {gather*} \log \left (3 \, x\right ) - \log \left (\log \left (3 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(3*x)-1)/x/log(3*x),x, algorithm="fricas")

[Out]

log(3*x) - log(log(3*x))

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giac [A]  time = 0.25, size = 10, normalized size = 1.11 \begin {gather*} \log \relax (x) - \log \left (\log \left (3 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(3*x)-1)/x/log(3*x),x, algorithm="giac")

[Out]

log(x) - log(log(3*x))

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maple [A]  time = 0.02, size = 11, normalized size = 1.22

method	result	size
risch	\(\ln \relax (x )-\ln \left (\ln \left (3 x \right )\right )\)	\(11\)
derivativedivides	\(\ln \left (3 x \right )-\ln \left (\ln \left (3 x \right )\right )\)	\(13\)
default	\(\ln \left (3 x \right )-\ln \left (\ln \left (3 x \right )\right )\)	\(13\)
norman	\(\ln \left (3 x \right )-\ln \left (\ln \left (3 x \right )\right )\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(3*x)-1)/x/ln(3*x),x,method=_RETURNVERBOSE)

[Out]

ln(x)-ln(ln(3*x))

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maxima [A]  time = 0.34, size = 12, normalized size = 1.33 \begin {gather*} \log \left (3 \, x\right ) - \log \left (\log \left (3 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(3*x)-1)/x/log(3*x),x, algorithm="maxima")

[Out]

log(3*x) - log(log(3*x))

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mupad [B]  time = 7.68, size = 10, normalized size = 1.11 \begin {gather*} \ln \relax (x)-\ln \left (\ln \left (3\,x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3*x) - 1)/(x*log(3*x)),x)

[Out]

log(x) - log(log(3*x))

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sympy [A]  time = 0.10, size = 8, normalized size = 0.89 \begin {gather*} \log {\relax (x )} - \log {\left (\log {\left (3 x \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(3*x)-1)/x/ln(3*x),x)

[Out]

log(x) - log(log(3*x))

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