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

Optimal. Leaf size=11 \[ \frac {\log (a+b \log (x))}{b} \]

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Rubi [A]  time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2302, 29} \[ \frac {\log (a+b \log (x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*Log[x])),x]

[Out]

Log[a + b*Log[x]]/b

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

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Mathematica [A]  time = 0.02, size = 11, normalized size = 1.00 \[ \frac {\log (a+b \log (x))}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*Log[x])),x]

[Out]

Log[a + b*Log[x]]/b

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

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 0.95, size = 11, normalized size = 1.00 \[ \frac {\log \left (b \log \relax (x) + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*log(x) + a)/b

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giac [B]  time = 0.60, size = 30, normalized size = 2.73 \[ \frac {\log \left (\frac {1}{4} \, \pi ^{2} b^{2} {\left (\mathrm {sgn}\relax (x) - 1\right )}^{2} + {\left (b \log \left ({\left | x \right |}\right ) + a\right )}^{2}\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(1/4*pi^2*b^2*(sgn(x) - 1)^2 + (b*log(abs(x)) + a)^2)/b

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maple [A]  time = 0.02, size = 12, normalized size = 1.09

method	result	size
derivativedivides	\(\frac {\ln \left (a +b \ln \relax (x )\right )}{b}\)	\(12\)
default	\(\frac {\ln \left (a +b \ln \relax (x )\right )}{b}\)	\(12\)
norman	\(\frac {\ln \left (a +b \ln \relax (x )\right )}{b}\)	\(12\)
risch	\(\frac {\ln \left (\ln \relax (x )+\frac {a}{b}\right )}{b}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(a+b*ln(x))/b

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maxima [A]  time = 0.42, size = 11, normalized size = 1.00 \[ \frac {\log \left (b \log \relax (x) + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*log(x) + a)/b

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mupad [B]  time = 0.29, size = 11, normalized size = 1.00 \[ \frac {\ln \left (a+b\,\ln \relax (x)\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*log(x))),x)

[Out]

log(a + b*log(x))/b

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sympy [A]  time = 0.13, size = 8, normalized size = 0.73 \[ \frac {\log {\left (\frac {a}{b} + \log {\relax (x )} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a/b + log(x))/b

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