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

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

[Out]

Log[a + b*Log[x]]/b

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Rubi [A]  time = 0.0237006, antiderivative size = 11, normalized size of antiderivative = 1., 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 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]

Rule 29

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

Mathematica [A]  time = 0.0147001, size = 11, normalized size = 1. \[ \frac{\log (a+b \log (x))}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*Log[x])),x]

[Out]

Log[a + b*Log[x]]/b

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Maple [A]  time = 0.001, size = 12, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+b\ln \left ( x \right ) \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.934022, size = 15, normalized size = 1.36 \begin{align*} \frac{\log \left (b \log \left (x\right ) + a\right )}{b} \end{align*}

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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Fricas [A]  time = 2.39609, size = 28, normalized size = 2.55 \begin{align*} \frac{\log \left (b \log \left (x\right ) + a\right )}{b} \end{align*}

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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Sympy [A]  time = 0.112607, size = 8, normalized size = 0.73 \begin{align*} \frac{\log{\left (\frac{a}{b} + \log{\left (x \right )} \right )}}{b} \end{align*}

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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Giac [B]  time = 1.08645, size = 41, normalized size = 3.73 \begin{align*} \frac{\log \left (\frac{1}{4} \, \pi ^{2} b^{2}{\left (\mathrm{sgn}\left (x\right ) - 1\right )}^{2} +{\left (b \log \left ({\left | x \right |}\right ) + a\right )}^{2}\right )}{2 \, b} \end{align*}

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