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

Optimal. Leaf size=8 \[ \text{PolyLog}\left (2,-\frac{b}{x}\right ) \]

[Out]

PolyLog[2, -(b/x)]

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Rubi [A]  time = 0.0077864, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2391} \[ \text{PolyLog}\left (2,-\frac{b}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[1 + b/x]/x,x]

[Out]

PolyLog[2, -(b/x)]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{\log \left (1+\frac{b}{x}\right )}{x} \, dx &=\text{Li}_2\left (-\frac{b}{x}\right )\\ \end{align*}

Mathematica [B]  time = 0.0033783, size = 34, normalized size = 4.25 \[ -\text{PolyLog}\left (2,-\frac{-b-x}{x}\right )-\log \left (-\frac{b}{x}\right ) \log \left (\frac{b+x}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[1 + b/x]/x,x]

[Out]

-(Log[-(b/x)]*Log[(b + x)/x]) - PolyLog[2, -((-b - x)/x)]

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Maple [A]  time = 0.059, size = 9, normalized size = 1.1 \begin{align*}{\it dilog} \left ( 1+{\frac{b}{x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

dilog(1+b/x)

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Maxima [B]  time = 1.04336, size = 47, normalized size = 5.88 \begin{align*} \log \left (b + x\right ) \log \left (x\right ) - \frac{1}{2} \, \log \left (x\right )^{2} - \log \left (x\right ) \log \left (\frac{x}{b} + 1\right ) -{\rm Li}_2\left (-\frac{x}{b}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b + x)*log(x) - 1/2*log(x)^2 - log(x)*log(x/b + 1) - dilog(-x/b)

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Fricas [A]  time = 2.24473, size = 31, normalized size = 3.88 \begin{align*}{\rm Li}_2\left (-\frac{b + x}{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

dilog(-(b + x)/x + 1)

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Sympy [C]  time = 4.31709, size = 8, normalized size = 1. \begin{align*} \operatorname{Li}_{2}\left (\frac{b e^{i \pi }}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

polylog(2, b*exp_polar(I*pi)/x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (\frac{b}{x} + 1\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(b/x + 1)/x, x)