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

Optimal. Leaf size=8 \[ \text {Li}_2\left (-\frac {b}{x}\right ) \]

[Out]

polylog(2,-b/x)

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Rubi [A]  time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.00, size = 34, normalized size = 4.25 \[ -\text {Li}_2\left (-\frac {-b-x}{x}\right )-\log \left (-\frac {b}{x}\right ) \log \left (\frac {b+x}{x}\right ) \]

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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fricas [A]  time = 0.45, size = 11, normalized size = 1.38 \[ {\rm Li}_2\left (-\frac {b + x}{x} + 1\right ) \]

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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giac [B]  time = 0.35, size = 110, normalized size = 13.75 \[ -\frac {b^{3} {\left (\frac {1}{\frac {b + x}{x} - 1} - \log \left (\frac {{\left | b + x \right |}}{{\left | x \right |}}\right ) + \log \left ({\left | \frac {b + x}{x} - 1 \right |}\right )\right )} + \frac {b^{3} \log \left (-b {\left (\frac {{\left (b - \frac {1}{\frac {1}{b} - \frac {b + x}{b x}}\right )} {\left (\frac {1}{b} - \frac {b + x}{b x}\right )}}{b} + \frac {1}{b}\right )} + 1\right )}{{\left (\frac {b + x}{x} - 1\right )}^{2}}}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^3*(1/((b + x)/x - 1) - log(abs(b + x)/abs(x)) + log(abs((b + x)/x - 1))) + b^3*log(-b*((b - 1/(1/b - (
b + x)/(b*x)))*(1/b - (b + x)/(b*x))/b + 1/b) + 1)/((b + x)/x - 1)^2)/b^2

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maple [A]  time = 0.05, size = 9, normalized size = 1.12 \[ \dilog \left (\frac {b}{x}+1\right ) \]

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 = 0.71, size = 35, normalized size = 4.38 \[ \log \left (b + x\right ) \log \relax (x) - \frac {1}{2} \, \log \relax (x)^{2} - \log \relax (x) \log \left (\frac {x}{b} + 1\right ) - {\rm Li}_2\left (-\frac {x}{b}\right ) \]

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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mupad [B]  time = 0.26, size = 8, normalized size = 1.00 \[ \mathrm {polylog}\left (2,-\frac {b}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(b/x + 1)/x,x)

[Out]

polylog(2, -b/x)

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sympy [C]  time = 2.78, size = 8, normalized size = 1.00 \[ \operatorname {Li}_{2}\left (\frac {b e^{i \pi }}{x}\right ) \]

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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