3.1.45 \(\int \frac {\log (1+\frac {b}{x})}{x} \, dx\) [45]

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 = {2438} \begin {gather*} \text {PolyLog}\left (2,-\frac {b}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[1 + b/x]/x,x]

[Out]

PolyLog[2, -(b/x)]

Rule 2438

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] Leaf count is larger than twice the leaf count of optimal. \(34\) vs. \(2(8)=16\).
time = 0.00, size = 34, normalized size = 4.25 \begin {gather*} -\log \left (-\frac {b}{x}\right ) \log \left (\frac {b+x}{x}\right )-\text {Li}_2\left (-\frac {-b-x}{x}\right ) \end {gather*}

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 = 0.33, size = 9, normalized size = 1.12

method	result	size
derivativedivides	\(\dilog \left (1+\frac {b}{x}\right )\)	\(9\)
default	\(\dilog \left (1+\frac {b}{x}\right )\)	\(9\)
risch	\(\dilog \left (1+\frac {b}{x}\right )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

dilog(1+b/x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (7) = 14\).
time = 0.28, size = 35, normalized size = 4.38 \begin {gather*} \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 {gather*}

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 = 0.39, size = 11, normalized size = 1.38 \begin {gather*} {\rm Li}_2\left (-\frac {b + x}{x} + 1\right ) \end {gather*}

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] Result contains complex when optimal does not.
time = 1.58, size = 8, normalized size = 1.00 \begin {gather*} \operatorname {Li}_{2}\left (\frac {b e^{i \pi }}{x}\right ) \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (7) = 14\).
time = 2.99, size = 110, normalized size = 13.75 \begin {gather*} -\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}} \end {gather*}

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

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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