3.75 \(\int \frac {\log (\frac {x}{c})}{c-x} \, dx\)

Optimal. Leaf size=10 \[ \text {Li}_2\left (1-\frac {x}{c}\right ) \]

[Out]

polylog(2,1-x/c)

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2315} \[ \text {PolyLog}\left (2,1-\frac {x}{c}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[x/c]/(c - x),x]

[Out]

PolyLog[2, 1 - x/c]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.10 \[ \text {Li}_2\left (\frac {c-x}{c}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[x/c]/(c - x),x]

[Out]

PolyLog[2, (c - x)/c]

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fricas [A]  time = 0.60, size = 9, normalized size = 0.90 \[ {\rm Li}_2\left (-\frac {x}{c} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x/c)/(c-x),x, algorithm="fricas")

[Out]

dilog(-x/c + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\frac {x}{c}\right )}{c - x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x/c)/(c-x),x, algorithm="giac")

[Out]

integrate(log(x/c)/(c - x), x)

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maple [A]  time = 0.03, size = 7, normalized size = 0.70 \[ \dilog \left (\frac {x}{c}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x/c)/(c-x),x)

[Out]

dilog(x/c)

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maxima [B]  time = 0.57, size = 45, normalized size = 4.50 \[ \log \left (c - x\right ) \log \relax (x) - \log \left (c - x\right ) \log \left (\frac {x}{c}\right ) - \log \relax (x) \log \left (-\frac {x}{c} + 1\right ) - {\rm Li}_2\left (\frac {x}{c}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x/c)/(c-x),x, algorithm="maxima")

[Out]

log(c - x)*log(x) - log(c - x)*log(x/c) - log(x)*log(-x/c + 1) - dilog(x/c)

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mupad [B]  time = 3.50, size = 6, normalized size = 0.60 \[ {\mathrm {Li}}_{\mathrm {2}}\left (\frac {x}{c}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x/c)/(c - x),x)

[Out]

dilog(x/c)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\log {\left (\frac {x}{c} \right )}}{- c + x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(x/c)/(c-x),x)

[Out]

-Integral(log(x/c)/(-c + x), x)

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