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

Optimal. Leaf size=10 \[ \text{PolyLog}\left (2,1-\frac{x}{c}\right ) \]

[Out]

PolyLog[2, 1 - x/c]

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Rubi [A]  time = 0.0105183, antiderivative size = 10, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0022178, size = 11, normalized size = 1.1 \[ \text{PolyLog}\left (2,\frac{c-x}{c}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[x/c]/(c - x),x]

[Out]

PolyLog[2, (c - x)/c]

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Maple [A]  time = 0.039, size = 7, normalized size = 0.7 \begin{align*}{\it dilog} \left ({\frac{x}{c}} \right ) \end{align*}

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 = 1.1079, size = 61, normalized size = 6.1 \begin{align*} \log \left (c - x\right ) \log \left (x\right ) - \log \left (c - x\right ) \log \left (\frac{x}{c}\right ) - \log \left (x\right ) \log \left (-\frac{x}{c} + 1\right ) -{\rm Li}_2\left (\frac{x}{c}\right ) \end{align*}

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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Fricas [A]  time = 0.934072, size = 23, normalized size = 2.3 \begin{align*}{\rm Li}_2\left (-\frac{x}{c} + 1\right ) \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\log{\left (\frac{x}{c} \right )}}{- c + x}\, dx \end{align*}

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

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)