3.268 \(\int \frac {\log (x)}{-1+x} \, dx\)

Optimal. Leaf size=9 \[ -\text {Li}_2(1-x) \]

[Out]

-polylog(2,1-x)

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Rubi [A]  time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2315} \[ -\text {PolyLog}(2,1-x) \]

Antiderivative was successfully verified.

[In]

Int[Log[x]/(-1 + x),x]

[Out]

-PolyLog[2, 1 - x]

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 (x)}{-1+x} \, dx &=-\text {Li}_2(1-x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 1.00 \[ -\text {Li}_2(1-x) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[x]/(-1 + x),x]

[Out]

-PolyLog[2, 1 - x]

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fricas [A]  time = 0.42, size = 8, normalized size = 0.89 \[ -{\rm Li}_2\left (-x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-dilog(-x + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \relax (x)}{x - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(x)/(x - 1), x)

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maple [A]  time = 0.06, size = 5, normalized size = 0.56 \[ -\dilog \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x)/(x-1),x)

[Out]

-dilog(x)

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maxima [A]  time = 0.64, size = 12, normalized size = 1.33 \[ \log \relax (x) \log \left (-x + 1\right ) + {\rm Li}_2\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)*log(-x + 1) + dilog(x)

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mupad [B]  time = 0.02, size = 4, normalized size = 0.44 \[ -{\mathrm {Li}}_{\mathrm {2}}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x)/(x - 1),x)

[Out]

-dilog(x)

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sympy [C]  time = 1.72, size = 10, normalized size = 1.11 \[ - \operatorname {Li}_{2}\left (\left (x - 1\right ) e^{i \pi }\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-polylog(2, (x - 1)*exp_polar(I*pi))

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