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

Optimal. Leaf size=9 $-\text{PolyLog}(2,1-x)$

[Out]

-PolyLog[2, 1 - x]

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Rubi [A]  time = 0.0089794, antiderivative size = 9, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0015296, size = 9, normalized size = 1. $-\text{PolyLog}(2,1-x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[x]/(-1 + x),x]

[Out]

-PolyLog[2, 1 - x]

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Maple [A]  time = 0.001, size = 5, normalized size = 0.6 \begin{align*} -{\it dilog} \left ( x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-dilog(x)

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Maxima [A]  time = 1.0358, size = 16, normalized size = 1.78 \begin{align*} \log \left (x\right ) \log \left (-x + 1\right ) +{\rm Li}_2\left (x\right ) \end{align*}

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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Fricas [A]  time = 2.06785, size = 22, normalized size = 2.44 \begin{align*} -{\rm Li}_2\left (-x + 1\right ) \end{align*}

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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Sympy [C]  time = 1.92516, size = 10, normalized size = 1.11 \begin{align*} - \operatorname{Li}_{2}\left (\left (x - 1\right ) e^{i \pi }\right ) \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (x\right )}{x - 1}\,{d x} \end{align*}

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)