3.74 \(\int \frac{\log (c x)}{1-c x} \, dx\)

Optimal. Leaf size=12 \[ \frac{\text{PolyLog}(2,1-c x)}{c} \]

[Out]

PolyLog[2, 1 - c*x]/c

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Rubi [A]  time = 0.0109987, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2315} \[ \frac{\text{PolyLog}(2,1-c x)}{c} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*x]/(1 - c*x),x]

[Out]

PolyLog[2, 1 - c*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 (c x)}{1-c x} \, dx &=\frac{\text{Li}_2(1-c x)}{c}\\ \end{align*}

Mathematica [A]  time = 0.0022577, size = 12, normalized size = 1. \[ \frac{\text{PolyLog}(2,1-c x)}{c} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*x]/(1 - c*x),x]

[Out]

PolyLog[2, 1 - c*x]/c

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Maple [A]  time = 0.038, size = 9, normalized size = 0.8 \begin{align*}{\frac{{\it dilog} \left ( cx \right ) }{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*dilog(c*x)

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Maxima [B]  time = 1.21025, size = 65, normalized size = 5.42 \begin{align*} -\frac{\log \left (c x - 1\right ) \log \left (c x\right )}{c} + \frac{\log \left (c x - 1\right ) \log \left (x\right )}{c} - \frac{\log \left (-c x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (c x\right )}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.977331, size = 26, normalized size = 2.17 \begin{align*} \frac{{\rm Li}_2\left (-c x + 1\right )}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

dilog(-c*x + 1)/c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(log(c*x)/(c*x - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-log(c*x)/(c*x - 1), x)