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

Optimal. Leaf size=12 \[ \frac {\text {Li}_2(1-c x)}{c} \]

[Out]

polylog(2,-c*x+1)/c

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \[ \frac {\text {Li}_2(1-c x)}{c} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*x]/(1 - c*x),x]

[Out]

PolyLog[2, 1 - c*x]/c

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fricas [A]  time = 0.84, size = 11, normalized size = 0.92 \[ \frac {{\rm Li}_2\left (-c x + 1\right )}{c} \]

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

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)

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maple [A]  time = 0.04, size = 9, normalized size = 0.75 \[ \frac {\dilog \left (c x \right )}{c} \]

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 = 0.54, size = 48, normalized size = 4.00 \[ -\frac {\log \left (c x - 1\right ) \log \left (c x\right )}{c} + \frac {\log \left (c x - 1\right ) \log \relax (x)}{c} - \frac {\log \left (-c x + 1\right ) \log \relax (x) + {\rm Li}_2\left (c x\right )}{c} \]

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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mupad [B]  time = 3.46, size = 8, normalized size = 0.67 \[ \frac {{\mathrm {Li}}_{\mathrm {2}}\left (c\,x\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-log(c*x)/(c*x - 1),x)

[Out]

dilog(c*x)/c

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\log {\left (c x \right )}}{c x - 1}\, dx \]

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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