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

Optimal. Leaf size=8 \[ \frac {\text {li}(c x)}{c} \]

[Out]

Li(c*x)/c

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Rubi [A]  time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2298} \[ \frac {\text {li}(c x)}{c} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*x]^(-1),x]

[Out]

LogIntegral[c*x]/c

Rule 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

Rubi steps

\begin {align*} \int \frac {1}{\log (c x)} \, dx &=\frac {\text {li}(c x)}{c}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 1.00 \[ \frac {\text {li}(c x)}{c} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*x]^(-1),x]

[Out]

LogIntegral[c*x]/c

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fricas [A]  time = 0.40, size = 8, normalized size = 1.00 \[ \frac {\operatorname {log\_integral}\left (c x\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log_integral(c*x)/c

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giac [A]  time = 0.18, size = 9, normalized size = 1.12 \[ \frac {{\rm Ei}\left (\log \left (c x\right )\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(log(c*x))/c

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maple [A]  time = 0.03, size = 14, normalized size = 1.75 \[ -\frac {\Ei \left (1, -\ln \left (c x \right )\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/c*Ei(1,-ln(c*x))

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maxima [A]  time = 0.89, size = 9, normalized size = 1.12 \[ \frac {{\rm Ei}\left (\log \left (c x\right )\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(log(c*x))/c

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mupad [B]  time = 3.43, size = 8, normalized size = 1.00 \[ \frac {\mathrm {logint}\left (c\,x\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/log(c*x),x)

[Out]

logint(c*x)/c

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sympy [A]  time = 0.48, size = 5, normalized size = 0.62 \[ \frac {\operatorname {li}{\left (c x \right )}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

li(c*x)/c

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