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

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

[Out]

LogIntegral[c*x]/c

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Rubi [A]  time = 0.0028761, antiderivative size = 8, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0045917, size = 8, normalized size = 1. \[ \frac{\text{li}(c x)}{c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

LogIntegral[c*x]/c

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Maple [A]  time = 0.037, size = 14, normalized size = 1.8 \begin{align*} -{\frac{{\it Ei} \left ( 1,-\ln \left ( cx \right ) \right ) }{c}} \end{align*}

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 = 1.18215, size = 12, normalized size = 1.5 \begin{align*} \frac{{\rm Ei}\left (\log \left (c x\right )\right )}{c} \end{align*}

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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Fricas [A]  time = 0.844042, size = 28, normalized size = 3.5 \begin{align*} \frac{\logintegral \left (c x\right )}{c} \end{align*}

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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Sympy [A]  time = 0.510081, size = 5, normalized size = 0.62 \begin{align*} \frac{\operatorname{li}{\left (c x \right )}}{c} \end{align*}

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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Giac [A]  time = 1.10455, size = 12, normalized size = 1.5 \begin{align*} \frac{{\rm Ei}\left (\log \left (c x\right )\right )}{c} \end{align*}

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