\(\int \frac {1}{\log (c x)} \, dx\) [25]

Optimal result

Integrand size = 6, antiderivative size = 8 \[ \int \frac {1}{\log (c x)} \, dx=\frac {\operatorname {LogIntegral}(c x)}{c} \]

[Out]

Li(c*x)/c

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2335} \[ \int \frac {1}{\log (c x)} \, dx=\frac {\operatorname {LogIntegral}(c x)}{c} \]

[In]

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

[Out]

LogIntegral[c*x]/c

Rule 2335

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {li}(c x)}{c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\log (c x)} \, dx=\frac {\operatorname {LogIntegral}(c x)}{c} \]

[In]

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

[Out]

LogIntegral[c*x]/c

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.75

method	result	size
derivativedivides	\(-\frac {\operatorname {Ei}_{1}\left (-\ln \left (x c \right )\right )}{c}\)	\(14\)
default	\(-\frac {\operatorname {Ei}_{1}\left (-\ln \left (x c \right )\right )}{c}\)	\(14\)
risch	\(-\frac {\operatorname {Ei}_{1}\left (-\ln \left (x c \right )\right )}{c}\)	\(14\)

[In]

int(1/ln(x*c),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\log (c x)} \, dx=\frac {\operatorname {log\_integral}\left (c x\right )}{c} \]

[In]

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

[Out]

log_integral(c*x)/c

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\log (c x)} \, dx=\frac {\operatorname {li}{\left (c x \right )}}{c} \]

[In]

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

[Out]

li(c*x)/c

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\log (c x)} \, dx=\frac {{\rm Ei}\left (\log \left (c x\right )\right )}{c} \]

[In]

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

[Out]

Ei(log(c*x))/c

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\log (c x)} \, dx=\frac {{\rm Ei}\left (\log \left (c x\right )\right )}{c} \]

[In]

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

[Out]

Ei(log(c*x))/c

Mupad [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\log (c x)} \, dx=\frac {\mathrm {logint}\left (c\,x\right )}{c} \]

[In]

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

[Out]

logint(c*x)/c