\(\int \frac {1}{x \log (c x)} \, dx\) [26]

Optimal result

Integrand size = 10, antiderivative size = 5 \[ \int \frac {1}{x \log (c x)} \, dx=\log (\log (c x)) \]

[Out]

ln(ln(c*x))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2339, 29} \[ \int \frac {1}{x \log (c x)} \, dx=\log (\log (c x)) \]

[In]

Int[1/(x*Log[c*x]),x]

[Out]

Log[Log[c*x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (c x)\right ) \\ & = \log (\log (c x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \log (c x)} \, dx=\log (\log (c x)) \]

[In]

Integrate[1/(x*Log[c*x]),x]

[Out]

Log[Log[c*x]]

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.20

method	result	size
derivativedivides	\(\ln \left (\ln \left (x c \right )\right )\)	\(6\)
default	\(\ln \left (\ln \left (x c \right )\right )\)	\(6\)
norman	\(\ln \left (\ln \left (x c \right )\right )\)	\(6\)
risch	\(\ln \left (\ln \left (x c \right )\right )\)	\(6\)
parallelrisch	\(\ln \left (\ln \left (x c \right )\right )\)	\(6\)

[In]

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

[Out]

ln(ln(x*c))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \log (c x)} \, dx=\log \left (\log \left (c x\right )\right ) \]

[In]

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

[Out]

log(log(c*x))

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \log (c x)} \, dx=\log {\left (\log {\left (c x \right )} \right )} \]

[In]

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

[Out]

log(log(c*x))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \log (c x)} \, dx=\log \left (\log \left (c x\right )\right ) \]

[In]

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

[Out]

log(log(c*x))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \log (c x)} \, dx=\log \left (\log \left (c x\right )\right ) \]

[In]

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

[Out]

log(log(c*x))

Mupad [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \log (c x)} \, dx=\ln \left (\ln \left (c\,x\right )\right ) \]

[In]

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

[Out]

log(log(c*x))