\(\int \frac {1}{\log ^3(c x)} \, dx\) [39]

Optimal result

Integrand size = 6, antiderivative size = 34 \[ \int \frac {1}{\log ^3(c x)} \, dx=-\frac {x}{2 \log ^2(c x)}-\frac {x}{2 \log (c x)}+\frac {\operatorname {LogIntegral}(c x)}{2 c} \]

[Out]

1/2*Li(c*x)/c-1/2*x/ln(c*x)^2-1/2*x/ln(c*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2334, 2335} \[ \int \frac {1}{\log ^3(c x)} \, dx=\frac {\operatorname {LogIntegral}(c x)}{2 c}-\frac {x}{2 \log ^2(c x)}-\frac {x}{2 \log (c x)} \]

[In]

Int[Log[c*x]^(-3),x]

[Out]

-1/2*x/Log[c*x]^2 - x/(2*Log[c*x]) + LogIntegral[c*x]/(2*c)

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x*((a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1)))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

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 {x}{2 \log ^2(c x)}+\frac {1}{2} \int \frac {1}{\log ^2(c x)} \, dx \\ & = -\frac {x}{2 \log ^2(c x)}-\frac {x}{2 \log (c x)}+\frac {1}{2} \int \frac {1}{\log (c x)} \, dx \\ & = -\frac {x}{2 \log ^2(c x)}-\frac {x}{2 \log (c x)}+\frac {\text {li}(c x)}{2 c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\log ^3(c x)} \, dx=-\frac {x}{2 \log ^2(c x)}-\frac {x}{2 \log (c x)}+\frac {\operatorname {LogIntegral}(c x)}{2 c} \]

[In]

Integrate[Log[c*x]^(-3),x]

[Out]

-1/2*x/Log[c*x]^2 - x/(2*Log[c*x]) + LogIntegral[c*x]/(2*c)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88

method	result	size
risch	\(-\frac {x \left (1+\ln \left (x c \right )\right )}{2 \ln \left (x c \right )^{2}}-\frac {\operatorname {Ei}_{1}\left (-\ln \left (x c \right )\right )}{2 c}\)	\(30\)
derivativedivides	\(\frac {-\frac {x c}{2 \ln \left (x c \right )^{2}}-\frac {x c}{2 \ln \left (x c \right )}-\frac {\operatorname {Ei}_{1}\left (-\ln \left (x c \right )\right )}{2}}{c}\)	\(36\)
default	\(\frac {-\frac {x c}{2 \ln \left (x c \right )^{2}}-\frac {x c}{2 \ln \left (x c \right )}-\frac {\operatorname {Ei}_{1}\left (-\ln \left (x c \right )\right )}{2}}{c}\)	\(36\)

[In]

int(1/ln(x*c)^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*x*(1+ln(x*c))/ln(x*c)^2-1/2/c*Ei(1,-ln(x*c))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\log ^3(c x)} \, dx=-\frac {c x \log \left (c x\right ) - \log \left (c x\right )^{2} \operatorname {log\_integral}\left (c x\right ) + c x}{2 \, c \log \left (c x\right )^{2}} \]

[In]

integrate(1/log(c*x)^3,x, algorithm="fricas")

[Out]

-1/2*(c*x*log(c*x) - log(c*x)^2*log_integral(c*x) + c*x)/(c*log(c*x)^2)

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\log ^3(c x)} \, dx=\frac {- x \log {\left (c x \right )} - x}{2 \log {\left (c x \right )}^{2}} + \frac {\operatorname {li}{\left (c x \right )}}{2 c} \]

[In]

integrate(1/ln(c*x)**3,x)

[Out]

(-x*log(c*x) - x)/(2*log(c*x)**2) + li(c*x)/(2*c)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.38 \[ \int \frac {1}{\log ^3(c x)} \, dx=-\frac {\Gamma \left (-2, -\log \left (c x\right )\right )}{c} \]

[In]

integrate(1/log(c*x)^3,x, algorithm="maxima")

[Out]

-gamma(-2, -log(c*x))/c

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\log ^3(c x)} \, dx=\frac {{\rm Ei}\left (\log \left (c x\right )\right )}{2 \, c} - \frac {x}{2 \, \log \left (c x\right )} - \frac {x}{2 \, \log \left (c x\right )^{2}} \]

[In]

integrate(1/log(c*x)^3,x, algorithm="giac")

[Out]

1/2*Ei(log(c*x))/c - 1/2*x/log(c*x) - 1/2*x/log(c*x)^2

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\log ^3(c x)} \, dx=\frac {\mathrm {logint}\left (c\,x\right )}{2\,c}-\frac {\frac {x}{2}+\frac {x\,\ln \left (c\,x\right )}{2}}{{\ln \left (c\,x\right )}^2} \]

[In]

int(1/log(c*x)^3,x)

[Out]

logint(c*x)/(2*c) - (x/2 + (x*log(c*x))/2)/log(c*x)^2