3.39 \(\int \frac {1}{\log ^3(c x)} \, dx\)

Optimal. Leaf size=34 \[ \frac {\text {li}(c x)}{2 c}-\frac {x}{2 \log ^2(c x)}-\frac {x}{2 \log (c x)} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2297, 2298} \[ \frac {\text {li}(c x)}{2 c}-\frac {x}{2 \log ^2(c x)}-\frac {x}{2 \log (c x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2297

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 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 ^3(c x)} \, dx &=-\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*}

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Mathematica [A]  time = 0.01, size = 34, normalized size = 1.00 \[ \frac {\text {li}(c x)}{2 c}-\frac {x}{2 \log ^2(c x)}-\frac {x}{2 \log (c x)} \]

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.40, size = 34, normalized size = 1.00 \[ -\frac {c x \log \left (c x\right ) - \log \left (c x\right )^{2} \operatorname {log\_integral}\left (c x\right ) + c x}{2 \, c \log \left (c x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.22, size = 29, normalized size = 0.85 \[ \frac {{\rm Ei}\left (\log \left (c x\right )\right )}{2 \, c} - \frac {x}{2 \, \log \left (c x\right )} - \frac {x}{2 \, \log \left (c x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.03, size = 33, normalized size = 0.97 \[ -\frac {\Ei \left (1, -\ln \left (c x \right )\right )}{2 c}-\frac {x}{2 \ln \left (c x \right )}-\frac {x}{2 \ln \left (c x \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.79, size = 13, normalized size = 0.38 \[ -\frac {\Gamma \left (-2, -\log \left (c x\right )\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-gamma(-2, -log(c*x))/c

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mupad [B]  time = 3.53, size = 29, normalized size = 0.85 \[ \frac {\mathrm {logint}\left (c\,x\right )}{2\,c}-\frac {\frac {x}{2}+\frac {x\,\ln \left (c\,x\right )}{2}}{{\ln \left (c\,x\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.50, size = 26, normalized size = 0.76 \[ \frac {- x \log {\left (c x \right )} - x}{2 \log {\left (c x \right )}^{2}} + \frac {\operatorname {li}{\left (c x \right )}}{2 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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