3.32 \(\int \frac {1}{\log ^2(c x)} \, dx\)

Optimal. Leaf size=18 \[ \frac {\text {li}(c x)}{c}-\frac {x}{\log (c x)} \]

[Out]

Li(c*x)/c-x/ln(c*x)

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2297, 2298} \[ \frac {\text {li}(c x)}{c}-\frac {x}{\log (c x)} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*x]^(-2),x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \[ \frac {\text {li}(c x)}{c}-\frac {x}{\log (c x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*x]^(-2),x]

[Out]

-(x/Log[c*x]) + LogIntegral[c*x]/c

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fricas [A]  time = 0.41, size = 25, normalized size = 1.39 \[ -\frac {c x - \log \left (c x\right ) \operatorname {log\_integral}\left (c x\right )}{c \log \left (c x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/log(c*x)^2,x, algorithm="fricas")

[Out]

-(c*x - log(c*x)*log_integral(c*x))/(c*log(c*x))

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giac [A]  time = 0.20, size = 19, normalized size = 1.06 \[ \frac {{\rm Ei}\left (\log \left (c x\right )\right )}{c} - \frac {x}{\log \left (c x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/log(c*x)^2,x, algorithm="giac")

[Out]

Ei(log(c*x))/c - x/log(c*x)

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maple [A]  time = 0.03, size = 24, normalized size = 1.33 \[ -\frac {\Ei \left (1, -\ln \left (c x \right )\right )}{c}-\frac {x}{\ln \left (c x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/ln(c*x)^2,x)

[Out]

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

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maxima [A]  time = 0.91, size = 12, normalized size = 0.67 \[ \frac {\Gamma \left (-1, -\log \left (c x\right )\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/log(c*x)^2,x, algorithm="maxima")

[Out]

gamma(-1, -log(c*x))/c

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mupad [B]  time = 3.34, size = 18, normalized size = 1.00 \[ \frac {\mathrm {logint}\left (c\,x\right )}{c}-\frac {x}{\ln \left (c\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/log(c*x)^2,x)

[Out]

logint(c*x)/c - x/log(c*x)

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sympy [A]  time = 0.50, size = 12, normalized size = 0.67 \[ - \frac {x}{\log {\left (c x \right )}} + \frac {\operatorname {li}{\left (c x \right )}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/ln(c*x)**2,x)

[Out]

-x/log(c*x) + li(c*x)/c

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