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]

-(x/Log[c*x]) + LogIntegral[c*x]/c

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Rubi [A]  time = 0.0050655, antiderivative size = 18, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0040804, size = 18, normalized size = 1. \[ \frac{\text{li}(c x)}{c}-\frac{x}{\log (c x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x/Log[c*x]) + LogIntegral[c*x]/c

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Maple [A]  time = 0.036, size = 24, normalized size = 1.3 \begin{align*} -{\frac{x}{\ln \left ( cx \right ) }}-{\frac{{\it Ei} \left ( 1,-\ln \left ( cx \right ) \right ) }{c}} \end{align*}

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 = 1.23525, size = 16, normalized size = 0.89 \begin{align*} \frac{\Gamma \left (-1, -\log \left (c x\right )\right )}{c} \end{align*}

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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Fricas [A]  time = 0.791876, size = 68, normalized size = 3.78 \begin{align*} -\frac{c x - \log \left (c x\right ) \logintegral \left (c x\right )}{c \log \left (c x\right )} \end{align*}

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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Sympy [A]  time = 0.487873, size = 12, normalized size = 0.67 \begin{align*} - \frac{x}{\log{\left (c x \right )}} + \frac{\operatorname{li}{\left (c x \right )}}{c} \end{align*}

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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Giac [A]  time = 1.11997, size = 26, normalized size = 1.44 \begin{align*} \frac{{\rm Ei}\left (\log \left (c x\right )\right )}{c} - \frac{x}{\log \left (c x\right )} \end{align*}

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)