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

Optimal. Leaf size=9 \[ c \text{Ei}(-\log (c x)) \]

[Out]

c*ExpIntegralEi[-Log[c*x]]

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Rubi [A]  time = 0.0236958, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2309, 2178} \[ c \text{Ei}(-\log (c x)) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Log[c*x]),x]

[Out]

c*ExpIntegralEi[-Log[c*x]]

Rule 2309

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

\begin{align*} \int \frac{1}{x^2 \log (c x)} \, dx &=c \operatorname{Subst}\left (\int \frac{e^{-x}}{x} \, dx,x,\log (c x)\right )\\ &=c \text{Ei}(-\log (c x))\\ \end{align*}

Mathematica [A]  time = 0.0137685, size = 9, normalized size = 1. \[ c \text{Ei}(-\log (c x)) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Log[c*x]),x]

[Out]

c*ExpIntegralEi[-Log[c*x]]

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Maple [A]  time = 0.038, size = 10, normalized size = 1.1 \begin{align*} -c{\it Ei} \left ( 1,\ln \left ( cx \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.21867, size = 12, normalized size = 1.33 \begin{align*} c{\rm Ei}\left (-\log \left (c x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*Ei(-log(c*x))

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Fricas [A]  time = 0.840993, size = 34, normalized size = 3.78 \begin{align*} c \logintegral \left (\frac{1}{c x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*log_integral(1/(c*x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \log{\left (c x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**2*log(c*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \log \left (c x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x^2*log(c*x)), x)