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

Optimal. Leaf size=11 \[ c^2 \text{Ei}(-2 \log (c x)) \]

[Out]

c^2*ExpIntegralEi[-2*Log[c*x]]

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Rubi [A]  time = 0.0241797, antiderivative size = 11, 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} \[ c^2 \text{Ei}(-2 \log (c x)) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Log[c*x]),x]

[Out]

c^2*ExpIntegralEi[-2*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^3 \log (c x)} \, dx &=c^2 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{x} \, dx,x,\log (c x)\right )\\ &=c^2 \text{Ei}(-2 \log (c x))\\ \end{align*}

Mathematica [A]  time = 0.013946, size = 11, normalized size = 1. \[ c^2 \text{Ei}(-2 \log (c x)) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Log[c*x]),x]

[Out]

c^2*ExpIntegralEi[-2*Log[c*x]]

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Maple [A]  time = 0.038, size = 14, normalized size = 1.3 \begin{align*} -{c}^{2}{\it Ei} \left ( 1,2\,\ln \left ( cx \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.16887, size = 15, normalized size = 1.36 \begin{align*} c^{2}{\rm Ei}\left (-2 \, \log \left (c x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.724369, size = 42, normalized size = 3.82 \begin{align*} c^{2} \logintegral \left (\frac{1}{c^{2} x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^2*log_integral(1/(c^2*x^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \log{\left (c x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**3*log(c*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \log \left (c x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x^3*log(c*x)), x)