3.24 \(\int \frac{x}{\log (c x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0174104, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2309, 2178} \[ \frac{\text{Ei}(2 \log (c x))}{c^2} \]

Antiderivative was successfully verified.

[In]

Int[x/Log[c*x],x]

[Out]

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

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

Mathematica [A]  time = 0.013095, size = 11, normalized size = 1. \[ \frac{\text{Ei}(2 \log (c x))}{c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Log[c*x],x]

[Out]

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

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Maple [A]  time = 0.043, size = 14, normalized size = 1.3 \begin{align*} -{\frac{{\it Ei} \left ( 1,-2\,\ln \left ( cx \right ) \right ) }{{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/ln(c*x),x)

[Out]

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

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Maxima [A]  time = 1.13168, size = 15, normalized size = 1.36 \begin{align*} \frac{{\rm Ei}\left (2 \, \log \left (c x\right )\right )}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.836101, size = 36, normalized size = 3.27 \begin{align*} \frac{\logintegral \left (c^{2} x^{2}\right )}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/log(c*x), x)

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Giac [A]  time = 1.10774, size = 15, normalized size = 1.36 \begin{align*} \frac{{\rm Ei}\left (2 \, \log \left (c x\right )\right )}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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