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

Optimal. Leaf size=37 \[ \frac{2 \text{Ei}(2 \log (c x))}{c^2}-\frac{x^2}{2 \log ^2(c x)}-\frac{x^2}{\log (c x)} \]

[Out]

(2*ExpIntegralEi[2*Log[c*x]])/c^2 - x^2/(2*Log[c*x]^2) - x^2/Log[c*x]

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Rubi [A]  time = 0.0314825, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2306, 2309, 2178} \[ \frac{2 \text{Ei}(2 \log (c x))}{c^2}-\frac{x^2}{2 \log ^2(c x)}-\frac{x^2}{\log (c x)} \]

Antiderivative was successfully verified.

[In]

Int[x/Log[c*x]^3,x]

[Out]

(2*ExpIntegralEi[2*Log[c*x]])/c^2 - x^2/(2*Log[c*x]^2) - x^2/Log[c*x]

Rule 2306

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log
[c*x^n])^(p + 1))/(b*d*n*(p + 1)), x] - Dist[(m + 1)/(b*n*(p + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x]
, x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]

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

Mathematica [A]  time = 0.0049546, size = 37, normalized size = 1. \[ \frac{2 \text{Ei}(2 \log (c x))}{c^2}-\frac{x^2}{2 \log ^2(c x)}-\frac{x^2}{\log (c x)} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Log[c*x]^3,x]

[Out]

(2*ExpIntegralEi[2*Log[c*x]])/c^2 - x^2/(2*Log[c*x]^2) - x^2/Log[c*x]

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Maple [A]  time = 0.045, size = 37, normalized size = 1. \begin{align*} -{\frac{{x}^{2}}{2\, \left ( \ln \left ( cx \right ) \right ) ^{2}}}-{\frac{{x}^{2}}{\ln \left ( cx \right ) }}-2\,{\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)^3,x)

[Out]

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

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Maxima [A]  time = 1.15327, size = 18, normalized size = 0.49 \begin{align*} -\frac{4 \, \Gamma \left (-2, -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)^3,x, algorithm="maxima")

[Out]

-4*gamma(-2, -2*log(c*x))/c^2

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Fricas [A]  time = 0.852558, size = 123, normalized size = 3.32 \begin{align*} -\frac{2 \, c^{2} x^{2} \log \left (c x\right ) + c^{2} x^{2} - 4 \, \log \left (c x\right )^{2} \logintegral \left (c^{2} x^{2}\right )}{2 \, c^{2} \log \left (c x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*c^2*x^2*log(c*x) + c^2*x^2 - 4*log(c*x)^2*log_integral(c^2*x^2))/(c^2*log(c*x)^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{- 2 x^{2} \log{\left (c x \right )} - x^{2}}{2 \log{\left (c x \right )}^{2}} + 2 \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)**3,x)

[Out]

(-2*x**2*log(c*x) - x**2)/(2*log(c*x)**2) + 2*Integral(x/log(c*x), x)

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Giac [A]  time = 1.13404, size = 47, normalized size = 1.27 \begin{align*} -\frac{x^{2}}{\log \left (c x\right )} - \frac{x^{2}}{2 \, \log \left (c x\right )^{2}} + \frac{2 \,{\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)^3,x, algorithm="giac")

[Out]

-x^2/log(c*x) - 1/2*x^2/log(c*x)^2 + 2*Ei(2*log(c*x))/c^2