3.37 \(\int \frac{x^2}{\log ^3(c x)} \, dx\)

Optimal. Leaf size=41 \[ \frac{9 \text{Ei}(3 \log (c x))}{2 c^3}-\frac{x^3}{2 \log ^2(c x)}-\frac{3 x^3}{2 \log (c x)} \]

[Out]

(9*ExpIntegralEi[3*Log[c*x]])/(2*c^3) - x^3/(2*Log[c*x]^2) - (3*x^3)/(2*Log[c*x])

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Rubi [A]  time = 0.05251, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2306, 2309, 2178} \[ \frac{9 \text{Ei}(3 \log (c x))}{2 c^3}-\frac{x^3}{2 \log ^2(c x)}-\frac{3 x^3}{2 \log (c x)} \]

Antiderivative was successfully verified.

[In]

Int[x^2/Log[c*x]^3,x]

[Out]

(9*ExpIntegralEi[3*Log[c*x]])/(2*c^3) - x^3/(2*Log[c*x]^2) - (3*x^3)/(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^2}{\log ^3(c x)} \, dx &=-\frac{x^3}{2 \log ^2(c x)}+\frac{3}{2} \int \frac{x^2}{\log ^2(c x)} \, dx\\ &=-\frac{x^3}{2 \log ^2(c x)}-\frac{3 x^3}{2 \log (c x)}+\frac{9}{2} \int \frac{x^2}{\log (c x)} \, dx\\ &=-\frac{x^3}{2 \log ^2(c x)}-\frac{3 x^3}{2 \log (c x)}+\frac{9 \operatorname{Subst}\left (\int \frac{e^{3 x}}{x} \, dx,x,\log (c x)\right )}{2 c^3}\\ &=\frac{9 \text{Ei}(3 \log (c x))}{2 c^3}-\frac{x^3}{2 \log ^2(c x)}-\frac{3 x^3}{2 \log (c x)}\\ \end{align*}

Mathematica [A]  time = 0.0148196, size = 41, normalized size = 1. \[ \frac{9 \text{Ei}(3 \log (c x))}{2 c^3}-\frac{x^3}{2 \log ^2(c x)}-\frac{3 x^3}{2 \log (c x)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Log[c*x]^3,x]

[Out]

(9*ExpIntegralEi[3*Log[c*x]])/(2*c^3) - x^3/(2*Log[c*x]^2) - (3*x^3)/(2*Log[c*x])

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Maple [A]  time = 0.034, size = 37, normalized size = 0.9 \begin{align*} -{\frac{{x}^{3}}{2\, \left ( \ln \left ( cx \right ) \right ) ^{2}}}-{\frac{3\,{x}^{3}}{2\,\ln \left ( cx \right ) }}-{\frac{9\,{\it Ei} \left ( 1,-3\,\ln \left ( cx \right ) \right ) }{2\,{c}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/ln(c*x)^3,x)

[Out]

-1/2*x^3/ln(c*x)^2-3/2*x^3/ln(c*x)-9/2/c^3*Ei(1,-3*ln(c*x))

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Maxima [A]  time = 1.12459, size = 18, normalized size = 0.44 \begin{align*} -\frac{9 \, \Gamma \left (-2, -3 \, \log \left (c x\right )\right )}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/log(c*x)^3,x, algorithm="maxima")

[Out]

-9*gamma(-2, -3*log(c*x))/c^3

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Fricas [A]  time = 0.755635, size = 123, normalized size = 3. \begin{align*} -\frac{3 \, c^{3} x^{3} \log \left (c x\right ) + c^{3} x^{3} - 9 \, \log \left (c x\right )^{2} \logintegral \left (c^{3} x^{3}\right )}{2 \, c^{3} \log \left (c x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/log(c*x)^3,x, algorithm="fricas")

[Out]

-1/2*(3*c^3*x^3*log(c*x) + c^3*x^3 - 9*log(c*x)^2*log_integral(c^3*x^3))/(c^3*log(c*x)^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{- 3 x^{3} \log{\left (c x \right )} - x^{3}}{2 \log{\left (c x \right )}^{2}} + \frac{9 \int \frac{x^{2}}{\log{\left (c x \right )}}\, dx}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/ln(c*x)**3,x)

[Out]

(-3*x**3*log(c*x) - x**3)/(2*log(c*x)**2) + 9*Integral(x**2/log(c*x), x)/2

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Giac [A]  time = 1.12121, size = 47, normalized size = 1.15 \begin{align*} -\frac{3 \, x^{3}}{2 \, \log \left (c x\right )} - \frac{x^{3}}{2 \, \log \left (c x\right )^{2}} + \frac{9 \,{\rm Ei}\left (3 \, \log \left (c x\right )\right )}{2 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/log(c*x)^3,x, algorithm="giac")

[Out]

-3/2*x^3/log(c*x) - 1/2*x^3/log(c*x)^2 + 9/2*Ei(3*log(c*x))/c^3