3.41 \(\int \frac{1}{x^2 \log ^3(c x)} \, dx\)

Optimal. Leaf size=39 \[ \frac{1}{2} c \text{Ei}(-\log (c x))-\frac{1}{2 x \log ^2(c x)}+\frac{1}{2 x \log (c x)} \]

[Out]

(c*ExpIntegralEi[-Log[c*x]])/2 - 1/(2*x*Log[c*x]^2) + 1/(2*x*Log[c*x])

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Rubi [A]  time = 0.0495498, antiderivative size = 39, 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} \[ \frac{1}{2} c \text{Ei}(-\log (c x))-\frac{1}{2 x \log ^2(c x)}+\frac{1}{2 x \log (c x)} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Log[c*x]^3),x]

[Out]

(c*ExpIntegralEi[-Log[c*x]])/2 - 1/(2*x*Log[c*x]^2) + 1/(2*x*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{1}{x^2 \log ^3(c x)} \, dx &=-\frac{1}{2 x \log ^2(c x)}-\frac{1}{2} \int \frac{1}{x^2 \log ^2(c x)} \, dx\\ &=-\frac{1}{2 x \log ^2(c x)}+\frac{1}{2 x \log (c x)}+\frac{1}{2} \int \frac{1}{x^2 \log (c x)} \, dx\\ &=-\frac{1}{2 x \log ^2(c x)}+\frac{1}{2 x \log (c x)}+\frac{1}{2} c \operatorname{Subst}\left (\int \frac{e^{-x}}{x} \, dx,x,\log (c x)\right )\\ &=\frac{1}{2} c \text{Ei}(-\log (c x))-\frac{1}{2 x \log ^2(c x)}+\frac{1}{2 x \log (c x)}\\ \end{align*}

Mathematica [A]  time = 0.0145465, size = 39, normalized size = 1. \[ \frac{1}{2} c \text{Ei}(-\log (c x))-\frac{1}{2 x \log ^2(c x)}+\frac{1}{2 x \log (c x)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Log[c*x]^3),x]

[Out]

(c*ExpIntegralEi[-Log[c*x]])/2 - 1/(2*x*Log[c*x]^2) + 1/(2*x*Log[c*x])

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Maple [A]  time = 0.034, size = 33, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,x \left ( \ln \left ( cx \right ) \right ) ^{2}}}+{\frac{1}{2\,x\ln \left ( cx \right ) }}-{\frac{c{\it Ei} \left ( 1,\ln \left ( cx \right ) \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/ln(c*x)^3,x)

[Out]

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

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Maxima [A]  time = 1.16639, size = 12, normalized size = 0.31 \begin{align*} -c \Gamma \left (-2, \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)^3,x, algorithm="maxima")

[Out]

-c*gamma(-2, log(c*x))

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Fricas [A]  time = 0.846329, size = 100, normalized size = 2.56 \begin{align*} \frac{c x \log \left (c x\right )^{2} \logintegral \left (\frac{1}{c x}\right ) + \log \left (c x\right ) - 1}{2 \, x \log \left (c x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/log(c*x)^3,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/ln(c*x)**3,x)

[Out]

Integral(1/(x**2*log(c*x)), x)/2 + (log(c*x) - 1)/(2*x*log(c*x)**2)

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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 )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/log(c*x)^3,x, algorithm="giac")

[Out]

integrate(1/(x^2*log(c*x)^3), x)