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

Optimal. Leaf size=10 \[ -\frac{1}{2 \log ^2(c x)} \]

[Out]

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

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Rubi [A]  time = 0.0121339, antiderivative size = 10, 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 = {2302, 30} \[ -\frac{1}{2 \log ^2(c x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{x \log ^3(c x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x^3} \, dx,x,\log (c x)\right )\\ &=-\frac{1}{2 \log ^2(c x)}\\ \end{align*}

Mathematica [A]  time = 0.0010784, size = 10, normalized size = 1. \[ -\frac{1}{2 \log ^2(c x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.033, size = 9, normalized size = 0.9 \begin{align*} -{\frac{1}{2\, \left ( \ln \left ( cx \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.10878, size = 11, normalized size = 1.1 \begin{align*} -\frac{1}{2 \, \log \left (c x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.860774, size = 23, normalized size = 2.3 \begin{align*} -\frac{1}{2 \, \log \left (c x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.090014, size = 10, normalized size = 1. \begin{align*} - \frac{1}{2 \log{\left (c x \right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(2*log(c*x)**2)

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Giac [A]  time = 1.11042, size = 11, normalized size = 1.1 \begin{align*} -\frac{1}{2 \, \log \left (c x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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