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

Optimal. Leaf size=8 \[ -\frac {1}{\log (c x)} \]

[Out]

-1/ln(c*x)

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Rubi [A]  time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2302, 30} \[ -\frac {1}{\log (c x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 30

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

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]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 8, normalized size = 1.00 \[ -\frac {1}{\log (c x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 8, normalized size = 1.00 \[ -\frac {1}{\log \left (c x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/log(c*x)

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giac [A]  time = 0.20, size = 8, normalized size = 1.00 \[ -\frac {1}{\log \left (c x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/log(c*x)

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maple [A]  time = 0.02, size = 9, normalized size = 1.12 \[ -\frac {1}{\ln \left (c x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/ln(c*x)

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maxima [A]  time = 0.64, size = 8, normalized size = 1.00 \[ -\frac {1}{\log \left (c x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/log(c*x)

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mupad [B]  time = 3.56, size = 8, normalized size = 1.00 \[ -\frac {1}{\ln \left (c\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*log(c*x)^2),x)

[Out]

-1/log(c*x)

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sympy [A]  time = 0.09, size = 7, normalized size = 0.88 \[ - \frac {1}{\log {\left (c x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/log(c*x)

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