3.139 \(\int \frac{1}{x \log ^{\frac{3}{2}}(a x^n)} \, dx\)

Optimal. Leaf size=15 \[ -\frac{2}{n \sqrt{\log \left (a x^n\right )}} \]

[Out]

-2/(n*Sqrt[Log[a*x^n]])

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Rubi [A]  time = 0.0140202, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2302, 30} \[ -\frac{2}{n \sqrt{\log \left (a x^n\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(n*Sqrt[Log[a*x^n]])

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

Mathematica [A]  time = 0.0018364, size = 15, normalized size = 1. \[ -\frac{2}{n \sqrt{\log \left (a x^n\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(n*Sqrt[Log[a*x^n]])

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Maple [A]  time = 0.039, size = 14, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{n\sqrt{\ln \left ( a{x}^{n} \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/n/ln(a*x^n)^(1/2)

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Maxima [A]  time = 1.13858, size = 18, normalized size = 1.2 \begin{align*} -\frac{2}{n \sqrt{\log \left (a x^{n}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(a*x^n)^(3/2),x, algorithm="maxima")

[Out]

-2/(n*sqrt(log(a*x^n)))

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Fricas [A]  time = 0.94426, size = 70, normalized size = 4.67 \begin{align*} -\frac{2 \, \sqrt{n \log \left (x\right ) + \log \left (a\right )}}{n^{2} \log \left (x\right ) + n \log \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(n*log(x) + log(a))/(n^2*log(x) + n*log(a))

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Sympy [A]  time = 165.01, size = 48, normalized size = 3.2 \begin{align*} \begin{cases} \tilde{\infty } \log{\left (x \right )} & \text{for}\: \left (a = 1 \vee a = e^{- n \log{\left (x \right )}}\right ) \wedge \left (a = e^{- n \log{\left (x \right )}} \vee n = 0\right ) \\\frac{\log{\left (x \right )}}{\log{\left (a \right )}^{\frac{3}{2}}} & \text{for}\: n = 0 \\- \frac{2}{n \sqrt{n \log{\left (x \right )} + \log{\left (a \right )}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/ln(a*x**n)**(3/2),x)

[Out]

Piecewise((zoo*log(x), (Eq(a, 1) | Eq(a, exp(-n*log(x)))) & (Eq(n, 0) | Eq(a, exp(-n*log(x))))), (log(x)/log(a
)**(3/2), Eq(n, 0)), (-2/(n*sqrt(n*log(x) + log(a))), True))

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Giac [A]  time = 1.19797, size = 19, normalized size = 1.27 \begin{align*} -\frac{2}{\sqrt{n \log \left (x\right ) + \log \left (a\right )} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt(n*log(x) + log(a))*n)