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

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 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 ^{\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*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ -\frac {2}{n \sqrt {\log \left (a x^n\right )}} \]

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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fricas [A]  time = 0.46, size = 24, normalized size = 1.60 \[ -\frac {2 \, \sqrt {n \log \relax (x) + \log \relax (a)}}{n^{2} \log \relax (x) + n \log \relax (a)} \]

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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giac [A]  time = 0.27, size = 14, normalized size = 0.93 \[ -\frac {2}{\sqrt {n \log \relax (x) + \log \relax (a)} n} \]

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)

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maple [A]  time = 0.03, size = 14, normalized size = 0.93 \[ -\frac {2}{n \sqrt {\ln \left (a \,x^{n}\right )}} \]

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 = 0.70, size = 13, normalized size = 0.87 \[ -\frac {2}{n \sqrt {\log \left (a x^{n}\right )}} \]

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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mupad [B]  time = 3.45, size = 13, normalized size = 0.87 \[ -\frac {2}{n\,\sqrt {\ln \left (a\,x^n\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 93.98, size = 41, normalized size = 2.73 \[ \begin {cases} \tilde {\infty } \log {\relax (x )} & \text {for}\: a = 1 \wedge n = 0 \\\frac {\log {\relax (x )}}{\log {\relax (a )}^{\frac {3}{2}}} & \text {for}\: n = 0 \\\tilde {\infty } \log {\relax (x )} & \text {for}\: a = e^{- n \log {\relax (x )}} \\- \frac {2}{n \sqrt {n \log {\relax (x )} + \log {\relax (a )}}} & \text {otherwise} \end {cases} \]

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

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