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

Optimal. Leaf size=58 \[ \frac{2 \sqrt{\pi } x \left (a x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{n^{3/2}}-\frac{2 x}{n \sqrt{\log \left (a x^n\right )}} \]

[Out]

(2*Sqrt[Pi]*x*Erfi[Sqrt[Log[a*x^n]]/Sqrt[n]])/(n^(3/2)*(a*x^n)^n^(-1)) - (2*x)/(n*Sqrt[Log[a*x^n]])

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Rubi [A]  time = 0.0285107, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2297, 2300, 2180, 2204} \[ \frac{2 \sqrt{\pi } x \left (a x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{n^{3/2}}-\frac{2 x}{n \sqrt{\log \left (a x^n\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[Pi]*x*Erfi[Sqrt[Log[a*x^n]]/Sqrt[n]])/(n^(3/2)*(a*x^n)^n^(-1)) - (2*x)/(n*Sqrt[Log[a*x^n]])

Rule 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2300

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

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{1}{\log ^{\frac{3}{2}}\left (a x^n\right )} \, dx &=-\frac{2 x}{n \sqrt{\log \left (a x^n\right )}}+\frac{2 \int \frac{1}{\sqrt{\log \left (a x^n\right )}} \, dx}{n}\\ &=-\frac{2 x}{n \sqrt{\log \left (a x^n\right )}}+\frac{\left (2 x \left (a x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{\sqrt{x}} \, dx,x,\log \left (a x^n\right )\right )}{n^2}\\ &=-\frac{2 x}{n \sqrt{\log \left (a x^n\right )}}+\frac{\left (4 x \left (a x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int e^{\frac{x^2}{n}} \, dx,x,\sqrt{\log \left (a x^n\right )}\right )}{n^2}\\ &=\frac{2 \sqrt{\pi } x \left (a x^n\right )^{-1/n} \text{erfi}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{n^{3/2}}-\frac{2 x}{n \sqrt{\log \left (a x^n\right )}}\\ \end{align*}

Mathematica [A]  time = 0.0343871, size = 69, normalized size = 1.19 \[ -\frac{2 x \left (a x^n\right )^{-1/n} \left (\left (a x^n\right )^{\frac{1}{n}}-\sqrt{-\frac{\log \left (a x^n\right )}{n}} \text{Gamma}\left (\frac{1}{2},-\frac{\log \left (a x^n\right )}{n}\right )\right )}{n \sqrt{\log \left (a x^n\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x*((a*x^n)^n^(-1) - Gamma[1/2, -(Log[a*x^n]/n)]*Sqrt[-(Log[a*x^n]/n)]))/(n*(a*x^n)^n^(-1)*Sqrt[Log[a*x^n]]
)

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Maple [F]  time = 0.175, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ( a{x}^{n} \right ) \right ) ^{-{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\log \left (a x^{n}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(a*x^n)^(-3/2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\log{\left (a x^{n} \right )}^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(a*x**n)**(-3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\log \left (a x^{n}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(a*x^n)^(-3/2), x)