3.145 \(\int \frac{1}{\log ^{\frac{5}{2}}(a x^n)} \, dx\)

Optimal. Leaf size=80 \[ \frac{4 \sqrt{\pi } x \left (a x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{3 n^{5/2}}-\frac{4 x}{3 n^2 \sqrt{\log \left (a x^n\right )}}-\frac{2 x}{3 n \log ^{\frac{3}{2}}\left (a x^n\right )} \]

[Out]

(4*Sqrt[Pi]*x*Erfi[Sqrt[Log[a*x^n]]/Sqrt[n]])/(3*n^(5/2)*(a*x^n)^n^(-1)) - (2*x)/(3*n*Log[a*x^n]^(3/2)) - (4*x
)/(3*n^2*Sqrt[Log[a*x^n]])

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Rubi [A]  time = 0.0372524, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2297, 2300, 2180, 2204} \[ \frac{4 \sqrt{\pi } x \left (a x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{3 n^{5/2}}-\frac{4 x}{3 n^2 \sqrt{\log \left (a x^n\right )}}-\frac{2 x}{3 n \log ^{\frac{3}{2}}\left (a x^n\right )} \]

Antiderivative was successfully verified.

[In]

Int[Log[a*x^n]^(-5/2),x]

[Out]

(4*Sqrt[Pi]*x*Erfi[Sqrt[Log[a*x^n]]/Sqrt[n]])/(3*n^(5/2)*(a*x^n)^n^(-1)) - (2*x)/(3*n*Log[a*x^n]^(3/2)) - (4*x
)/(3*n^2*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{5}{2}}\left (a x^n\right )} \, dx &=-\frac{2 x}{3 n \log ^{\frac{3}{2}}\left (a x^n\right )}+\frac{2 \int \frac{1}{\log ^{\frac{3}{2}}\left (a x^n\right )} \, dx}{3 n}\\ &=-\frac{2 x}{3 n \log ^{\frac{3}{2}}\left (a x^n\right )}-\frac{4 x}{3 n^2 \sqrt{\log \left (a x^n\right )}}+\frac{4 \int \frac{1}{\sqrt{\log \left (a x^n\right )}} \, dx}{3 n^2}\\ &=-\frac{2 x}{3 n \log ^{\frac{3}{2}}\left (a x^n\right )}-\frac{4 x}{3 n^2 \sqrt{\log \left (a x^n\right )}}+\frac{\left (4 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 )}{3 n^3}\\ &=-\frac{2 x}{3 n \log ^{\frac{3}{2}}\left (a x^n\right )}-\frac{4 x}{3 n^2 \sqrt{\log \left (a x^n\right )}}+\frac{\left (8 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 )}{3 n^3}\\ &=\frac{4 \sqrt{\pi } x \left (a x^n\right )^{-1/n} \text{erfi}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{3 n^{5/2}}-\frac{2 x}{3 n \log ^{\frac{3}{2}}\left (a x^n\right )}-\frac{4 x}{3 n^2 \sqrt{\log \left (a x^n\right )}}\\ \end{align*}

Mathematica [A]  time = 0.0482157, size = 83, normalized size = 1.04 \[ -\frac{2 x \left (a x^n\right )^{-1/n} \left (2 n \left (-\frac{\log \left (a x^n\right )}{n}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{\log \left (a x^n\right )}{n}\right )+\left (a x^n\right )^{\frac{1}{n}} \left (2 \log \left (a x^n\right )+n\right )\right )}{3 n^2 \log ^{\frac{3}{2}}\left (a x^n\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[a*x^n]^(-5/2),x]

[Out]

(-2*x*(2*n*Gamma[1/2, -(Log[a*x^n]/n)]*(-(Log[a*x^n]/n))^(3/2) + (a*x^n)^n^(-1)*(n + 2*Log[a*x^n])))/(3*n^2*(a
*x^n)^n^(-1)*Log[a*x^n]^(3/2))

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Maple [F]  time = 0.174, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ( a{x}^{n} \right ) \right ) ^{-{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/ln(a*x^n)^(5/2),x)

[Out]

int(1/ln(a*x^n)^(5/2),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{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/log(a*x^n)^(5/2),x, algorithm="maxima")

[Out]

integrate(log(a*x^n)^(-5/2), 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)^(5/2),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{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/ln(a*x**n)**(5/2),x)

[Out]

Integral(log(a*x**n)**(-5/2), 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{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/log(a*x^n)^(5/2),x, algorithm="giac")

[Out]

integrate(log(a*x^n)^(-5/2), x)