3.130 \(\int \frac{x}{\sqrt{\log (a x^n)}} \, dx\)

Optimal. Leaf size=51 \[ \frac{\sqrt{\frac{\pi }{2}} x^2 \left (a x^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{\sqrt{n}} \]

[Out]

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

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Rubi [A]  time = 0.0326668, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2310, 2180, 2204} \[ \frac{\sqrt{\frac{\pi }{2}} x^2 \left (a x^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{\sqrt{n}} \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[Log[a*x^n]],x]

[Out]

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

Rule 2310

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, 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{x}{\sqrt{\log \left (a x^n\right )}} \, dx &=\frac{\left (x^2 \left (a x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{\sqrt{x}} \, dx,x,\log \left (a x^n\right )\right )}{n}\\ &=\frac{\left (2 x^2 \left (a x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int e^{\frac{2 x^2}{n}} \, dx,x,\sqrt{\log \left (a x^n\right )}\right )}{n}\\ &=\frac{\sqrt{\frac{\pi }{2}} x^2 \left (a x^n\right )^{-2/n} \text{erfi}\left (\frac{\sqrt{2} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{\sqrt{n}}\\ \end{align*}

Mathematica [A]  time = 0.0089993, size = 51, normalized size = 1. \[ \frac{\sqrt{\frac{\pi }{2}} x^2 \left (a x^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{\sqrt{n}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[Log[a*x^n]],x]

[Out]

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

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Maple [F]  time = 0.181, size = 0, normalized size = 0. \begin{align*} \int{x{\frac{1}{\sqrt{\ln \left ( a{x}^{n} \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/sqrt(log(a*x**n)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/sqrt(log(a*x^n)), x)