3.119 \(\int \frac{\sqrt{\log (a x^n)}}{x^2} \, dx\)

Optimal. Leaf size=59 \[ \frac{\sqrt{\pi } \sqrt{n} \left (a x^n\right )^{\frac{1}{n}} \text{Erf}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{2 x}-\frac{\sqrt{\log \left (a x^n\right )}}{x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0502454, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2305, 2310, 2180, 2205} \[ \frac{\sqrt{\pi } \sqrt{n} \left (a x^n\right )^{\frac{1}{n}} \text{Erf}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{2 x}-\frac{\sqrt{\log \left (a x^n\right )}}{x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Log[a*x^n]]/x^2,x]

[Out]

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

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

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 2205

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

Rubi steps

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

Mathematica [A]  time = 0.0471729, size = 65, normalized size = 1.1 \[ -\frac{n \left (a x^n\right )^{\frac{1}{n}} \sqrt{\frac{\log \left (a x^n\right )}{n}} \text{Gamma}\left (\frac{1}{2},\frac{\log \left (a x^n\right )}{n}\right )+2 \log \left (a x^n\right )}{2 x \sqrt{\log \left (a x^n\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Log[a*x^n]]/x^2,x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.178, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\sqrt{\ln \left ( a{x}^{n} \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(log(a*x^n))/x^2, x)

________________________________________________________________________________________

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(log(a*x**n))/x**2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(log(a*x^n))/x^2, x)