\(\int \frac {\sqrt {\log (a x^n)}}{x^3} \, dx\) [120]

Optimal result

Integrand size = 14, antiderivative size = 72 \[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x^3} \, dx=\frac {\sqrt {n} \sqrt {\frac {\pi }{2}} \left (a x^n\right )^{2/n} \text {erf}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{4 x^2}-\frac {\sqrt {\log \left (a x^n\right )}}{2 x^2} \]

[Out]

1/8*(a*x^n)^(2/n)*erf(2^(1/2)*ln(a*x^n)^(1/2)/n^(1/2))*n^(1/2)*2^(1/2)*Pi^(1/2)/x^2-1/2*ln(a*x^n)^(1/2)/x^2

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2342, 2347, 2211, 2236} \[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x^3} \, dx=\frac {\sqrt {\frac {\pi }{2}} \sqrt {n} \left (a x^n\right )^{2/n} \text {erf}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{4 x^2}-\frac {\sqrt {\log \left (a x^n\right )}}{2 x^2} \]

[In]

Int[Sqrt[Log[a*x^n]]/x^3,x]

[Out]

(Sqrt[n]*Sqrt[Pi/2]*(a*x^n)^(2/n)*Erf[(Sqrt[2]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(4*x^2) - Sqrt[Log[a*x^n]]/(2*x^2)

Rule 2211

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] &&  !TrueQ[$UseGamma]

Rule 2236

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]

Rule 2342

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 2347

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)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x^3} \, dx=-\frac {4 \log \left (a x^n\right )+\sqrt {2} n \left (a x^n\right )^{2/n} \Gamma \left (\frac {1}{2},\frac {2 \log \left (a x^n\right )}{n}\right ) \sqrt {\frac {\log \left (a x^n\right )}{n}}}{8 x^2 \sqrt {\log \left (a x^n\right )}} \]

[In]

Integrate[Sqrt[Log[a*x^n]]/x^3,x]

[Out]

-1/8*(4*Log[a*x^n] + Sqrt[2]*n*(a*x^n)^(2/n)*Gamma[1/2, (2*Log[a*x^n])/n]*Sqrt[Log[a*x^n]/n])/(x^2*Sqrt[Log[a*
x^n]])

Maple [F]

\[\int \frac {\sqrt {\ln \left (a \,x^{n}\right )}}{x^{3}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x^3} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x^3} \, dx=\int \frac {\sqrt {\log {\left (a x^{n} \right )}}}{x^{3}}\, dx \]

[In]

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

[Out]

Integral(sqrt(log(a*x**n))/x**3, x)

Maxima [F]

\[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x^3} \, dx=\int { \frac {\sqrt {\log \left (a x^{n}\right )}}{x^{3}} \,d x } \]

[In]

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

[Out]

integrate(sqrt(log(a*x^n))/x^3, x)

Giac [F]

\[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x^3} \, dx=\int { \frac {\sqrt {\log \left (a x^{n}\right )}}{x^{3}} \,d x } \]

[In]

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

[Out]

integrate(sqrt(log(a*x^n))/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x^3} \, dx=\int \frac {\sqrt {\ln \left (a\,x^n\right )}}{x^3} \,d x \]

[In]

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

[Out]

int(log(a*x^n)^(1/2)/x^3, x)