\(\int \frac {x^m}{\sqrt {\log (a x^n)}} \, dx\) [164]

Optimal result

Integrand size = 14, antiderivative size = 61 \[ \int \frac {x^m}{\sqrt {\log \left (a x^n\right )}} \, dx=\frac {\sqrt {\pi } x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {\sqrt {1+m} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {1+m} \sqrt {n}} \]

[Out]

x^(1+m)*erfi((1+m)^(1/2)*ln(a*x^n)^(1/2)/n^(1/2))*Pi^(1/2)/((a*x^n)^((1+m)/n))/(1+m)^(1/2)/n^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2347, 2211, 2235} \[ \int \frac {x^m}{\sqrt {\log \left (a x^n\right )}} \, dx=\frac {\sqrt {\pi } x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \text {erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {m+1} \sqrt {n}} \]

[In]

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

[Out]

(Sqrt[Pi]*x^(1 + m)*Erfi[(Sqrt[1 + m]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(Sqrt[1 + m]*Sqrt[n]*(a*x^n)^((1 + m)/n))

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 2235

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]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \frac {x^m}{\sqrt {\log \left (a x^n\right )}} \, dx=\frac {\sqrt {\pi } x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {\sqrt {1+m} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {1+m} \sqrt {n}} \]

[In]

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

[Out]

(Sqrt[Pi]*x^(1 + m)*Erfi[(Sqrt[1 + m]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(Sqrt[1 + m]*Sqrt[n]*(a*x^n)^((1 + m)/n))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(x^m/sqrt(log(a*x^n)), x)

Sympy [F]

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

[In]

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

[Out]

Integral(x**m/sqrt(log(a*x**n)), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^m/sqrt(log(a*x^n)), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^m/sqrt(log(a*x^n)), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^m/log(a*x^n)^(1/2), x)