\(\int \sqrt {\log (a x^n)} \, dx\) [117]

Optimal result

Integrand size = 10, antiderivative size = 56 \[ \int \sqrt {\log \left (a x^n\right )} \, dx=-\frac {1}{2} \sqrt {n} \sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )+x \sqrt {\log \left (a x^n\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2333, 2337, 2211, 2235} \[ \int \sqrt {\log \left (a x^n\right )} \, dx=x \sqrt {\log \left (a x^n\right )}-\frac {1}{2} \sqrt {\pi } \sqrt {n} x \left (a x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right ) \]

[In]

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

[Out]

-1/2*(Sqrt[n]*Sqrt[Pi]*x*Erfi[Sqrt[Log[a*x^n]]/Sqrt[n]])/(a*x^n)^n^(-1) + x*Sqrt[Log[a*x^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 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2337

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]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \sqrt {\log \left (a x^n\right )} \, dx=-\frac {1}{2} \sqrt {n} \sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )+x \sqrt {\log \left (a x^n\right )} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(log(a*x**n)), x)

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(log(a*x^n)), x)

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(log(a*x^n)), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(log(a*x^n)^(1/2), x)