\(\int \frac {1}{\sqrt {-\log (a x^n)}} \, dx\) [265]

Optimal result

Integrand size = 12, antiderivative size = 43 \[ \int \frac {1}{\sqrt {-\log \left (a x^n\right )}} \, dx=-\frac {\sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {erf}\left (\frac {\sqrt {-\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {n}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2337, 2211, 2236} \[ \int \frac {1}{\sqrt {-\log \left (a x^n\right )}} \, dx=-\frac {\sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {erf}\left (\frac {\sqrt {-\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {n}} \]

[In]

Int[1/Sqrt[-Log[a*x^n]],x]

[Out]

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

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 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}& = \frac {\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 )}{n} \\ & = -\frac {\left (2 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 )}{n} \\ & = -\frac {\sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {erf}\left (\frac {\sqrt {-\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {n}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\sqrt {-\log \left (a x^n\right )}} \, dx=\frac {\sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right ) \sqrt {\log \left (a x^n\right )}}{\sqrt {n} \sqrt {-\log \left (a x^n\right )}} \]

[In]

Integrate[1/Sqrt[-Log[a*x^n]],x]

[Out]

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

Maple [F]

\[\int \frac {1}{\sqrt {-\ln \left (a \,x^{n}\right )}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {-\log \left (a x^n\right )}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(-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]

\[ \int \frac {1}{\sqrt {-\log \left (a x^n\right )}} \, dx=\int \frac {1}{\sqrt {- \log {\left (a x^{n} \right )}}}\, dx \]

[In]

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

[Out]

Integral(1/sqrt(-log(a*x**n)), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {-\log \left (a x^n\right )}} \, dx=\int { \frac {1}{\sqrt {-\log \left (a x^{n}\right )}} \,d x } \]

[In]

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

[Out]

integrate(1/sqrt(-log(a*x^n)), x)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\sqrt {-\log \left (a x^n\right )}} \, dx=\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {-n \log \left (x\right ) - \log \left (a\right )}}{\sqrt {n}}\right )}{a^{\left (\frac {1}{n}\right )} \sqrt {n}} \]

[In]

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

[Out]

sqrt(pi)*erf(-sqrt(-n*log(x) - log(a))/sqrt(n))/(a^(1/n)*sqrt(n))

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {-\log \left (a x^n\right )}} \, dx=\int \frac {1}{\sqrt {-\ln \left (a\,x^n\right )}} \,d x \]

[In]

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

[Out]

int(1/(-log(a*x^n))^(1/2), x)