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

Optimal. Leaf size=43 \[ -\frac {\sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {erf}\left (\frac {\sqrt {-\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {n}} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2300, 2180, 2205} \[ -\frac {\sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {Erf}\left (\frac {\sqrt {-\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {n}} \]

Antiderivative was successfully verified.

[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 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]

Rule 2300

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*} \int \frac {1}{\sqrt {-\log \left (a x^n\right )}} \, dx &=\frac {\left (x \left (a x^n\right )^{-1/n}\right ) \operatorname {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 ) \operatorname {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*}

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Mathematica [A]  time = 0.01, size = 62, normalized size = 1.44 \[ \frac {\sqrt {\pi } x \left (a x^n\right )^{-1/n} \sqrt {\log \left (a x^n\right )} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {n} \sqrt {-\log \left (a x^n\right )}} \]

Antiderivative was successfully verified.

[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]])

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.20, size = 32, normalized size = 0.74 \[ \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {-n \log \relax (x) - \log \relax (a)}}{\sqrt {n}}\right )}{a^{\left (\frac {1}{n}\right )} \sqrt {n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [F]  time = 1.32, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-\ln \left (a \,x^{n}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-\log \left (a x^{n}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-log(a*x^n)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {-\ln \left (a\,x^n\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(-log(a*x^n))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- \log {\left (a x^{n} \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-log(a*x**n)), x)

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