3.632 \(\int \frac{\log ^n(\log (x))}{x} \, dx\)

Optimal. Leaf size=24 \[ (-\log (\log (x)))^{-n} \log ^n(\log (x)) \text{Gamma}(n+1,-\log (\log (x))) \]

[Out]

(Gamma[1 + n, -Log[Log[x]]]*Log[Log[x]]^n)/(-Log[Log[x]])^n

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Rubi [A]  time = 0.032174, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2299, 2181} \[ (-\log (\log (x)))^{-n} \log ^n(\log (x)) \text{Gamma}(n+1,-\log (\log (x))) \]

Antiderivative was successfully verified.

[In]

Int[Log[Log[x]]^n/x,x]

[Out]

(Gamma[1 + n, -Log[Log[x]]]*Log[Log[x]]^n)/(-Log[Log[x]])^n

Rule 2299

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[1/(n*c^(1/n)), Subst[Int[E^(x/n)*(a + b*x)^p
, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[1/n]

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rubi steps

\begin{align*} \int \frac{\log ^n(\log (x))}{x} \, dx &=\operatorname{Subst}\left (\int \log ^n(x) \, dx,x,\log (x)\right )\\ &=\operatorname{Subst}\left (\int e^x x^n \, dx,x,\log (\log (x))\right )\\ &=\Gamma (1+n,-\log (\log (x))) (-\log (\log (x)))^{-n} \log ^n(\log (x))\\ \end{align*}

Mathematica [A]  time = 0.0171608, size = 24, normalized size = 1. \[ (-\log (\log (x)))^{-n} \log ^n(\log (x)) \text{Gamma}(n+1,-\log (\log (x))) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[Log[x]]^n/x,x]

[Out]

(Gamma[1 + n, -Log[Log[x]]]*Log[Log[x]]^n)/(-Log[Log[x]])^n

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( \ln \left ( x \right ) \right ) \right ) ^{n}}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(ln(x))^n/x,x)

[Out]

int(ln(ln(x))^n/x,x)

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Maxima [A]  time = 1.05007, size = 39, normalized size = 1.62 \begin{align*} -\left (-\log \left (\log \left (x\right )\right )\right )^{-n - 1} \log \left (\log \left (x\right )\right )^{n + 1} \Gamma \left (n + 1, -\log \left (\log \left (x\right )\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(log(x))^n/x,x, algorithm="maxima")

[Out]

-(-log(log(x)))^(-n - 1)*log(log(x))^(n + 1)*gamma(n + 1, -log(log(x)))

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Fricas [A]  time = 2.77683, size = 51, normalized size = 2.12 \begin{align*} \cos \left (\pi n\right ) \Gamma \left (n + 1, -\log \left (\log \left (x\right )\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(log(x))^n/x,x, algorithm="fricas")

[Out]

cos(pi*n)*gamma(n + 1, -log(log(x)))

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Sympy [A]  time = 2.76667, size = 24, normalized size = 1. \begin{align*} \left (- \log{\left (\log{\left (x \right )} \right )}\right )^{- n} \log{\left (\log{\left (x \right )} \right )}^{n} \Gamma \left (n + 1, - \log{\left (\log{\left (x \right )} \right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(ln(x))**n/x,x)

[Out]

(-log(log(x)))**(-n)*log(log(x))**n*uppergamma(n + 1, -log(log(x)))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (\log \left (x\right )\right )^{n}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(log(x))^n/x,x, algorithm="giac")

[Out]

integrate(log(log(x))^n/x, x)