\(\int \frac {\log ^n(\log (x))}{x} \, dx\) [632]

Optimal result

Integrand size = 9, antiderivative size = 24 \[ \int \frac {\log ^n(\log (x))}{x} \, dx=\Gamma (1+n,-\log (\log (x))) (-\log (\log (x)))^{-n} \log ^n(\log (x)) \]

[Out]

GAMMA(1+n,-ln(ln(x)))*ln(ln(x))^n/((-ln(ln(x)))^n)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2336, 2212} \[ \int \frac {\log ^n(\log (x))}{x} \, dx=(-\log (\log (x)))^{-n} \log ^n(\log (x)) \Gamma (n+1,-\log (\log (x))) \]

[In]

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

[Out]

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

Rule 2212

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

Rule 2336

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]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \log ^n(x) \, dx,x,\log (x)\right ) \\ & = \text {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] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^n(\log (x))}{x} \, dx=\Gamma (1+n,-\log (\log (x))) (-\log (\log (x)))^{-n} \log ^n(\log (x)) \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {\log ^n(\log (x))}{x} \, dx=e^{\left (-i \, \pi n\right )} \Gamma \left (n + 1, -\log \left (\log \left (x\right )\right )\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.70 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^n(\log (x))}{x} \, dx=\left (- \log {\left (\log {\left (x \right )} \right )}\right )^{- n} \log {\left (\log {\left (x \right )} \right )}^{n} \Gamma \left (n + 1, - \log {\left (\log {\left (x \right )} \right )}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {\log ^n(\log (x))}{x} \, dx=-\left (-\log \left (\log \left (x\right )\right )\right )^{-n - 1} \log \left (\log \left (x\right )\right )^{n + 1} \Gamma \left (n + 1, -\log \left (\log \left (x\right )\right )\right ) \]

[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)))

Giac [F]

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

[In]

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

[Out]

integrate(log(log(x))^n/x, x)

Mupad [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^n(\log (x))}{x} \, dx=\frac {{\ln \left (\ln \left (x\right )\right )}^n\,\Gamma \left (n+1,-\ln \left (\ln \left (x\right )\right )\right )}{{\left (-\ln \left (\ln \left (x\right )\right )\right )}^n} \]

[In]

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

[Out]

(log(log(x))^n*igamma(n + 1, -log(log(x))))/(-log(log(x)))^n