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

3.7.32.1 Optimal result

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

output

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

3.7.32.2 Mathematica [A] (verified)

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

input

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

output

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

3.7.32.3 Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3039, 2736, 2612}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3039

\(\displaystyle \int \log ^n(\log (x))d\log (x)\)

\(\Big \downarrow \) 2736

\(\displaystyle \int x \log ^n(\log (x))d\log (\log (x))\)

\(\Big \downarrow \) 2612

\(\displaystyle (-\log (\log (x)))^{-n} \log ^n(\log (x)) \Gamma (n+1,-\log (\log (x)))\)

input

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

output

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

3.7.32.3.1 Defintions of rubi rules used

rule 2612

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 2736

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[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 3039

Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst 
[[3]]   Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /;  !FalseQ[lst]] /; 
NonsumQ[u]
 

3.7.32.4 Maple [F]

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

input

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

output

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

3.7.32.5 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 \[ \int \frac {\log ^n(\log (x))}{x} \, dx=e^{\left (-i \, \pi n\right )} \Gamma \left (n + 1, -\log \left (\log \left (x\right )\right )\right ) \]

input

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

output

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

3.7.32.6 Sympy [A] (verification not implemented)

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

input

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

output

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

3.7.32.7 Maxima [A] (verification not implemented)

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

input

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

output

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

3.7.32.8 Giac [F]

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

input

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

output

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

3.7.32.9 Mupad [B] (verification not implemented)

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

input

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

output

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