\(\int \frac {\log ^n(x)}{x} \, dx\) [618]

Optimal result

Integrand size = 8, antiderivative size = 12 \[ \int \frac {\log ^n(x)}{x} \, dx=\frac {\log ^{1+n}(x)}{1+n} \]

[Out]

ln(x)^(1+n)/(1+n)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2339, 30} \[ \int \frac {\log ^n(x)}{x} \, dx=\frac {\log ^{n+1}(x)}{n+1} \]

[In]

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

[Out]

Log[x]^(1 + n)/(1 + n)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2339

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

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int x^n \, dx,x,\log (x)\right ) \\ & = \frac {\log ^{1+n}(x)}{1+n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^n(x)}{x} \, dx=\frac {\log ^{1+n}(x)}{1+n} \]

[In]

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

[Out]

Log[x]^(1 + n)/(1 + n)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(\frac {\ln \left (x \right )^{1+n}}{1+n}\)	\(13\)
default	\(\frac {\ln \left (x \right )^{1+n}}{1+n}\)	\(13\)
risch	\(\frac {\ln \left (x \right ) \ln \left (x \right )^{n}}{1+n}\)	\(13\)
norman	\(\frac {\ln \left (x \right ) {\mathrm e}^{n \ln \left (\ln \left (x \right )\right )}}{1+n}\)	\(15\)

[In]

int(ln(x)^n/x,x,method=_RETURNVERBOSE)

[Out]

ln(x)^(1+n)/(1+n)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^n(x)}{x} \, dx=\frac {\log \left (x\right )^{n} \log \left (x\right )}{n + 1} \]

[In]

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

[Out]

log(x)^n*log(x)/(n + 1)

Sympy [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.25 \[ \int \frac {\log ^n(x)}{x} \, dx=\begin {cases} \frac {\log {\left (x \right )}^{n + 1}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (\log {\left (x \right )} \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((log(x)**(n + 1)/(n + 1), Ne(n, -1)), (log(log(x)), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^n(x)}{x} \, dx=\frac {\log \left (x\right )^{n + 1}}{n + 1} \]

[In]

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

[Out]

log(x)^(n + 1)/(n + 1)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^n(x)}{x} \, dx=\frac {\log \left (x\right )^{n + 1}}{n + 1} \]

[In]

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

[Out]

log(x)^(n + 1)/(n + 1)

Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.83 \[ \int \frac {\log ^n(x)}{x} \, dx=\left \{\begin {array}{cl} \ln \left (\ln \left (x\right )\right ) & \text {\ if\ \ }n=-1\\ \frac {{\ln \left (x\right )}^{n+1}}{n+1} & \text {\ if\ \ }n\neq -1 \end {array}\right . \]

[In]

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

[Out]

piecewise(n == -1, log(log(x)), n ~= -1, log(x)^(n + 1)/(n + 1))