\(\int \frac {\log ^p(x)}{x} \, dx\) [64]

Optimal result

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

[Out]

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

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} \[ \int \frac {\log ^p(x)}{x} \, dx=\frac {\log ^{p+1}(x)}{p+1} \]

[In]

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

[Out]

Log[x]^(1 + p)/(1 + p)

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^p \, dx,x,\log (x)\right ) \\ & = \frac {\log ^{1+p}(x)}{1+p} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

Log[x]^(1 + p)/(1 + p)

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^p(x)}{x} \, dx=\frac {\log \left (x\right )^{p} \log \left (x\right )}{p + 1} \]

[In]

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

[Out]

log(x)^p*log(x)/(p + 1)

Sympy [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.25 \[ \int \frac {\log ^p(x)}{x} \, dx=\begin {cases} \frac {\log {\left (x \right )}^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (\log {\left (x \right )} \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^p(x)}{x} \, dx=\frac {\log \left (x\right )^{p + 1}}{p + 1} \]

[In]

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

[Out]

log(x)^(p + 1)/(p + 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^p(x)}{x} \, dx=\frac {\log \left (x\right )^{p + 1}}{p + 1} \]

[In]

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

[Out]

log(x)^(p + 1)/(p + 1)

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.83 \[ \int \frac {\log ^p(x)}{x} \, dx=\left \{\begin {array}{cl} \ln \left (\ln \left (x\right )\right ) & \text {\ if\ \ }p=-1\\ \frac {{\ln \left (x\right )}^{p+1}}{p+1} & \text {\ if\ \ }p\neq -1 \end {array}\right . \]

[In]

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

[Out]

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