3.64 \(\int \frac {\log ^p(x)}{x} \, dx\)

Optimal. Leaf size=12 \[ \frac {\log ^{p+1}(x)}{p+1} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2302, 30} \[ \frac {\log ^{p+1}(x)}{p+1} \]

Antiderivative was successfully verified.

[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 2302

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

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \[ \frac {\log ^{p+1}(x)}{p+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x]^(1 + p)/(1 + p)

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fricas [A]  time = 0.44, size = 12, normalized size = 1.00 \[ \frac {\log \relax (x)^{p} \log \relax (x)}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)^p*log(x)/(p + 1)

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giac [A]  time = 1.23, size = 12, normalized size = 1.00 \[ \frac {\log \relax (x)^{p + 1}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)^(p + 1)/(p + 1)

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maple [A]  time = 0.00, size = 13, normalized size = 1.08 \[ \frac {\ln \relax (x )^{p +1}}{p +1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.41, size = 12, normalized size = 1.00 \[ \frac {\log \relax (x)^{p + 1}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)^(p + 1)/(p + 1)

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mupad [B]  time = 0.15, size = 22, normalized size = 1.83 \[ \left \{\begin {array}{cl} \ln \left (\ln \relax (x)\right ) & \text {\ if\ \ }p=-1\\ \frac {{\ln \relax (x)}^{p+1}}{p+1} & \text {\ if\ \ }p\neq -1 \end {array}\right . \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.88, size = 15, normalized size = 1.25 \[ \begin {cases} \frac {\log {\relax (x )}^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (\log {\relax (x )} \right )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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