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

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

[Out]

Log[x]^(1 + n)/(1 + n)

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Rubi [A]  time = 0.0174533, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2302, 30} \[ \frac{\log ^{n+1}(x)}{n+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x]^(1 + n)/(1 + n)

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]

Rule 30

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

Rubi steps

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

Mathematica [A]  time = 0.0033826, size = 12, normalized size = 1. \[ \frac{\log ^{n+1}(x)}{n+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x]^(1 + n)/(1 + n)

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Maple [A]  time = 0.002, size = 13, normalized size = 1.1 \begin{align*}{\frac{ \left ( \ln \left ( x \right ) \right ) ^{1+n}}{1+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.40761, size = 34, normalized size = 2.83 \begin{align*} \frac{\log \left (x\right )^{n} \log \left (x\right )}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)^n*log(x)/(n + 1)

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Sympy [A]  time = 0.865165, size = 15, normalized size = 1.25 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13295, size = 16, normalized size = 1.33 \begin{align*} \frac{\log \left (x\right )^{n + 1}}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)^(n + 1)/(n + 1)