### 3.300 $$\int \log ^m(x)^p \, dx$$

Optimal. Leaf size=26 $(-\log (x))^{-m p} \log ^m(x)^p \text{Gamma}(m p+1,-\log (x))$

[Out]

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

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Rubi [A]  time = 0.0252434, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, Rules used = {6720, 2299, 2181} $(-\log (x))^{-m p} \log ^m(x)^p \text{Gamma}(m p+1,-\log (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[(Log[x]^m)^p,x]

[Out]

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

Rule 6720

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m)^FracPart[p])/v^(m*FracPart[p]), Int
[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rule 2299

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[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 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rubi steps

\begin{align*} \int \log ^m(x)^p \, dx &=\left (\log ^{-m p}(x) \log ^m(x)^p\right ) \int \log ^{m p}(x) \, dx\\ &=\left (\log ^{-m p}(x) \log ^m(x)^p\right ) \operatorname{Subst}\left (\int e^x x^{m p} \, dx,x,\log (x)\right )\\ &=\Gamma (1+m p,-\log (x)) (-\log (x))^{-m p} \log ^m(x)^p\\ \end{align*}

Mathematica [A]  time = 0.0137504, size = 26, normalized size = 1. $(-\log (x))^{-m p} \log ^m(x)^p \text{Gamma}(m p+1,-\log (x))$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Log[x]^m)^p,x]

[Out]

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

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int \left ( \left ( \ln \left ( x \right ) \right ) ^{m} \right ) ^{p}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\log \left (x\right )^{m}\right )}^{p}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((log(x)^m)^p, x)

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Fricas [A]  time = 2.10844, size = 50, normalized size = 1.92 \begin{align*} \cos \left (\pi m p\right ) \Gamma \left (m p + 1, -\log \left (x\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(pi*m*p)*gamma(m*p + 1, -log(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\log{\left (x \right )}^{m}\right )^{p}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((log(x)**m)**p, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\log \left (x\right )^{m}\right )}^{p}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((log(x)^m)^p, x)