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

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

[Out]

GAMMA(m*p+1,-ln(x))*(ln(x)^m)^p/((-ln(x))^(m*p))

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Rubi [A]  time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 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]

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 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])

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*}

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Mathematica [A]  time = 0.01, size = 26, normalized size = 1.00 \[ (-\log (x))^{-m p} \log ^m(x)^p \Gamma (m p+1,-\log (x)) \]

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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fricas [A]  time = 0.46, size = 16, normalized size = 0.62 \[ \cos \left (\pi m p\right ) \Gamma \left (m p + 1, -\log \relax (x)\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\log \relax (x)^{m}\right )}^{p}\,{d x} \]

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)

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maple [F]  time = 0.28, size = 0, normalized size = 0.00 \[ \int \left (\ln \relax (x )^{m}\right )^{p}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\log \relax (x)^{m}\right )}^{p}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int {\left ({\ln \relax (x)}^m\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((log(x)^m)^p, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\log {\relax (x )}^{m}\right )^{p}\, dx \]

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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