\(\int \log ^m(x)^p \, dx\) [300]

Optimal result

Integrand size = 6, antiderivative size = 26 \[ \int \log ^m(x)^p \, dx=\Gamma (1+m p,-\log (x)) (-\log (x))^{-m p} \log ^m(x)^p \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6852, 2336, 2212} \[ \int \log ^m(x)^p \, dx=(-\log (x))^{-m p} \log ^m(x)^p \Gamma (m p+1,-\log (x)) \]

[In]

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

[Out]

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

Rule 2212

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

Rule 2336

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 6852

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*} \text {integral}& = \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 ) \text {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] (verified)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \log ^m(x)^p \, dx=\Gamma (1+m p,-\log (x)) (-\log (x))^{-m p} \log ^m(x)^p \]

[In]

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

[Out]

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

Maple [F]

\[\int \left (\ln \left (x \right )^{m}\right )^{p}d x\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int \log ^m(x)^p \, dx=e^{\left (-i \, \pi m p\right )} \Gamma \left (m p + 1, -\log \left (x\right )\right ) \]

[In]

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

[Out]

e^(-I*pi*m*p)*gamma(m*p + 1, -log(x))

Sympy [F]

\[ \int \log ^m(x)^p \, dx=\int \left (\log {\left (x \right )}^{m}\right )^{p}\, dx \]

[In]

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

[Out]

Integral((log(x)**m)**p, x)

Maxima [F]

\[ \int \log ^m(x)^p \, dx=\int { {\left (\log \left (x\right )^{m}\right )}^{p} \,d x } \]

[In]

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

[Out]

integrate((log(x)^m)^p, x)

Giac [F]

\[ \int \log ^m(x)^p \, dx=\int { {\left (\log \left (x\right )^{m}\right )}^{p} \,d x } \]

[In]

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

[Out]

integrate((log(x)^m)^p, x)

Mupad [F(-1)]

Timed out. \[ \int \log ^m(x)^p \, dx=\int {\left ({\ln \left (x\right )}^m\right )}^p \,d x \]

[In]

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

[Out]

int((log(x)^m)^p, x)