\(\int e^{-24 x^2} (12 x^3 \log ^3(x^3)+(4 x^3-48 x^5) \log ^4(x^3)) \, dx\) [6882]

Optimal result

Integrand size = 38, antiderivative size = 17 \[ \int e^{-24 x^2} \left (12 x^3 \log ^3\left (x^3\right )+\left (4 x^3-48 x^5\right ) \log ^4\left (x^3\right )\right ) \, dx=e^{-24 x^2} x^4 \log ^4\left (x^3\right ) \]

[Out]

ln(x^3)^4/exp(3*x^2)^8*x^4

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {6873, 12, 2326} \[ \int e^{-24 x^2} \left (12 x^3 \log ^3\left (x^3\right )+\left (4 x^3-48 x^5\right ) \log ^4\left (x^3\right )\right ) \, dx=e^{-24 x^2} x^4 \log ^4\left (x^3\right ) \]

[In]

Int[(12*x^3*Log[x^3]^3 + (4*x^3 - 48*x^5)*Log[x^3]^4)/E^(24*x^2),x]

[Out]

(x^4*Log[x^3]^4)/E^(24*x^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps \begin{align*} \text {integral}& = \int 4 e^{-24 x^2} x^3 \log ^3\left (x^3\right ) \left (3+\log \left (x^3\right )-12 x^2 \log \left (x^3\right )\right ) \, dx \\ & = 4 \int e^{-24 x^2} x^3 \log ^3\left (x^3\right ) \left (3+\log \left (x^3\right )-12 x^2 \log \left (x^3\right )\right ) \, dx \\ & = e^{-24 x^2} x^4 \log ^4\left (x^3\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int e^{-24 x^2} \left (12 x^3 \log ^3\left (x^3\right )+\left (4 x^3-48 x^5\right ) \log ^4\left (x^3\right )\right ) \, dx=e^{-24 x^2} x^4 \log ^4\left (x^3\right ) \]

[In]

Integrate[(12*x^3*Log[x^3]^3 + (4*x^3 - 48*x^5)*Log[x^3]^4)/E^(24*x^2),x]

[Out]

(x^4*Log[x^3]^4)/E^(24*x^2)

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12

method	result	size
parallelrisch	\(\ln \left (x^{3}\right )^{4} {\mathrm e}^{-24 x^{2}} x^{4}\)	\(19\)
risch	\(\text {Expression too large to display}\)	\(3066\)

[In]

int(((-48*x^5+4*x^3)*ln(x^3)^4+12*x^3*ln(x^3)^3)/exp(3*x^2)^8,x,method=_RETURNVERBOSE)

[Out]

ln(x^3)^4/exp(3*x^2)^8*x^4

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int e^{-24 x^2} \left (12 x^3 \log ^3\left (x^3\right )+\left (4 x^3-48 x^5\right ) \log ^4\left (x^3\right )\right ) \, dx=x^{4} e^{\left (-24 \, x^{2}\right )} \log \left (x^{3}\right )^{4} \]

[In]

integrate(((-48*x^5+4*x^3)*log(x^3)^4+12*x^3*log(x^3)^3)/exp(3*x^2)^8,x, algorithm="fricas")

[Out]

x^4*e^(-24*x^2)*log(x^3)^4

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int e^{-24 x^2} \left (12 x^3 \log ^3\left (x^3\right )+\left (4 x^3-48 x^5\right ) \log ^4\left (x^3\right )\right ) \, dx=x^{4} e^{- 24 x^{2}} \log {\left (x^{3} \right )}^{4} \]

[In]

integrate(((-48*x**5+4*x**3)*ln(x**3)**4+12*x**3*ln(x**3)**3)/exp(3*x**2)**8,x)

[Out]

x**4*exp(-24*x**2)*log(x**3)**4

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int e^{-24 x^2} \left (12 x^3 \log ^3\left (x^3\right )+\left (4 x^3-48 x^5\right ) \log ^4\left (x^3\right )\right ) \, dx=81 \, x^{4} e^{\left (-24 \, x^{2}\right )} \log \left (x\right )^{4} \]

[In]

integrate(((-48*x^5+4*x^3)*log(x^3)^4+12*x^3*log(x^3)^3)/exp(3*x^2)^8,x, algorithm="maxima")

[Out]

81*x^4*e^(-24*x^2)*log(x)^4

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int e^{-24 x^2} \left (12 x^3 \log ^3\left (x^3\right )+\left (4 x^3-48 x^5\right ) \log ^4\left (x^3\right )\right ) \, dx=x^{4} e^{\left (-24 \, x^{2}\right )} \log \left (x^{3}\right )^{4} \]

[In]

integrate(((-48*x^5+4*x^3)*log(x^3)^4+12*x^3*log(x^3)^3)/exp(3*x^2)^8,x, algorithm="giac")

[Out]

x^4*e^(-24*x^2)*log(x^3)^4

Mupad [B] (verification not implemented)

Time = 12.50 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int e^{-24 x^2} \left (12 x^3 \log ^3\left (x^3\right )+\left (4 x^3-48 x^5\right ) \log ^4\left (x^3\right )\right ) \, dx=x^4\,{\ln \left (x^3\right )}^4\,{\mathrm {e}}^{-24\,x^2} \]

[In]

int(exp(-24*x^2)*(log(x^3)^4*(4*x^3 - 48*x^5) + 12*x^3*log(x^3)^3),x)

[Out]

x^4*log(x^3)^4*exp(-24*x^2)