3.69.82 \(\int e^{-24 x^2} (12 x^3 \log ^3(x^3)+(4 x^3-48 x^5) \log ^4(x^3)) \, dx\)
Optimal. Leaf size=17 \[ e^{-24 x^2} x^4 \log ^4\left (x^3\right ) \]
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Rubi [A] time = 0.25, antiderivative size = 17, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used =
{6741, 12, 2288} \begin {gather*} e^{-24 x^2} x^4 \log ^4\left (x^3\right ) \end {gather*}
Antiderivative was successfully verified.
[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 2288
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 6741
Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]
Rubi steps
\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}
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Mathematica [A] time = 0.09, size = 17, normalized size = 1.00 \begin {gather*} e^{-24 x^2} x^4 \log ^4\left (x^3\right ) \end {gather*}
Antiderivative was successfully verified.
[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)
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fricas [A] time = 0.77, size = 16, normalized size = 0.94 \begin {gather*} x^{4} e^{\left (-24 \, x^{2}\right )} \log \left (x^{3}\right )^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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
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giac [A] time = 0.26, size = 16, normalized size = 0.94 \begin {gather*} x^{4} e^{\left (-24 \, x^{2}\right )} \log \left (x^{3}\right )^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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
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maple [C] time = 0.27, size = 3066, normalized size = 180.35
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Verification of antiderivative is not currently implemented for this CAS.
[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]
81*x^4*exp(-24*x^2)*ln(x)^4-54*I*Pi*(csgn(I*x^2)*csgn(I*x)^2-2*csgn(I*x)*csgn(I*x^2)^2+csgn(I*x)*csgn(I*x^2)*c
sgn(I*x^3)-csgn(I*x)*csgn(I*x^3)^2+csgn(I*x^2)^3-csgn(I*x^2)*csgn(I*x^3)^2+csgn(I*x^3)^3)*x^4*exp(-24*x^2)*ln(
x)^3-27/2*Pi^2*(csgn(I*x)^4*csgn(I*x^2)^2-4*csgn(I*x)^3*csgn(I*x^2)^3+2*csgn(I*x)^3*csgn(I*x^2)^2*csgn(I*x^3)-
2*csgn(I*x)^3*csgn(I*x^2)*csgn(I*x^3)^2+6*csgn(I*x)^2*csgn(I*x^2)^4-4*csgn(I*x)^2*csgn(I*x^2)^3*csgn(I*x^3)+3*
csgn(I*x)^2*csgn(I*x^2)^2*csgn(I*x^3)^2+csgn(I*x)^2*csgn(I*x^3)^4-4*csgn(I*x)*csgn(I*x^2)^5+2*csgn(I*x^2)^4*cs
gn(I*x)*csgn(I*x^3)+2*csgn(I*x)*csgn(I*x^2)^3*csgn(I*x^3)^2-6*csgn(I*x)*csgn(I*x^2)^2*csgn(I*x^3)^3+4*csgn(I*x
)*csgn(I*x^2)*csgn(I*x^3)^4-2*csgn(I*x)*csgn(I*x^3)^5+csgn(I*x^2)^6-2*csgn(I*x^2)^4*csgn(I*x^3)^2+2*csgn(I*x^2
)^3*csgn(I*x^3)^3+csgn(I*x^2)^2*csgn(I*x^3)^4-2*csgn(I*x^2)*csgn(I*x^3)^5+csgn(I*x^3)^6)*x^4*exp(-24*x^2)*ln(x
)^2+3/2*I*Pi^3*(3*csgn(I*x)^5*csgn(I*x^2)^3*csgn(I*x^3)-3*csgn(I*x)^5*csgn(I*x^2)^2*csgn(I*x^3)^2-12*csgn(I*x)
^4*csgn(I*x^2)^4*csgn(I*x^3)+12*csgn(I*x)^4*csgn(I*x^2)^3*csgn(I*x^3)^2-3*csgn(I*x)^4*csgn(I*x^2)^2*csgn(I*x^3
)^3+3*csgn(I*x)^4*csgn(I*x^2)*csgn(I*x^3)^4+18*csgn(I*x)^3*csgn(I*x^2)^5*csgn(I*x^3)-12*csgn(I*x)^3*csgn(I*x^2
)^4*csgn(I*x^3)^2-5*csgn(I*x)^3*csgn(I*x^2)^3*csgn(I*x^3)^3+3*csgn(I*x)^3*csgn(I*x^2)^2*csgn(I*x^3)^4-3*csgn(I
*x)^3*csgn(I*x^2)*csgn(I*x^3)^5-12*csgn(I*x)^2*csgn(I*x^2)^6*csgn(I*x^3)-3*csgn(I*x)^2*csgn(I*x^2)^5*csgn(I*x^
3)^2+24*csgn(I*x)^2*csgn(I*x^2)^4*csgn(I*x^3)^3-21*csgn(I*x)^2*csgn(I*x^2)^3*csgn(I*x^3)^4+15*csgn(I*x)^2*csgn
(I*x^2)^2*csgn(I*x^3)^5-6*csgn(I*x)^2*csgn(I*x^2)*csgn(I*x^3)^6+3*csgn(I*x)*csgn(I*x^2)^7*csgn(I*x^3)+9*csgn(I
*x)*csgn(I*x^2)^6*csgn(I*x^3)^2-18*csgn(I*x)*csgn(I*x^2)^5*csgn(I*x^3)^3+6*csgn(I*x)*csgn(I*x^2)^4*csgn(I*x^3)
^4+9*csgn(I*x)*csgn(I*x^2)^3*csgn(I*x^3)^5-15*csgn(I*x)*csgn(I*x^2)^2*csgn(I*x^3)^6+9*csgn(I*x)*csgn(I*x^2)*cs
gn(I*x^3)^7+csgn(I*x)^6*csgn(I*x^2)^3-6*csgn(I*x)^5*csgn(I*x^2)^4+15*csgn(I*x)^4*csgn(I*x^2)^5-20*csgn(I*x)^3*
csgn(I*x^2)^6-csgn(I*x)^3*csgn(I*x^3)^6+15*csgn(I*x)^2*csgn(I*x^2)^7+3*csgn(I*x)^2*csgn(I*x^3)^7-6*csgn(I*x)*c
sgn(I*x^2)^8-3*csgn(I*x)*csgn(I*x^3)^8-3*csgn(I*x^2)^7*csgn(I*x^3)^2+3*csgn(I*x^2)^6*csgn(I*x^3)^3+3*csgn(I*x^
2)^5*csgn(I*x^3)^4-6*csgn(I*x^2)^4*csgn(I*x^3)^5+2*csgn(I*x^2)^3*csgn(I*x^3)^6+3*csgn(I*x^2)^2*csgn(I*x^3)^7-3
*csgn(I*x^2)*csgn(I*x^3)^8+csgn(I*x^2)^9+csgn(I*x^3)^9)*x^4*exp(-24*x^2)*ln(x)+1/16*Pi^4*(-41*csgn(I*x)^4*csgn
(I*x^2)^4*csgn(I*x^3)^4+44*csgn(I*x)^4*csgn(I*x^2)^3*csgn(I*x^3)^5-16*csgn(I*x)^4*csgn(I*x^2)^2*csgn(I*x^3)^6+
8*csgn(I*x)^4*csgn(I*x^2)*csgn(I*x^3)^7+60*csgn(I*x)^3*csgn(I*x^2)^8*csgn(I*x^3)-4*csgn(I*x)^3*csgn(I*x^2)^7*c
sgn(I*x^3)^2-100*csgn(I*x)^3*csgn(I*x^2)^6*csgn(I*x^3)^3+108*csgn(I*x)^3*csgn(I*x^2)^5*csgn(I*x^3)^4-76*csgn(I
*x)^3*csgn(I*x^2)^4*csgn(I*x^3)^5+24*csgn(I*x)^3*csgn(I*x^2)^3*csgn(I*x^3)^6-12*csgn(I*x)^3*csgn(I*x^2)^2*csgn
(I*x^3)^7+4*csgn(I*x)^3*csgn(I*x^2)*csgn(I*x^3)^8-24*csgn(I*x)^2*csgn(I*x^2)^9*csgn(I*x^3)-30*csgn(I*x)^2*csgn
(I*x^2)^8*csgn(I*x^3)^2+96*csgn(I*x)^2*csgn(I*x^2)^7*csgn(I*x^3)^3-66*csgn(I*x)^2*csgn(I*x^2)^6*csgn(I*x^3)^4-
12*csgn(I*x)^2*csgn(I*x^2)^5*csgn(I*x^3)^5+74*csgn(I*x)^2*csgn(I*x^2)^4*csgn(I*x^3)^6-72*csgn(I*x)^2*csgn(I*x^
2)^3*csgn(I*x^3)^7+48*csgn(I*x)^2*csgn(I*x^2)^2*csgn(I*x^3)^8-20*csgn(I*x)^2*csgn(I*x^2)*csgn(I*x^3)^9+4*csgn(
I*x)*csgn(I*x^2)^10*csgn(I*x^3)+20*csgn(I*x)*csgn(I*x^2)^9*csgn(I*x^3)^2-36*csgn(I*x)*csgn(I*x^2)^8*csgn(I*x^3
)^3+48*csgn(I*x)*csgn(I*x^2)^6*csgn(I*x^3)^5-52*csgn(I*x)*csgn(I*x^2)^5*csgn(I*x^3)^6+8*csgn(I*x)*csgn(I*x^2)^
4*csgn(I*x^3)^7+28*csgn(I*x)*csgn(I*x^2)^3*csgn(I*x^3)^8-32*csgn(I*x)*csgn(I*x^2)^2*csgn(I*x^3)^9+16*csgn(I*x)
*csgn(I*x^2)*csgn(I*x^3)^10+4*csgn(I*x)^7*csgn(I*x^2)^4*csgn(I*x^3)-4*csgn(I*x)^7*csgn(I*x^2)^3*csgn(I*x^3)^2-
24*csgn(I*x)^6*csgn(I*x^2)^5*csgn(I*x^3)+26*csgn(I*x)^6*csgn(I*x^2)^4*csgn(I*x^3)^2-8*csgn(I*x)^6*csgn(I*x^2)^
3*csgn(I*x^3)^3+6*csgn(I*x)^6*csgn(I*x^2)^2*csgn(I*x^3)^4+60*csgn(I*x)^5*csgn(I*x^2)^6*csgn(I*x^3)-60*csgn(I*x
)^5*csgn(I*x^2)^5*csgn(I*x^3)^2+16*csgn(I*x)^5*csgn(I*x^2)^4*csgn(I*x^3)^3-12*csgn(I*x)^5*csgn(I*x^2)^3*csgn(I
*x^3)^4-4*csgn(I*x)^5*csgn(I*x^2)*csgn(I*x^3)^6-80*csgn(I*x)^4*csgn(I*x^2)^7*csgn(I*x^3)+56*csgn(I*x)^4*csgn(I
*x^2)^6*csgn(I*x^3)^2+28*csgn(I*x)^4*csgn(I*x^2)^5*csgn(I*x^3)^3+csgn(I*x)^8*csgn(I*x^2)^4-8*csgn(I*x)^7*csgn(
I*x^2)^5+28*csgn(I*x)^6*csgn(I*x^2)^6-56*csgn(I*x)^5*csgn(I*x^2)^7+70*csgn(I*x)^4*csgn(I*x^2)^8+csgn(I*x)^4*cs
gn(I*x^3)^8-56*csgn(I*x)^3*csgn(I*x^2)^9-4*csgn(I*x)^3*csgn(I*x^3)^9+28*csgn(I*x)^2*csgn(I*x^2)^10+6*csgn(I*x)
^2*csgn(I*x^3)^10-8*csgn(I*x)*csgn(I*x^2)^11-4*csgn(I*x)*csgn(I*x^3)^11-4*csgn(I*x^2)^10*csgn(I*x^3)^2+4*csgn(
I*x^2)^9*csgn(I*x^3)^3+6*csgn(I*x^2)^8*csgn(I*x^3)^4-12*csgn(I*x^2)^7*csgn(I*x^3)^5+2*csgn(I*x^2)^6*csgn(I*x^3
)^6+12*csgn(I*x^2)^5*csgn(I*x^3)^7-11*csgn(I*x^2)^4*csgn(I*x^3)^8+6*csgn(I*x^2)^2*csgn(I*x^3)^10-4*csgn(I*x^2)
*csgn(I*x^3)^11+csgn(I*x^2)^12+csgn(I*x^3)^12)*x^4*exp(-24*x^2)
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maxima [A] time = 0.43, size = 15, normalized size = 0.88 \begin {gather*} 81 \, x^{4} e^{\left (-24 \, x^{2}\right )} \log \relax (x)^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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
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mupad [B] time = 4.22, size = 16, normalized size = 0.94 \begin {gather*} x^4\,{\ln \left (x^3\right )}^4\,{\mathrm {e}}^{-24\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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sympy [A] time = 0.41, size = 15, normalized size = 0.88 \begin {gather*} x^{4} e^{- 24 x^{2}} \log {\left (x^{3} \right )}^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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
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