3.69.82 \(\int e^{-24 x^2} (12 x^3 \log ^3(x^3)+(4 x^3-48 x^5) \log ^4(x^3)) \, dx\) [6882]
Optimal. Leaf size=17 \[ e^{-24 x^2} x^4 \log ^4\left (x^3\right ) \]
[Out]
ln(x^3)^4/exp(3*x^2)^8*x^4
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Rubi [A]
time = 0.10, 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 = {6873, 12, 2326}
\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 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 {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 = 0.05, 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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.36, size = 3066, normalized size = 180.35
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result |
size |
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| risch |
\(\text {Expression too large to display}\) |
\(3066\) |
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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^3)^3-csgn(I*x^3)^2*csgn(I*x)-csgn(I*x^3)^2*csgn(I*x^2)+csgn(I*x^
3)*csgn(I*x)*csgn(I*x^2)+csgn(I*x)^2*csgn(I*x^2)-2*csgn(I*x)*csgn(I*x^2)^2+csgn(I*x^2)^3)*x^4*exp(-24*x^2)*ln(
x)^3-27/2*Pi^2*(csgn(I*x^3)^6-2*csgn(I*x^3)^5*csgn(I*x)-2*csgn(I*x^3)^5*csgn(I*x^2)+csgn(I*x^3)^4*csgn(I*x)^2+
4*csgn(I*x^3)^4*csgn(I*x)*csgn(I*x^2)+csgn(I*x^3)^4*csgn(I*x^2)^2-6*csgn(I*x)*csgn(I*x^2)^2*csgn(I*x^3)^3+2*cs
gn(I*x^2)^3*csgn(I*x^3)^3-2*csgn(I*x)^3*csgn(I*x^2)*csgn(I*x^3)^2+3*csgn(I*x)^2*csgn(I*x^2)^2*csgn(I*x^3)^2+2*
csgn(I*x)*csgn(I*x^2)^3*csgn(I*x^3)^2-2*csgn(I*x^2)^4*csgn(I*x^3)^2+2*csgn(I*x)^3*csgn(I*x^2)^2*csgn(I*x^3)-4*
csgn(I*x)^2*csgn(I*x^2)^3*csgn(I*x^3)+2*csgn(I*x^2)^4*csgn(I*x^3)*csgn(I*x)+csgn(I*x)^4*csgn(I*x^2)^2-4*csgn(I
*x)^3*csgn(I*x^2)^3+6*csgn(I*x)^2*csgn(I*x^2)^4-4*csgn(I*x)*csgn(I*x^2)^5+csgn(I*x^2)^6)*x^4*exp(-24*x^2)*ln(x
)^2+3/2*I*Pi^3*(9*csgn(I*x^3)^7*csgn(I*x)*csgn(I*x^2)-6*csgn(I*x^3)^6*csgn(I*x)^2*csgn(I*x^2)-15*csgn(I*x^3)^6
*csgn(I*x)*csgn(I*x^2)^2-3*csgn(I*x^3)^5*csgn(I*x)^3*csgn(I*x^2)+15*csgn(I*x^3)^5*csgn(I*x)^2*csgn(I*x^2)^2+9*
csgn(I*x^3)^5*csgn(I*x)*csgn(I*x^2)^3+3*csgn(I*x^3)^4*csgn(I*x)^4*csgn(I*x^2)+3*csgn(I*x^3)^4*csgn(I*x)^3*csgn
(I*x^2)^2-21*csgn(I*x^3)^4*csgn(I*x)^2*csgn(I*x^2)^3+6*csgn(I*x^3)^4*csgn(I*x)*csgn(I*x^2)^4-3*csgn(I*x^3)^3*c
sgn(I*x)^4*csgn(I*x^2)^2-5*csgn(I*x^3)^3*csgn(I*x)^3*csgn(I*x^2)^3+24*csgn(I*x^3)^3*csgn(I*x)^2*csgn(I*x^2)^4-
18*csgn(I*x^3)^3*csgn(I*x)*csgn(I*x^2)^5-3*csgn(I*x^3)^2*csgn(I*x)^5*csgn(I*x^2)^2+12*csgn(I*x^3)^2*csgn(I*x)^
4*csgn(I*x^2)^3-12*csgn(I*x^3)^2*csgn(I*x)^3*csgn(I*x^2)^4-3*csgn(I*x^3)^2*csgn(I*x)^2*csgn(I*x^2)^5+9*csgn(I*
x^3)^2*csgn(I*x)*csgn(I*x^2)^6+3*csgn(I*x^3)*csgn(I*x)^5*csgn(I*x^2)^3-12*csgn(I*x^3)*csgn(I*x)^4*csgn(I*x^2)^
4+18*csgn(I*x^3)*csgn(I*x)^3*csgn(I*x^2)^5-12*csgn(I*x^3)*csgn(I*x)^2*csgn(I*x^2)^6+3*csgn(I*x^3)*csgn(I*x)*cs
gn(I*x^2)^7+csgn(I*x^3)^9+csgn(I*x^2)^9-3*csgn(I*x^3)^8*csgn(I*x)-3*csgn(I*x^3)^8*csgn(I*x^2)+3*csgn(I*x^3)^7*
csgn(I*x)^2+3*csgn(I*x^3)^7*csgn(I*x^2)^2-csgn(I*x^3)^6*csgn(I*x)^3+2*csgn(I*x^3)^6*csgn(I*x^2)^3-6*csgn(I*x^3
)^5*csgn(I*x^2)^4+3*csgn(I*x^3)^4*csgn(I*x^2)^5+3*csgn(I*x^3)^3*csgn(I*x^2)^6-3*csgn(I*x^3)^2*csgn(I*x^2)^7+cs
gn(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+
15*csgn(I*x)^2*csgn(I*x^2)^7-6*csgn(I*x)*csgn(I*x^2)^8)*x^4*exp(-24*x^2)*ln(x)+1/16*Pi^4*(16*csgn(I*x^3)^10*cs
gn(I*x)*csgn(I*x^2)-20*csgn(I*x^3)^9*csgn(I*x)^2*csgn(I*x^2)-32*csgn(I*x^3)^9*csgn(I*x)*csgn(I*x^2)^2+4*csgn(I
*x^3)^8*csgn(I*x)^3*csgn(I*x^2)+48*csgn(I*x^3)^8*csgn(I*x)^2*csgn(I*x^2)^2+28*csgn(I*x^3)^8*csgn(I*x)*csgn(I*x
^2)^3+8*csgn(I*x^3)^7*csgn(I*x)^4*csgn(I*x^2)-12*csgn(I*x^3)^7*csgn(I*x)^3*csgn(I*x^2)^2-72*csgn(I*x^3)^7*csgn
(I*x)^2*csgn(I*x^2)^3+8*csgn(I*x^3)^7*csgn(I*x)*csgn(I*x^2)^4-4*csgn(I*x^3)^6*csgn(I*x)^5*csgn(I*x^2)-16*csgn(
I*x^3)^6*csgn(I*x)^4*csgn(I*x^2)^2+24*csgn(I*x^3)^6*csgn(I*x)^3*csgn(I*x^2)^3+74*csgn(I*x^3)^6*csgn(I*x)^2*csg
n(I*x^2)^4-52*csgn(I*x^3)^6*csgn(I*x)*csgn(I*x^2)^5+44*csgn(I*x^3)^5*csgn(I*x)^4*csgn(I*x^2)^3-76*csgn(I*x^3)^
5*csgn(I*x)^3*csgn(I*x^2)^4-12*csgn(I*x^3)^5*csgn(I*x)^2*csgn(I*x^2)^5+48*csgn(I*x^3)^5*csgn(I*x)*csgn(I*x^2)^
6+6*csgn(I*x^3)^4*csgn(I*x)^6*csgn(I*x^2)^2-12*csgn(I*x^3)^4*csgn(I*x)^5*csgn(I*x^2)^3-41*csgn(I*x^3)^4*csgn(I
*x)^4*csgn(I*x^2)^4+108*csgn(I*x^3)^4*csgn(I*x)^3*csgn(I*x^2)^5-66*csgn(I*x^3)^4*csgn(I*x)^2*csgn(I*x^2)^6-8*c
sgn(I*x^3)^3*csgn(I*x)^6*csgn(I*x^2)^3+16*csgn(I*x^3)^3*csgn(I*x)^5*csgn(I*x^2)^4+28*csgn(I*x^3)^3*csgn(I*x)^4
*csgn(I*x^2)^5-100*csgn(I*x^3)^3*csgn(I*x)^3*csgn(I*x^2)^6+96*csgn(I*x^3)^3*csgn(I*x)^2*csgn(I*x^2)^7-36*csgn(
I*x^3)^3*csgn(I*x)*csgn(I*x^2)^8-4*csgn(I*x^3)^2*csgn(I*x)^7*csgn(I*x^2)^3+csgn(I*x^3)^12+csgn(I*x^2)^12-4*csg
n(I*x^3)^11*csgn(I*x)-4*csgn(I*x^3)^11*csgn(I*x^2)+6*csgn(I*x^3)^10*csgn(I*x)^2+6*csgn(I*x^3)^10*csgn(I*x^2)^2
-4*csgn(I*x^3)^9*csgn(I*x)^3+csgn(I*x^3)^8*csgn(I*x)^4-11*csgn(I*x^3)^8*csgn(I*x^2)^4+12*csgn(I*x^3)^7*csgn(I*
x^2)^5+2*csgn(I*x^3)^6*csgn(I*x^2)^6-12*csgn(I*x^3)^5*csgn(I*x^2)^7+6*csgn(I*x^3)^4*csgn(I*x^2)^8+4*csgn(I*x^3
)^3*csgn(I*x^2)^9-4*csgn(I*x^3)^2*csgn(I*x^2)^10+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)^8-56*csgn(I*x)^3*csgn(I*x^2)^9+2
8*csgn(I*x)^2*csgn(I*x^2)^10-8*csgn(I*x)*csgn(I*x^2)^11+26*csgn(I*x^3)^2*csgn(I*x)^6*csgn(I*x^2)^4-60*csgn(I*x
^3)^2*csgn(I*x)^5*csgn(I*x^2)^5+56*csgn(I*x^3)^2*csgn(I*x)^4*csgn(I*x^2)^6-4*csgn(I*x^3)^2*csgn(I*x)^3*csgn(I*
x^2)^7-30*csgn(I*x^3)^2*csgn(I*x)^2*csgn(I*x^2)^8+20*csgn(I*x^3)^2*csgn(I*x)*csgn(I*x^2)^9+4*csgn(I*x^3)*csgn(
I*x)^7*csgn(I*x^2)^4-24*csgn(I*x^3)*csgn(I*x)^6*csgn(I*x^2)^5+60*csgn(I*x^3)*csgn(I*x)^5*csgn(I*x^2)^6-80*csgn
(I*x^3)*csgn(I*x)^4*csgn(I*x^2)^7+60*csgn(I*x^3)*csgn(I*x)^3*csgn(I*x^2)^8-24*csgn(I*x^3)*csgn(I*x)^2*csgn(I*x
^2)^9+4*csgn(I*x^3)*csgn(I*x)*csgn(I*x^2)^10)*x^4*exp(-24*x^2)
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Maxima [A]
time = 0.33, size = 15, normalized size = 0.88 \begin {gather*} 81 \, x^{4} e^{\left (-24 \, x^{2}\right )} \log \left (x\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="maxima")
[Out]
81*x^4*e^(-24*x^2)*log(x)^4
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Fricas [A]
time = 0.37, 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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Sympy [A]
time = 0.12, 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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Giac [A]
time = 0.40, 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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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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