3.84.99 \(\int e^{-4 x^2-8 x \log ^2(5)-4 \log ^4(5)} (-52488 x+e^{4 x^2+8 x \log ^2(5)+4 \log ^4(5)} (2 x-6 x^2+4 x^3)-52488 \log ^2(5)+e^{2 x^2+4 x \log ^2(5)+2 \log ^4(5)} (162-324 x-648 x^2+648 x^3+(-648 x+648 x^2) \log ^2(5))) \, dx\)

Optimal. Leaf size=23 \[ \left (81 e^{-2 \left (x+\log ^2(5)\right )^2}+x-x^2\right )^2 \]

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Rubi [B]  time = 1.44, antiderivative size = 118, normalized size of antiderivative = 5.13, number of steps used = 14, number of rules used = 8, integrand size = 119, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {6688, 12, 6742, 74, 2209, 2226, 2212, 2205} \begin {gather*} (1-x)^2 x^2+6561 e^{-4 \left (x+\log ^2(5)\right )^2}-81 e^{-2 \left (x+\log ^2(5)\right )^2}-162 e^{-2 \left (x+\log ^2(5)\right )^2} \left (x+\log ^2(5)\right )^2+162 \left (1+2 \log ^2(5)\right ) e^{-2 \left (x+\log ^2(5)\right )^2} \left (x+\log ^2(5)\right )+81 \left (1-2 \log ^4(5)-2 \log ^2(5)\right ) e^{-2 \left (x+\log ^2(5)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-4*x^2 - 8*x*Log[5]^2 - 4*Log[5]^4)*(-52488*x + E^(4*x^2 + 8*x*Log[5]^2 + 4*Log[5]^4)*(2*x - 6*x^2 + 4*
x^3) - 52488*Log[5]^2 + E^(2*x^2 + 4*x*Log[5]^2 + 2*Log[5]^4)*(162 - 324*x - 648*x^2 + 648*x^3 + (-648*x + 648
*x^2)*Log[5]^2)),x]

[Out]

6561/E^(4*(x + Log[5]^2)^2) - 81/E^(2*(x + Log[5]^2)^2) + (1 - x)^2*x^2 - (162*(x + Log[5]^2)^2)/E^(2*(x + Log
[5]^2)^2) + (162*(x + Log[5]^2)*(1 + 2*Log[5]^2))/E^(2*(x + Log[5]^2)^2) + (81*(1 - 2*Log[5]^2 - 2*Log[5]^4))/
E^(2*(x + Log[5]^2)^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 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rule 6688

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int 2 e^{-4 \left (x+\log ^2(5)\right )^2} \left (81-e^{2 \left (x+\log ^2(5)\right )^2} (-1+x) x\right ) \left (-e^{2 \left (x+\log ^2(5)\right )^2} (-1+2 x)-324 \left (x+\log ^2(5)\right )\right ) \, dx\\ &=2 \int e^{-4 \left (x+\log ^2(5)\right )^2} \left (81-e^{2 \left (x+\log ^2(5)\right )^2} (-1+x) x\right ) \left (-e^{2 \left (x+\log ^2(5)\right )^2} (-1+2 x)-324 \left (x+\log ^2(5)\right )\right ) \, dx\\ &=2 \int \left ((-1+x) x (-1+2 x)-26244 e^{-4 \left (x+\log ^2(5)\right )^2} \left (x+\log ^2(5)\right )+81 e^{-2 \left (x+\log ^2(5)\right )^2} \left (1+4 x^3-4 x^2 \left (1-\log ^2(5)\right )-2 x \left (1+2 \log ^2(5)\right )\right )\right ) \, dx\\ &=2 \int (-1+x) x (-1+2 x) \, dx+162 \int e^{-2 \left (x+\log ^2(5)\right )^2} \left (1+4 x^3-4 x^2 \left (1-\log ^2(5)\right )-2 x \left (1+2 \log ^2(5)\right )\right ) \, dx-52488 \int e^{-4 \left (x+\log ^2(5)\right )^2} \left (x+\log ^2(5)\right ) \, dx\\ &=6561 e^{-4 \left (x+\log ^2(5)\right )^2}+(1-x)^2 x^2+162 \int \left (4 e^{-2 \left (x+\log ^2(5)\right )^2} \left (x+\log ^2(5)\right )^3+e^{-2 \left (x+\log ^2(5)\right )^2} \left (1+2 \log ^2(5)\right )-4 e^{-2 \left (x+\log ^2(5)\right )^2} \left (x+\log ^2(5)\right )^2 \left (1+2 \log ^2(5)\right )+2 e^{-2 \left (x+\log ^2(5)\right )^2} \left (x+\log ^2(5)\right ) \left (-1+2 \log ^2(5)+2 \log ^4(5)\right )\right ) \, dx\\ &=6561 e^{-4 \left (x+\log ^2(5)\right )^2}+(1-x)^2 x^2+648 \int e^{-2 \left (x+\log ^2(5)\right )^2} \left (x+\log ^2(5)\right )^3 \, dx+\left (162 \left (1+2 \log ^2(5)\right )\right ) \int e^{-2 \left (x+\log ^2(5)\right )^2} \, dx-\left (648 \left (1+2 \log ^2(5)\right )\right ) \int e^{-2 \left (x+\log ^2(5)\right )^2} \left (x+\log ^2(5)\right )^2 \, dx-\left (324 \left (1-2 \log ^2(5)-2 \log ^4(5)\right )\right ) \int e^{-2 \left (x+\log ^2(5)\right )^2} \left (x+\log ^2(5)\right ) \, dx\\ &=6561 e^{-4 \left (x+\log ^2(5)\right )^2}+(1-x)^2 x^2-162 e^{-2 \left (x+\log ^2(5)\right )^2} \left (x+\log ^2(5)\right )^2+81 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \left (x+\log ^2(5)\right )\right ) \left (1+2 \log ^2(5)\right )+162 e^{-2 \left (x+\log ^2(5)\right )^2} \left (x+\log ^2(5)\right ) \left (1+2 \log ^2(5)\right )+81 e^{-2 \left (x+\log ^2(5)\right )^2} \left (1-2 \log ^2(5)-2 \log ^4(5)\right )+324 \int e^{-2 \left (x+\log ^2(5)\right )^2} \left (x+\log ^2(5)\right ) \, dx-\left (162 \left (1+2 \log ^2(5)\right )\right ) \int e^{-2 \left (x+\log ^2(5)\right )^2} \, dx\\ &=6561 e^{-4 \left (x+\log ^2(5)\right )^2}-81 e^{-2 \left (x+\log ^2(5)\right )^2}+(1-x)^2 x^2-162 e^{-2 \left (x+\log ^2(5)\right )^2} \left (x+\log ^2(5)\right )^2+162 e^{-2 \left (x+\log ^2(5)\right )^2} \left (x+\log ^2(5)\right ) \left (1+2 \log ^2(5)\right )+81 e^{-2 \left (x+\log ^2(5)\right )^2} \left (1-2 \log ^2(5)-2 \log ^4(5)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.41, size = 34, normalized size = 1.48 \begin {gather*} e^{-4 \left (x+\log ^2(5)\right )^2} \left (-81+e^{2 \left (x+\log ^2(5)\right )^2} (-1+x) x\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-4*x^2 - 8*x*Log[5]^2 - 4*Log[5]^4)*(-52488*x + E^(4*x^2 + 8*x*Log[5]^2 + 4*Log[5]^4)*(2*x - 6*x^
2 + 4*x^3) - 52488*Log[5]^2 + E^(2*x^2 + 4*x*Log[5]^2 + 2*Log[5]^4)*(162 - 324*x - 648*x^2 + 648*x^3 + (-648*x
+ 648*x^2)*Log[5]^2)),x]

[Out]

(-81 + E^(2*(x + Log[5]^2)^2)*(-1 + x)*x)^2/E^(4*(x + Log[5]^2)^2)

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fricas [B]  time = 0.66, size = 85, normalized size = 3.70 \begin {gather*} {\left ({\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{\left (4 \, \log \relax (5)^{4} + 8 \, x \log \relax (5)^{2} + 4 \, x^{2}\right )} - 162 \, {\left (x^{2} - x\right )} e^{\left (2 \, \log \relax (5)^{4} + 4 \, x \log \relax (5)^{2} + 2 \, x^{2}\right )} + 6561\right )} e^{\left (-4 \, \log \relax (5)^{4} - 8 \, x \log \relax (5)^{2} - 4 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-6*x^2+2*x)*exp(log(5)^4+2*x*log(5)^2+x^2)^4+((648*x^2-648*x)*log(5)^2+648*x^3-648*x^2-324*x+
162)*exp(log(5)^4+2*x*log(5)^2+x^2)^2-52488*log(5)^2-52488*x)/exp(log(5)^4+2*x*log(5)^2+x^2)^4,x, algorithm="f
ricas")

[Out]

((x^4 - 2*x^3 + x^2)*e^(4*log(5)^4 + 8*x*log(5)^2 + 4*x^2) - 162*(x^2 - x)*e^(2*log(5)^4 + 4*x*log(5)^2 + 2*x^
2) + 6561)*e^(-4*log(5)^4 - 8*x*log(5)^2 - 4*x^2)

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giac [B]  time = 0.23, size = 82, normalized size = 3.57 \begin {gather*} x^{4} - 2 \, x^{3} - 162 \, x^{2} e^{\left (-2 \, \log \relax (5)^{4} - 4 \, x \log \relax (5)^{2} - 2 \, x^{2}\right )} + x^{2} + 162 \, x e^{\left (-2 \, \log \relax (5)^{4} - 4 \, x \log \relax (5)^{2} - 2 \, x^{2}\right )} + 6561 \, e^{\left (-4 \, \log \relax (5)^{4} - 8 \, x \log \relax (5)^{2} - 4 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-6*x^2+2*x)*exp(log(5)^4+2*x*log(5)^2+x^2)^4+((648*x^2-648*x)*log(5)^2+648*x^3-648*x^2-324*x+
162)*exp(log(5)^4+2*x*log(5)^2+x^2)^2-52488*log(5)^2-52488*x)/exp(log(5)^4+2*x*log(5)^2+x^2)^4,x, algorithm="g
iac")

[Out]

x^4 - 2*x^3 - 162*x^2*e^(-2*log(5)^4 - 4*x*log(5)^2 - 2*x^2) + x^2 + 162*x*e^(-2*log(5)^4 - 4*x*log(5)^2 - 2*x
^2) + 6561*e^(-4*log(5)^4 - 8*x*log(5)^2 - 4*x^2)

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maple [B]  time = 0.09, size = 47, normalized size = 2.04

method	result	size
risch	\(x^{4}-2 x^{3}+x^{2}+\left (-162 x^{2}+162 x \right ) {\mathrm e}^{-2 \left (\ln \relax (5)^{2}+x \right )^{2}}+6561 \,{\mathrm e}^{-4 \left (\ln \relax (5)^{2}+x \right )^{2}}\)	\(47\)
default	\(x^{4}-2 x^{3}+x^{2}+6561 \,{\mathrm e}^{-4 \ln \relax (5)^{4}-8 x \ln \relax (5)^{2}-4 x^{2}}+162 x \,{\mathrm e}^{-2 \ln \relax (5)^{4}-4 x \ln \relax (5)^{2}-2 x^{2}}-162 x^{2} {\mathrm e}^{-2 \ln \relax (5)^{4}-4 x \ln \relax (5)^{2}-2 x^{2}}\)	\(83\)
norman	\(\left (6561+x^{4} {\mathrm e}^{4 \ln \relax (5)^{4}+8 x \ln \relax (5)^{2}+4 x^{2}}+{\mathrm e}^{4 \ln \relax (5)^{4}+8 x \ln \relax (5)^{2}+4 x^{2}} x^{2}+162 \,{\mathrm e}^{2 \ln \relax (5)^{4}+4 x \ln \relax (5)^{2}+2 x^{2}} x -162 \,{\mathrm e}^{2 \ln \relax (5)^{4}+4 x \ln \relax (5)^{2}+2 x^{2}} x^{2}-2 \,{\mathrm e}^{4 \ln \relax (5)^{4}+8 x \ln \relax (5)^{2}+4 x^{2}} x^{3}\right ) {\mathrm e}^{-4 \ln \relax (5)^{4}-8 x \ln \relax (5)^{2}-4 x^{2}}\)	\(133\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3-6*x^2+2*x)*exp(ln(5)^4+2*x*ln(5)^2+x^2)^4+((648*x^2-648*x)*ln(5)^2+648*x^3-648*x^2-324*x+162)*exp(
ln(5)^4+2*x*ln(5)^2+x^2)^2-52488*ln(5)^2-52488*x)/exp(ln(5)^4+2*x*ln(5)^2+x^2)^4,x,method=_RETURNVERBOSE)

[Out]

x^4-2*x^3+x^2+(-162*x^2+162*x)*exp(-2*(ln(5)^2+x)^2)+6561*exp(-4*(ln(5)^2+x)^2)

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maxima [C]  time = 0.66, size = 565, normalized size = 24.57 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-6*x^2+2*x)*exp(log(5)^4+2*x*log(5)^2+x^2)^4+((648*x^2-648*x)*log(5)^2+648*x^3-648*x^2-324*x+
162)*exp(log(5)^4+2*x*log(5)^2+x^2)^2-52488*log(5)^2-52488*x)/exp(log(5)^4+2*x*log(5)^2+x^2)^4,x, algorithm="m
axima")

[Out]

x^4 + 13122*sqrt(pi)*(log(5)^2 + x)*(erf(2*sqrt((log(5)^2 + x)^2)) - 1)*log(5)^2/sqrt((log(5)^2 + x)^2) - 81*I
*sqrt(2)*(2*I*sqrt(pi)*(log(5)^2 + x)*(erf(sqrt(2)*sqrt((log(5)^2 + x)^2)) - 1)*log(5)^4/sqrt((log(5)^2 + x)^2
) + 2*I*sqrt(2)*e^(-2*(log(5)^2 + x)^2)*log(5)^2 - I*(log(5)^2 + x)^3*gamma(3/2, 2*(log(5)^2 + x)^2)/((log(5)^
2 + x)^2)^(3/2))*log(5)^2 + 81*I*sqrt(2)*(-2*I*sqrt(pi)*(log(5)^2 + x)*(erf(sqrt(2)*sqrt((log(5)^2 + x)^2)) -
1)*log(5)^2/sqrt((log(5)^2 + x)^2) - I*sqrt(2)*e^(-2*(log(5)^2 + x)^2))*log(5)^2 - 13122*sqrt(pi)*erf(2*log(5)
^2 + 2*x)*log(5)^2 - 2*x^3 + x^2 + 81/2*sqrt(2)*sqrt(pi)*erf(sqrt(2)*log(5)^2 + sqrt(2)*x) - 81/2*I*sqrt(2)*(-
4*I*sqrt(pi)*(log(5)^2 + x)*(erf(sqrt(2)*sqrt((log(5)^2 + x)^2)) - 1)*log(5)^6/sqrt((log(5)^2 + x)^2) - 6*I*sq
rt(2)*e^(-2*(log(5)^2 + x)^2)*log(5)^4 + 6*I*(log(5)^2 + x)^3*gamma(3/2, 2*(log(5)^2 + x)^2)*log(5)^2/((log(5)
^2 + x)^2)^(3/2) - I*sqrt(2)*gamma(2, 2*(log(5)^2 + x)^2)) + 81*I*sqrt(2)*(2*I*sqrt(pi)*(log(5)^2 + x)*(erf(sq
rt(2)*sqrt((log(5)^2 + x)^2)) - 1)*log(5)^4/sqrt((log(5)^2 + x)^2) + 2*I*sqrt(2)*e^(-2*(log(5)^2 + x)^2)*log(5
)^2 - I*(log(5)^2 + x)^3*gamma(3/2, 2*(log(5)^2 + x)^2)/((log(5)^2 + x)^2)^(3/2)) + 81/2*I*sqrt(2)*(-2*I*sqrt(
pi)*(log(5)^2 + x)*(erf(sqrt(2)*sqrt((log(5)^2 + x)^2)) - 1)*log(5)^2/sqrt((log(5)^2 + x)^2) - I*sqrt(2)*e^(-2
*(log(5)^2 + x)^2)) + 6561*e^(-4*(log(5)^2 + x)^2)

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mupad [B]  time = 5.44, size = 82, normalized size = 3.57 \begin {gather*} 6561\,{\mathrm {e}}^{-4\,x^2-8\,{\ln \relax (5)}^2\,x-4\,{\ln \relax (5)}^4}-162\,x^2\,{\mathrm {e}}^{-2\,x^2-4\,{\ln \relax (5)}^2\,x-2\,{\ln \relax (5)}^4}+162\,x\,{\mathrm {e}}^{-2\,x^2-4\,{\ln \relax (5)}^2\,x-2\,{\ln \relax (5)}^4}+x^2-2\,x^3+x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(- 8*x*log(5)^2 - 4*log(5)^4 - 4*x^2)*(52488*x + exp(4*x*log(5)^2 + 2*log(5)^4 + 2*x^2)*(324*x + log(5
)^2*(648*x - 648*x^2) + 648*x^2 - 648*x^3 - 162) + 52488*log(5)^2 - exp(8*x*log(5)^2 + 4*log(5)^4 + 4*x^2)*(2*
x - 6*x^2 + 4*x^3)),x)

[Out]

6561*exp(- 8*x*log(5)^2 - 4*log(5)^4 - 4*x^2) - 162*x^2*exp(- 4*x*log(5)^2 - 2*log(5)^4 - 2*x^2) + 162*x*exp(-
4*x*log(5)^2 - 2*log(5)^4 - 2*x^2) + x^2 - 2*x^3 + x^4

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sympy [B]  time = 0.22, size = 68, normalized size = 2.96 \begin {gather*} x^{4} - 2 x^{3} + x^{2} + \left (- 162 x^{2} + 162 x\right ) e^{- 2 x^{2} - 4 x \log {\relax (5 )}^{2} - 2 \log {\relax (5 )}^{4}} + 6561 e^{- 4 x^{2} - 8 x \log {\relax (5 )}^{2} - 4 \log {\relax (5 )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3-6*x**2+2*x)*exp(ln(5)**4+2*x*ln(5)**2+x**2)**4+((648*x**2-648*x)*ln(5)**2+648*x**3-648*x**2
-324*x+162)*exp(ln(5)**4+2*x*ln(5)**2+x**2)**2-52488*ln(5)**2-52488*x)/exp(ln(5)**4+2*x*ln(5)**2+x**2)**4,x)

[Out]

x**4 - 2*x**3 + x**2 + (-162*x**2 + 162*x)*exp(-2*x**2 - 4*x*log(5)**2 - 2*log(5)**4) + 6561*exp(-4*x**2 - 8*x
*log(5)**2 - 4*log(5)**4)

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