3.99.98 \(\int \frac {1}{3} e^{-\frac {x^2}{3}} (-3+e^{\frac {72+6 x \log (\log (4))}{\log (\log (4))}} (54-6 x)+2 x^2) \, dx\)

Optimal. Leaf size=31 \[ e^{-\frac {x^2}{3}} \left (3 e^{3 \left (2 x+\frac {24}{\log (\log (4))}\right )}-x\right ) \]

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Rubi [A]  time = 0.24, antiderivative size = 35, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 5, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {12, 6742, 2205, 2236, 2212} \begin {gather*} 3 e^{-\frac {x^2}{3}+6 x+\frac {72}{\log (\log (4))}}-e^{-\frac {x^2}{3}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + E^((72 + 6*x*Log[Log[4]])/Log[Log[4]])*(54 - 6*x) + 2*x^2)/(3*E^(x^2/3)),x]

[Out]

3*E^(6*x - x^2/3 + 72/Log[Log[4]]) - x/E^(x^2/3)

Rule 12

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

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 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 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int e^{-\frac {x^2}{3}} \left (-3+e^{\frac {72+6 x \log (\log (4))}{\log (\log (4))}} (54-6 x)+2 x^2\right ) \, dx\\ &=\frac {1}{3} \int \left (-3 e^{-\frac {x^2}{3}}-6 e^{6 x-\frac {x^2}{3}+\frac {72}{\log (\log (4))}} (-9+x)+2 e^{-\frac {x^2}{3}} x^2\right ) \, dx\\ &=\frac {2}{3} \int e^{-\frac {x^2}{3}} x^2 \, dx-2 \int e^{6 x-\frac {x^2}{3}+\frac {72}{\log (\log (4))}} (-9+x) \, dx-\int e^{-\frac {x^2}{3}} \, dx\\ &=3 e^{6 x-\frac {x^2}{3}+\frac {72}{\log (\log (4))}}-e^{-\frac {x^2}{3}} x-\frac {1}{2} \sqrt {3 \pi } \text {erf}\left (\frac {x}{\sqrt {3}}\right )+\int e^{-\frac {x^2}{3}} \, dx\\ &=3 e^{6 x-\frac {x^2}{3}+\frac {72}{\log (\log (4))}}-e^{-\frac {x^2}{3}} x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 32, normalized size = 1.03 \begin {gather*} \frac {1}{3} e^{-\frac {x^2}{3}} \left (9 e^{6 x+\frac {72}{\log (\log (4))}}-3 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + E^((72 + 6*x*Log[Log[4]])/Log[Log[4]])*(54 - 6*x) + 2*x^2)/(3*E^(x^2/3)),x]

[Out]

(9*E^(6*x + 72/Log[Log[4]]) - 3*x)/(3*E^(x^2/3))

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fricas [A]  time = 1.39, size = 37, normalized size = 1.19 \begin {gather*} -x e^{\left (-\frac {1}{3} \, x^{2}\right )} + 3 \, e^{\left (-\frac {1}{3} \, x^{2} + \frac {6 \, {\left (x \log \left (2 \, \log \relax (2)\right ) + 12\right )}}{\log \left (2 \, \log \relax (2)\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-6*x+54)*exp((6*x*log(2*log(2))+72)/log(2*log(2)))+2*x^2-3)/exp(1/3*x^2),x, algorithm="fricas"
)

[Out]

-x*e^(-1/3*x^2) + 3*e^(-1/3*x^2 + 6*(x*log(2*log(2)) + 12)/log(2*log(2)))

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giac [A]  time = 0.24, size = 41, normalized size = 1.32 \begin {gather*} -x e^{\left (-\frac {1}{3} \, x^{2}\right )} + 3 \, e^{\left (-\frac {x^{2} \log \left (2 \, \log \relax (2)\right ) - 18 \, x \log \left (2 \, \log \relax (2)\right ) - 216}{3 \, \log \left (2 \, \log \relax (2)\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-6*x+54)*exp((6*x*log(2*log(2))+72)/log(2*log(2)))+2*x^2-3)/exp(1/3*x^2),x, algorithm="giac")

[Out]

-x*e^(-1/3*x^2) + 3*e^(-1/3*(x^2*log(2*log(2)) - 18*x*log(2*log(2)) - 216)/log(2*log(2)))

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maple [A]  time = 0.08, size = 35, normalized size = 1.13

method	result	size
norman	\(\left (-x +3 \,{\mathrm e}^{\frac {6 x \ln \left (2 \ln \relax (2)\right )+72}{\ln \left (2 \ln \relax (2)\right )}}\right ) {\mathrm e}^{-\frac {x^{2}}{3}}\)	\(35\)
risch	\(-x \,{\mathrm e}^{-\frac {x^{2}}{3}}+3 \,{\mathrm e}^{-\frac {x^{2} \ln \relax (2)+x^{2} \ln \left (\ln \relax (2)\right )-18 x \ln \relax (2)-18 x \ln \left (\ln \relax (2)\right )-216}{3 \left (\ln \relax (2)+\ln \left (\ln \relax (2)\right )\right )}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((-6*x+54)*exp((6*x*ln(2*ln(2))+72)/ln(2*ln(2)))+2*x^2-3)/exp(1/3*x^2),x,method=_RETURNVERBOSE)

[Out]

(-x+3*exp((6*x*ln(2*ln(2))+72)/ln(2*ln(2))))/exp(1/3*x^2)

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maxima [C]  time = 0.50, size = 110, normalized size = 3.55 \begin {gather*} 9 \, \sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{3} \, \sqrt {3} x - 3 \, \sqrt {3}\right ) e^{\left (\frac {72}{\log \relax (2) + \log \left (\log \relax (2)\right )} + 27\right )} - i \, \sqrt {3} {\left (-\frac {9 i \, \sqrt {3} \sqrt {\frac {1}{3}} \sqrt {\pi } {\left (x - 9\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{3}} \sqrt {{\left (x - 9\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (x - 9\right )}^{2}}} + i \, \sqrt {3} e^{\left (-\frac {1}{3} \, {\left (x - 9\right )}^{2}\right )}\right )} e^{\left (\frac {72}{\log \left (2 \, \log \relax (2)\right )} + 27\right )} - x e^{\left (-\frac {1}{3} \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-6*x+54)*exp((6*x*log(2*log(2))+72)/log(2*log(2)))+2*x^2-3)/exp(1/3*x^2),x, algorithm="maxima"
)

[Out]

9*sqrt(3)*sqrt(pi)*erf(1/3*sqrt(3)*x - 3*sqrt(3))*e^(72/(log(2) + log(log(2))) + 27) - I*sqrt(3)*(-9*I*sqrt(3)
*sqrt(1/3)*sqrt(pi)*(x - 9)*(erf(sqrt(1/3)*sqrt((x - 9)^2)) - 1)/sqrt((x - 9)^2) + I*sqrt(3)*e^(-1/3*(x - 9)^2
))*e^(72/log(2*log(2)) + 27) - x*e^(-1/3*x^2)

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mupad [B]  time = 0.11, size = 50, normalized size = 1.61 \begin {gather*} 3\,{64}^{\frac {x}{\ln \left (2\,\ln \relax (2)\right )}}\,{\mathrm {e}}^{\frac {72}{\ln \left (2\,\ln \relax (2)\right )}}\,{\mathrm {e}}^{-\frac {x^2}{3}}\,{\ln \relax (2)}^{\frac {6\,x}{\ln \left (\ln \relax (4)\right )}}-x\,{\mathrm {e}}^{-\frac {x^2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-x^2/3)*((exp((6*x*log(2*log(2)) + 72)/log(2*log(2)))*(6*x - 54))/3 - (2*x^2)/3 + 1),x)

[Out]

3*64^(x/log(2*log(2)))*exp(72/log(2*log(2)))*exp(-x^2/3)*log(2)^((6*x)/log(log(4))) - x*exp(-x^2/3)

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sympy [A]  time = 1.05, size = 36, normalized size = 1.16 \begin {gather*} - x e^{- \frac {x^{2}}{3}} + 3 e^{- \frac {x^{2}}{3}} e^{\frac {6 x \log {\left (2 \log {\relax (2 )} \right )} + 72}{\log {\left (2 \log {\relax (2 )} \right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-6*x+54)*exp((6*x*ln(2*ln(2))+72)/ln(2*ln(2)))+2*x**2-3)/exp(1/3*x**2),x)

[Out]

-x*exp(-x**2/3) + 3*exp(-x**2/3)*exp((6*x*log(2*log(2)) + 72)/log(2*log(2)))

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