3.60.29 \(\int e^{-x^2} (-4 e^{x^2} x^3+e^{2 e^{-x^2} (3 e^{x^2}+x)} (-2 e^{x^2} x-2 x^2+4 x^4)+e^{e^{-x^2} (3 e^{x^2}+x)} (6 e^{x^2} x^2+2 x^3-4 x^5)) \, dx\)

Optimal. Leaf size=26 \[ 6-\left (e^{3+e^{-x^2} x}-x\right )^2 x^2 \]

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Rubi [A]  time = 0.88, antiderivative size = 24, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 3, integrand size = 100, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {6741, 12, 6687} \begin {gather*} -\left (e^{e^{-x^2} x+3}-x\right )^2 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*E^x^2*x^3 + E^((2*(3*E^x^2 + x))/E^x^2)*(-2*E^x^2*x - 2*x^2 + 4*x^4) + E^((3*E^x^2 + x)/E^x^2)*(6*E^x^
2*x^2 + 2*x^3 - 4*x^5))/E^x^2,x]

[Out]

-((E^(3 + x/E^x^2) - x)^2*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 6687

Int[(u_)*(y_)^(m_.)*(z_)^(n_.), x_Symbol] :> With[{q = DerivativeDivides[y*z, u*z^(n - m), x]}, Simp[(q*y^(m +
1)*z^(m + 1))/(m + 1), x] /;  !FalseQ[q]] /; FreeQ[{m, n}, x] && NeQ[m, -1]

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 2 e^{-x^2} \left (e^{3+e^{-x^2} x}-x\right ) x \left (-e^{3+e^{-x^2} x+x^2}+2 e^{x^2} x-e^{3+e^{-x^2} x} x+2 e^{3+e^{-x^2} x} x^3\right ) \, dx\\ &=2 \int e^{-x^2} \left (e^{3+e^{-x^2} x}-x\right ) x \left (-e^{3+e^{-x^2} x+x^2}+2 e^{x^2} x-e^{3+e^{-x^2} x} x+2 e^{3+e^{-x^2} x} x^3\right ) \, dx\\ &=-\left (e^{3+e^{-x^2} x}-x\right )^2 x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.70, size = 24, normalized size = 0.92 \begin {gather*} -\left (e^{3+e^{-x^2} x}-x\right )^2 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*E^x^2*x^3 + E^((2*(3*E^x^2 + x))/E^x^2)*(-2*E^x^2*x - 2*x^2 + 4*x^4) + E^((3*E^x^2 + x)/E^x^2)*(
6*E^x^2*x^2 + 2*x^3 - 4*x^5))/E^x^2,x]

[Out]

-((E^(3 + x/E^x^2) - x)^2*x^2)

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fricas [B]  time = 0.61, size = 49, normalized size = 1.88 \begin {gather*} -x^{4} + 2 \, x^{3} e^{\left ({\left (x + 3 \, e^{\left (x^{2}\right )}\right )} e^{\left (-x^{2}\right )}\right )} - x^{2} e^{\left (2 \, {\left (x + 3 \, e^{\left (x^{2}\right )}\right )} e^{\left (-x^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(x^2)*x+4*x^4-2*x^2)*exp((3*exp(x^2)+x)/exp(x^2))^2+(6*x^2*exp(x^2)-4*x^5+2*x^3)*exp((3*exp(
x^2)+x)/exp(x^2))-4*x^3*exp(x^2))/exp(x^2),x, algorithm="fricas")

[Out]

-x^4 + 2*x^3*e^((x + 3*e^(x^2))*e^(-x^2)) - x^2*e^(2*(x + 3*e^(x^2))*e^(-x^2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -2 \, {\left (2 \, x^{3} e^{\left (x^{2}\right )} - {\left (2 \, x^{4} - x^{2} - x e^{\left (x^{2}\right )}\right )} e^{\left (2 \, {\left (x + 3 \, e^{\left (x^{2}\right )}\right )} e^{\left (-x^{2}\right )}\right )} + {\left (2 \, x^{5} - x^{3} - 3 \, x^{2} e^{\left (x^{2}\right )}\right )} e^{\left ({\left (x + 3 \, e^{\left (x^{2}\right )}\right )} e^{\left (-x^{2}\right )}\right )}\right )} e^{\left (-x^{2}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(x^2)*x+4*x^4-2*x^2)*exp((3*exp(x^2)+x)/exp(x^2))^2+(6*x^2*exp(x^2)-4*x^5+2*x^3)*exp((3*exp(
x^2)+x)/exp(x^2))-4*x^3*exp(x^2))/exp(x^2),x, algorithm="giac")

[Out]

integrate(-2*(2*x^3*e^(x^2) - (2*x^4 - x^2 - x*e^(x^2))*e^(2*(x + 3*e^(x^2))*e^(-x^2)) + (2*x^5 - x^3 - 3*x^2*
e^(x^2))*e^((x + 3*e^(x^2))*e^(-x^2)))*e^(-x^2), x)

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maple [B]  time = 0.08, size = 50, normalized size = 1.92

method	result	size
risch	\(-x^{4}+2 \,{\mathrm e}^{\left (3 \,{\mathrm e}^{x^{2}}+x \right ) {\mathrm e}^{-x^{2}}} x^{3}-{\mathrm e}^{2 \left (3 \,{\mathrm e}^{x^{2}}+x \right ) {\mathrm e}^{-x^{2}}} x^{2}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*exp(x^2)*x+4*x^4-2*x^2)*exp((3*exp(x^2)+x)/exp(x^2))^2+(6*x^2*exp(x^2)-4*x^5+2*x^3)*exp((3*exp(x^2)+x
)/exp(x^2))-4*x^3*exp(x^2))/exp(x^2),x,method=_RETURNVERBOSE)

[Out]

-x^4+2*exp((3*exp(x^2)+x)*exp(-x^2))*x^3-exp(2*(3*exp(x^2)+x)*exp(-x^2))*x^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -x^{4} - 2 \, \int -{\left (2 \, x^{4} e^{6} - x^{2} e^{6} - x e^{\left (x^{2} + 6\right )}\right )} e^{\left (-x^{2} + 2 \, x e^{\left (-x^{2}\right )}\right )}\,{d x} + 2 \, \int -{\left (2 \, x^{5} e^{3} - x^{3} e^{3} - 3 \, x^{2} e^{\left (x^{2} + 3\right )}\right )} e^{\left (-x^{2} + x e^{\left (-x^{2}\right )}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(x^2)*x+4*x^4-2*x^2)*exp((3*exp(x^2)+x)/exp(x^2))^2+(6*x^2*exp(x^2)-4*x^5+2*x^3)*exp((3*exp(
x^2)+x)/exp(x^2))-4*x^3*exp(x^2))/exp(x^2),x, algorithm="maxima")

[Out]

-x^4 - 2*integrate(-(2*x^4*e^6 - x^2*e^6 - x*e^(x^2 + 6))*e^(-x^2 + 2*x*e^(-x^2)), x) + 2*integrate(-(2*x^5*e^
3 - x^3*e^3 - 3*x^2*e^(x^2 + 3))*e^(-x^2 + x*e^(-x^2)), x)

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mupad [B]  time = 4.65, size = 39, normalized size = 1.50 \begin {gather*} 2\,x^3\,{\mathrm {e}}^3\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-x^2}}-x^4-x^2\,{\mathrm {e}}^6\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-x^2)*(exp(2*exp(-x^2)*(x + 3*exp(x^2)))*(2*x*exp(x^2) + 2*x^2 - 4*x^4) + 4*x^3*exp(x^2) - exp(exp(-x
^2)*(x + 3*exp(x^2)))*(6*x^2*exp(x^2) + 2*x^3 - 4*x^5)),x)

[Out]

2*x^3*exp(3)*exp(x*exp(-x^2)) - x^4 - x^2*exp(6)*exp(2*x*exp(-x^2))

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sympy [B]  time = 16.64, size = 42, normalized size = 1.62 \begin {gather*} - x^{4} + 2 x^{3} e^{\left (x + 3 e^{x^{2}}\right ) e^{- x^{2}}} - x^{2} e^{2 \left (x + 3 e^{x^{2}}\right ) e^{- x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(x**2)*x+4*x**4-2*x**2)*exp((3*exp(x**2)+x)/exp(x**2))**2+(6*x**2*exp(x**2)-4*x**5+2*x**3)*e
xp((3*exp(x**2)+x)/exp(x**2))-4*x**3*exp(x**2))/exp(x**2),x)

[Out]

-x**4 + 2*x**3*exp((x + 3*exp(x**2))*exp(-x**2)) - x**2*exp(2*(x + 3*exp(x**2))*exp(-x**2))

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