3.24.36 \(\int \frac {-36 x+24 e^{3 e^x} x-4 e^{6 e^x} x+e^{-\frac {x^2}{-6+2 e^{3 e^x}}} (6 x+e^{3 e^x} (-2 x+3 e^x x^2))}{18-12 e^{3 e^x}+2 e^{6 e^x}} \, dx\)

Optimal. Leaf size=28 \[ e^{\frac {x^2}{2 \left (3-e^{3 e^x}\right )}}-x^2 \]

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Rubi [A]  time = 2.55, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 92, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {6741, 12, 6742, 6706} \begin {gather*} e^{\frac {x^2}{2 \left (3-e^{3 e^x}\right )}}-x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-36*x + 24*E^(3*E^x)*x - 4*E^(6*E^x)*x + (6*x + E^(3*E^x)*(-2*x + 3*E^x*x^2))/E^(x^2/(-6 + 2*E^(3*E^x))))
/(18 - 12*E^(3*E^x) + 2*E^(6*E^x)),x]

[Out]

E^(x^2/(2*(3 - E^(3*E^x)))) - 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 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rule 6741

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

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 \frac {-36 x+24 e^{3 e^x} x-4 e^{6 e^x} x+e^{-\frac {x^2}{-6+2 e^{3 e^x}}} \left (6 x+e^{3 e^x} \left (-2 x+3 e^x x^2\right )\right )}{2 \left (3-e^{3 e^x}\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {-36 x+24 e^{3 e^x} x-4 e^{6 e^x} x+e^{-\frac {x^2}{-6+2 e^{3 e^x}}} \left (6 x+e^{3 e^x} \left (-2 x+3 e^x x^2\right )\right )}{\left (3-e^{3 e^x}\right )^2} \, dx\\ &=\frac {1}{2} \int \left (-4 x+\frac {e^{-\frac {x^2}{2 \left (-3+e^{3 e^x}\right )}} x \left (6-2 e^{3 e^x}+3 e^{3 e^x+x} x\right )}{\left (3-e^{3 e^x}\right )^2}\right ) \, dx\\ &=-x^2+\frac {1}{2} \int \frac {e^{-\frac {x^2}{2 \left (-3+e^{3 e^x}\right )}} x \left (6-2 e^{3 e^x}+3 e^{3 e^x+x} x\right )}{\left (3-e^{3 e^x}\right )^2} \, dx\\ &=e^{\frac {x^2}{2 \left (3-e^{3 e^x}\right )}}-x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.43, size = 32, normalized size = 1.14 \begin {gather*} \frac {1}{2} \left (2 e^{-\frac {x^2}{2 \left (-3+e^{3 e^x}\right )}}-2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-36*x + 24*E^(3*E^x)*x - 4*E^(6*E^x)*x + (6*x + E^(3*E^x)*(-2*x + 3*E^x*x^2))/E^(x^2/(-6 + 2*E^(3*E
^x))))/(18 - 12*E^(3*E^x) + 2*E^(6*E^x)),x]

[Out]

(2/E^(x^2/(2*(-3 + E^(3*E^x)))) - 2*x^2)/2

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fricas [A]  time = 0.75, size = 21, normalized size = 0.75 \begin {gather*} -x^{2} + e^{\left (-\frac {x^{2}}{2 \, {\left (e^{\left (3 \, e^{x}\right )} - 3\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*exp(x)*x^2-2*x)*exp(3*exp(x))+6*x)*exp(-x^2/(2*exp(3*exp(x))-6))-4*x*exp(3*exp(x))^2+24*x*exp(3
*exp(x))-36*x)/(2*exp(3*exp(x))^2-12*exp(3*exp(x))+18),x, algorithm="fricas")

[Out]

-x^2 + e^(-1/2*x^2/(e^(3*e^x) - 3))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (3 \, x^{2} e^{x} - 2 \, x\right )} e^{\left (3 \, e^{x}\right )} + 6 \, x\right )} e^{\left (-\frac {x^{2}}{2 \, {\left (e^{\left (3 \, e^{x}\right )} - 3\right )}}\right )} - 4 \, x e^{\left (6 \, e^{x}\right )} + 24 \, x e^{\left (3 \, e^{x}\right )} - 36 \, x}{2 \, {\left (e^{\left (6 \, e^{x}\right )} - 6 \, e^{\left (3 \, e^{x}\right )} + 9\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*exp(x)*x^2-2*x)*exp(3*exp(x))+6*x)*exp(-x^2/(2*exp(3*exp(x))-6))-4*x*exp(3*exp(x))^2+24*x*exp(3
*exp(x))-36*x)/(2*exp(3*exp(x))^2-12*exp(3*exp(x))+18),x, algorithm="giac")

[Out]

integrate(1/2*(((3*x^2*e^x - 2*x)*e^(3*e^x) + 6*x)*e^(-1/2*x^2/(e^(3*e^x) - 3)) - 4*x*e^(6*e^x) + 24*x*e^(3*e^
x) - 36*x)/(e^(6*e^x) - 6*e^(3*e^x) + 9), x)

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maple [A]  time = 0.05, size = 22, normalized size = 0.79

method	result	size
risch	\(-x^{2}+{\mathrm e}^{-\frac {x^{2}}{2 \left ({\mathrm e}^{3 \,{\mathrm e}^{x}}-3\right )}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((3*exp(x)*x^2-2*x)*exp(3*exp(x))+6*x)*exp(-x^2/(2*exp(3*exp(x))-6))-4*x*exp(3*exp(x))^2+24*x*exp(3*exp(x
))-36*x)/(2*exp(3*exp(x))^2-12*exp(3*exp(x))+18),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 1.00, size = 38, normalized size = 1.36 \begin {gather*} -{\left (x^{2} e^{\left (\frac {x^{2}}{2 \, {\left (e^{\left (3 \, e^{x}\right )} - 3\right )}}\right )} - 1\right )} e^{\left (-\frac {x^{2}}{2 \, {\left (e^{\left (3 \, e^{x}\right )} - 3\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*exp(x)*x^2-2*x)*exp(3*exp(x))+6*x)*exp(-x^2/(2*exp(3*exp(x))-6))-4*x*exp(3*exp(x))^2+24*x*exp(3
*exp(x))-36*x)/(2*exp(3*exp(x))^2-12*exp(3*exp(x))+18),x, algorithm="maxima")

[Out]

-(x^2*e^(1/2*x^2/(e^(3*e^x) - 3)) - 1)*e^(-1/2*x^2/(e^(3*e^x) - 3))

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mupad [B]  time = 1.62, size = 23, normalized size = 0.82 \begin {gather*} {\mathrm {e}}^{-\frac {x^2}{2\,{\mathrm {e}}^{3\,{\mathrm {e}}^x}-6}}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(36*x - exp(-x^2/(2*exp(3*exp(x)) - 6))*(6*x - exp(3*exp(x))*(2*x - 3*x^2*exp(x))) - 24*x*exp(3*exp(x)) +
4*x*exp(6*exp(x)))/(2*exp(6*exp(x)) - 12*exp(3*exp(x)) + 18),x)

[Out]

exp(-x^2/(2*exp(3*exp(x)) - 6)) - x^2

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sympy [A]  time = 0.52, size = 19, normalized size = 0.68 \begin {gather*} - x^{2} + e^{- \frac {x^{2}}{2 e^{3 e^{x}} - 6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*exp(x)*x**2-2*x)*exp(3*exp(x))+6*x)*exp(-x**2/(2*exp(3*exp(x))-6))-4*x*exp(3*exp(x))**2+24*x*ex
p(3*exp(x))-36*x)/(2*exp(3*exp(x))**2-12*exp(3*exp(x))+18),x)

[Out]

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

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