3.30.71 \(\int e^{4 x+e^{2 e^{4-3 x^2} x^2} x} (4+e^{2 e^{4-3 x^2} x^2} (1+e^{4-3 x^2} (4 x^2-12 x^4))) \, dx\)

Optimal. Leaf size=22 \[ e^{\left (4+e^{2 e^{4-3 x^2} x^2}\right ) x} \]

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Rubi [A]  time = 0.34, antiderivative size = 24, normalized size of antiderivative = 1.09, number of steps used = 1, number of rules used = 1, integrand size = 67, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {6706} \begin {gather*} e^{e^{2 e^{4-3 x^2} x^2} x+4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(4*x + E^(2*E^(4 - 3*x^2)*x^2)*x)*(4 + E^(2*E^(4 - 3*x^2)*x^2)*(1 + E^(4 - 3*x^2)*(4*x^2 - 12*x^4))),x]

[Out]

E^(4*x + E^(2*E^(4 - 3*x^2)*x^2)*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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^{4 x+e^{2 e^{4-3 x^2} x^2} x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.68, size = 24, normalized size = 1.09 \begin {gather*} e^{4 x+e^{2 e^{4-3 x^2} x^2} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(4*x + E^(2*E^(4 - 3*x^2)*x^2)*x)*(4 + E^(2*E^(4 - 3*x^2)*x^2)*(1 + E^(4 - 3*x^2)*(4*x^2 - 12*x^4)
)),x]

[Out]

E^(4*x + E^(2*E^(4 - 3*x^2)*x^2)*x)

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fricas [A]  time = 0.67, size = 21, normalized size = 0.95 \begin {gather*} e^{\left (x e^{\left (2 \, x^{2} e^{\left (-3 \, x^{2} + 4\right )}\right )} + 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-12*x^4+4*x^2)*exp(-3*x^2+4)+1)*exp(2*x^2*exp(-3*x^2+4))+4)*exp(x*exp(2*x^2*exp(-3*x^2+4))+4*x),x
, algorithm="fricas")

[Out]

e^(x*e^(2*x^2*e^(-3*x^2 + 4)) + 4*x)

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giac [A]  time = 0.37, size = 21, normalized size = 0.95 \begin {gather*} e^{\left (x e^{\left (2 \, x^{2} e^{\left (-3 \, x^{2} + 4\right )}\right )} + 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-12*x^4+4*x^2)*exp(-3*x^2+4)+1)*exp(2*x^2*exp(-3*x^2+4))+4)*exp(x*exp(2*x^2*exp(-3*x^2+4))+4*x),x
, algorithm="giac")

[Out]

e^(x*e^(2*x^2*e^(-3*x^2 + 4)) + 4*x)

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maple [A]  time = 0.06, size = 20, normalized size = 0.91

method	result	size
risch	\({\mathrm e}^{\left ({\mathrm e}^{2 x^{2} {\mathrm e}^{-3 x^{2}+4}}+4\right ) x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-12*x^4+4*x^2)*exp(-3*x^2+4)+1)*exp(2*x^2*exp(-3*x^2+4))+4)*exp(x*exp(2*x^2*exp(-3*x^2+4))+4*x),x,metho
d=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.49, size = 21, normalized size = 0.95 \begin {gather*} e^{\left (x e^{\left (2 \, x^{2} e^{\left (-3 \, x^{2} + 4\right )}\right )} + 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-12*x^4+4*x^2)*exp(-3*x^2+4)+1)*exp(2*x^2*exp(-3*x^2+4))+4)*exp(x*exp(2*x^2*exp(-3*x^2+4))+4*x),x
, algorithm="maxima")

[Out]

e^(x*e^(2*x^2*e^(-3*x^2 + 4)) + 4*x)

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mupad [B]  time = 1.90, size = 22, normalized size = 1.00 \begin {gather*} {\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{2\,x^2\,{\mathrm {e}}^4\,{\mathrm {e}}^{-3\,x^2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(4*x + x*exp(2*x^2*exp(4 - 3*x^2)))*(exp(2*x^2*exp(4 - 3*x^2))*(exp(4 - 3*x^2)*(4*x^2 - 12*x^4) + 1) +
4),x)

[Out]

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

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sympy [A]  time = 5.96, size = 20, normalized size = 0.91 \begin {gather*} e^{x e^{2 x^{2} e^{4 - 3 x^{2}}} + 4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-12*x**4+4*x**2)*exp(-3*x**2+4)+1)*exp(2*x**2*exp(-3*x**2+4))+4)*exp(x*exp(2*x**2*exp(-3*x**2+4))
+4*x),x)

[Out]

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

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