3.7.46 \(\int \frac {1}{3} e^{22-2 e^{e^{\frac {1}{3} (-6-2 x)}}+x^2} (4 e^{e^{\frac {1}{3} (-6-2 x)}+\frac {1}{3} (-6-2 x)}+6 x) \, dx\)

Optimal. Leaf size=20 \[ e^{22-2 e^{e^{-\frac {2}{3} (3+x)}}+x^2} \]

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Rubi [A]  time = 0.20, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {12, 6706} \begin {gather*} e^{x^2-2 e^{e^{-\frac {2}{3} (x+3)}}+22} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(22 - 2*E^E^((-6 - 2*x)/3) + x^2)*(4*E^(E^((-6 - 2*x)/3) + (-6 - 2*x)/3) + 6*x))/3,x]

[Out]

E^(22 - 2*E^E^((-2*(3 + x))/3) + 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int e^{22-2 e^{e^{\frac {1}{3} (-6-2 x)}}+x^2} \left (4 e^{e^{\frac {1}{3} (-6-2 x)}+\frac {1}{3} (-6-2 x)}+6 x\right ) \, dx\\ &=e^{22-2 e^{e^{-\frac {2}{3} (3+x)}}+x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.31, size = 20, normalized size = 1.00 \begin {gather*} e^{22-2 e^{e^{-2-\frac {2 x}{3}}}+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(22 - 2*E^E^((-6 - 2*x)/3) + x^2)*(4*E^(E^((-6 - 2*x)/3) + (-6 - 2*x)/3) + 6*x))/3,x]

[Out]

E^(22 - 2*E^E^(-2 - (2*x)/3) + x^2)

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fricas [A]  time = 1.01, size = 15, normalized size = 0.75 \begin {gather*} e^{\left (x^{2} - 2 \, e^{\left (e^{\left (-\frac {2}{3} \, x - 2\right )}\right )} + 22\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(4*exp(-2/3*x-2)*exp(exp(-2/3*x-2))+6*x)*exp(-2*exp(exp(-2/3*x-2))+x^2+22),x, algorithm="fricas"
)

[Out]

e^(x^2 - 2*e^(e^(-2/3*x - 2)) + 22)

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giac [A]  time = 0.53, size = 15, normalized size = 0.75 \begin {gather*} e^{\left (x^{2} - 2 \, e^{\left (e^{\left (-\frac {2}{3} \, x - 2\right )}\right )} + 22\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(4*exp(-2/3*x-2)*exp(exp(-2/3*x-2))+6*x)*exp(-2*exp(exp(-2/3*x-2))+x^2+22),x, algorithm="giac")

[Out]

e^(x^2 - 2*e^(e^(-2/3*x - 2)) + 22)

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maple [A]  time = 0.07, size = 16, normalized size = 0.80

method	result	size
norman	\({\mathrm e}^{-2 \,{\mathrm e}^{{\mathrm e}^{-\frac {2 x}{3}-2}}+x^{2}+22}\)	\(16\)
risch	\({\mathrm e}^{-2 \,{\mathrm e}^{{\mathrm e}^{-\frac {2 x}{3}-2}}+x^{2}+22}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(4*exp(-2/3*x-2)*exp(exp(-2/3*x-2))+6*x)*exp(-2*exp(exp(-2/3*x-2))+x^2+22),x,method=_RETURNVERBOSE)

[Out]

exp(-2*exp(exp(-2/3*x-2))+x^2+22)

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maxima [A]  time = 0.52, size = 15, normalized size = 0.75 \begin {gather*} e^{\left (x^{2} - 2 \, e^{\left (e^{\left (-\frac {2}{3} \, x - 2\right )}\right )} + 22\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(4*exp(-2/3*x-2)*exp(exp(-2/3*x-2))+6*x)*exp(-2*exp(exp(-2/3*x-2))+x^2+22),x, algorithm="maxima"
)

[Out]

e^(x^2 - 2*e^(e^(-2/3*x - 2)) + 22)

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mupad [B]  time = 0.60, size = 18, normalized size = 0.90 \begin {gather*} {\mathrm {e}}^{-2\,{\mathrm {e}}^{{\mathrm {e}}^{-\frac {2\,x}{3}}\,{\mathrm {e}}^{-2}}}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{22} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^2 - 2*exp(exp(- (2*x)/3 - 2)) + 22)*(6*x + 4*exp(exp(- (2*x)/3 - 2))*exp(- (2*x)/3 - 2)))/3,x)

[Out]

exp(-2*exp(exp(-(2*x)/3)*exp(-2)))*exp(x^2)*exp(22)

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sympy [A]  time = 0.25, size = 19, normalized size = 0.95 \begin {gather*} e^{x^{2} - 2 e^{e^{- \frac {2 x}{3} - 2}} + 22} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(4*exp(-2/3*x-2)*exp(exp(-2/3*x-2))+6*x)*exp(-2*exp(exp(-2/3*x-2))+x**2+22),x)

[Out]

exp(x**2 - 2*exp(exp(-2*x/3 - 2)) + 22)

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