\(\int \frac {1}{3} e^{22-2 e^{e^{\frac {1}{3} (-6-2 x)}}+x^2} (4 e^{e^{\frac {1}{3} (-6-2 x)}+\frac {1}{3} (-6-2 x)}+6 x) \, dx\) [646]

Optimal result

Integrand size = 55, antiderivative size = 20 \[ \int \frac {1}{3} e^{22-2 e^{e^{\frac {1}{3} (-6-2 x)}}+x^2} \left (4 e^{e^{\frac {1}{3} (-6-2 x)}+\frac {1}{3} (-6-2 x)}+6 x\right ) \, dx=e^{22-2 e^{e^{-\frac {2}{3} (3+x)}}+x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {12, 6838} \[ \int \frac {1}{3} e^{22-2 e^{e^{\frac {1}{3} (-6-2 x)}}+x^2} \left (4 e^{e^{\frac {1}{3} (-6-2 x)}+\frac {1}{3} (-6-2 x)}+6 x\right ) \, dx=e^{x^2-2 e^{e^{-\frac {2}{3} (x+3)}}+22} \]

[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 6838

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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{3} e^{22-2 e^{e^{\frac {1}{3} (-6-2 x)}}+x^2} \left (4 e^{e^{\frac {1}{3} (-6-2 x)}+\frac {1}{3} (-6-2 x)}+6 x\right ) \, dx=e^{22-2 e^{e^{-2-\frac {2 x}{3}}}+x^2} \]

[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)

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {1}{3} e^{22-2 e^{e^{\frac {1}{3} (-6-2 x)}}+x^2} \left (4 e^{e^{\frac {1}{3} (-6-2 x)}+\frac {1}{3} (-6-2 x)}+6 x\right ) \, dx=e^{\left (x^{2} - 2 \, e^{\left (e^{\left (-\frac {2}{3} \, x - 2\right )}\right )} + 22\right )} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {1}{3} e^{22-2 e^{e^{\frac {1}{3} (-6-2 x)}}+x^2} \left (4 e^{e^{\frac {1}{3} (-6-2 x)}+\frac {1}{3} (-6-2 x)}+6 x\right ) \, dx=e^{x^{2} - 2 e^{e^{- \frac {2 x}{3} - 2}} + 22} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {1}{3} e^{22-2 e^{e^{\frac {1}{3} (-6-2 x)}}+x^2} \left (4 e^{e^{\frac {1}{3} (-6-2 x)}+\frac {1}{3} (-6-2 x)}+6 x\right ) \, dx=e^{\left (x^{2} - 2 \, e^{\left (e^{\left (-\frac {2}{3} \, x - 2\right )}\right )} + 22\right )} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {1}{3} e^{22-2 e^{e^{\frac {1}{3} (-6-2 x)}}+x^2} \left (4 e^{e^{\frac {1}{3} (-6-2 x)}+\frac {1}{3} (-6-2 x)}+6 x\right ) \, dx=e^{\left (x^{2} - 2 \, e^{\left (e^{\left (-\frac {2}{3} \, x - 2\right )}\right )} + 22\right )} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {1}{3} e^{22-2 e^{e^{\frac {1}{3} (-6-2 x)}}+x^2} \left (4 e^{e^{\frac {1}{3} (-6-2 x)}+\frac {1}{3} (-6-2 x)}+6 x\right ) \, dx={\mathrm {e}}^{-2\,{\mathrm {e}}^{{\mathrm {e}}^{-\frac {2\,x}{3}}\,{\mathrm {e}}^{-2}}}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{22} \]

[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)