\(\int e^{-e^{x^3}} (e^{20 e^{400-e^{x^3}-39 x+x^2}} (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x))+e^{40 e^{400-e^{x^3}-39 x+x^2}} (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x))) \, dx\) [6740]

Optimal result

Integrand size = 125, antiderivative size = 26 \[ \int e^{-e^{x^3}} \left (e^{20 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )+e^{40 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )\right ) \, dx=\left (1+e^{20 e^{-e^{x^3}+(20-x)^2+x}}\right )^2 \]

[Out]

(1+exp(20/exp(exp(x^3))*exp(x+(-x+20)^2)))^2

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6820, 12, 6818} \[ \int e^{-e^{x^3}} \left (e^{20 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )+e^{40 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )\right ) \, dx=\left (e^{20 e^{-e^{x^3}+x^2-39 x+400}}+1\right )^2 \]

[In]

Int[(E^(20*E^(400 - E^x^3 - 39*x + x^2))*(-120*E^(400 - 39*x + x^2 + x^3)*x^2 + E^(400 - 39*x + x^2)*(-1560 +
80*x)) + E^(40*E^(400 - E^x^3 - 39*x + x^2))*(-120*E^(400 - 39*x + x^2 + x^3)*x^2 + E^(400 - 39*x + x^2)*(-156
0 + 80*x)))/E^E^x^3,x]

[Out]

(1 + E^(20*E^(400 - E^x^3 - 39*x + x^2)))^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 6818

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

Rule 6820

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

Rubi steps \begin{align*} \text {integral}& = \int 40 e^{400-e^{x^3}+20 e^{400-e^{x^3}-39 x+x^2}-39 x+x^2} \left (1+e^{20 e^{400-e^{x^3}-39 x+x^2}}\right ) \left (-39+2 x-3 e^{x^3} x^2\right ) \, dx \\ & = 40 \int e^{400-e^{x^3}+20 e^{400-e^{x^3}-39 x+x^2}-39 x+x^2} \left (1+e^{20 e^{400-e^{x^3}-39 x+x^2}}\right ) \left (-39+2 x-3 e^{x^3} x^2\right ) \, dx \\ & = \left (1+e^{20 e^{400-e^{x^3}-39 x+x^2}}\right )^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 1.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int e^{-e^{x^3}} \left (e^{20 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )+e^{40 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )\right ) \, dx=\left (1+e^{20 e^{400-e^{x^3}-39 x+x^2}}\right )^2 \]

[In]

Integrate[(E^(20*E^(400 - E^x^3 - 39*x + x^2))*(-120*E^(400 - 39*x + x^2 + x^3)*x^2 + E^(400 - 39*x + x^2)*(-1
560 + 80*x)) + E^(40*E^(400 - E^x^3 - 39*x + x^2))*(-120*E^(400 - 39*x + x^2 + x^3)*x^2 + E^(400 - 39*x + x^2)
*(-1560 + 80*x)))/E^E^x^3,x]

[Out]

(1 + E^(20*E^(400 - E^x^3 - 39*x + x^2)))^2

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54

method	result	size
risch	\({\mathrm e}^{40 \,{\mathrm e}^{x^{2}-39 x +400-{\mathrm e}^{x^{3}}}}+2 \,{\mathrm e}^{20 \,{\mathrm e}^{x^{2}-39 x +400-{\mathrm e}^{x^{3}}}}\)	\(40\)
parallelrisch	\({\mathrm e}^{40 \,{\mathrm e}^{x^{2}-39 x +400-{\mathrm e}^{x^{3}}}}+2 \,{\mathrm e}^{20 \,{\mathrm e}^{x^{2}-39 x +400} {\mathrm e}^{-{\mathrm e}^{x^{3}}}}\)	\(44\)

[In]

int(((-120*x^2*exp(x^2-39*x+400)*exp(x^3)+(80*x-1560)*exp(x^2-39*x+400))*exp(20*exp(x^2-39*x+400)/exp(exp(x^3)
))^2+(-120*x^2*exp(x^2-39*x+400)*exp(x^3)+(80*x-1560)*exp(x^2-39*x+400))*exp(20*exp(x^2-39*x+400)/exp(exp(x^3)
)))/exp(exp(x^3)),x,method=_RETURNVERBOSE)

[Out]

exp(40*exp(x^2-39*x+400-exp(x^3)))+2*exp(20*exp(x^2-39*x+400-exp(x^3)))

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int e^{-e^{x^3}} \left (e^{20 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )+e^{40 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )\right ) \, dx=e^{\left (40 \, e^{\left (x^{2} - 39 \, x - e^{\left (x^{3}\right )} + 400\right )}\right )} + 2 \, e^{\left (20 \, e^{\left (x^{2} - 39 \, x - e^{\left (x^{3}\right )} + 400\right )}\right )} \]

[In]

integrate(((-120*x^2*exp(x^2-39*x+400)*exp(x^3)+(80*x-1560)*exp(x^2-39*x+400))*exp(20*exp(x^2-39*x+400)/exp(ex
p(x^3)))^2+(-120*x^2*exp(x^2-39*x+400)*exp(x^3)+(80*x-1560)*exp(x^2-39*x+400))*exp(20*exp(x^2-39*x+400)/exp(ex
p(x^3))))/exp(exp(x^3)),x, algorithm="fricas")

[Out]

e^(40*e^(x^2 - 39*x - e^(x^3) + 400)) + 2*e^(20*e^(x^2 - 39*x - e^(x^3) + 400))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).

Time = 3.37 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int e^{-e^{x^3}} \left (e^{20 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )+e^{40 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )\right ) \, dx=e^{40 e^{x^{2} - 39 x + 400} e^{- e^{x^{3}}}} + 2 e^{20 e^{x^{2} - 39 x + 400} e^{- e^{x^{3}}}} \]

[In]

integrate(((-120*x**2*exp(x**2-39*x+400)*exp(x**3)+(80*x-1560)*exp(x**2-39*x+400))*exp(20*exp(x**2-39*x+400)/e
xp(exp(x**3)))**2+(-120*x**2*exp(x**2-39*x+400)*exp(x**3)+(80*x-1560)*exp(x**2-39*x+400))*exp(20*exp(x**2-39*x
+400)/exp(exp(x**3))))/exp(exp(x**3)),x)

[Out]

exp(40*exp(x**2 - 39*x + 400)*exp(-exp(x**3))) + 2*exp(20*exp(x**2 - 39*x + 400)*exp(-exp(x**3)))

Maxima [A] (verification not implemented)

none

Time = 0.55 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int e^{-e^{x^3}} \left (e^{20 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )+e^{40 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )\right ) \, dx=e^{\left (40 \, e^{\left (x^{2} - 39 \, x - e^{\left (x^{3}\right )} + 400\right )}\right )} + 2 \, e^{\left (20 \, e^{\left (x^{2} - 39 \, x - e^{\left (x^{3}\right )} + 400\right )}\right )} \]

[In]

integrate(((-120*x^2*exp(x^2-39*x+400)*exp(x^3)+(80*x-1560)*exp(x^2-39*x+400))*exp(20*exp(x^2-39*x+400)/exp(ex
p(x^3)))^2+(-120*x^2*exp(x^2-39*x+400)*exp(x^3)+(80*x-1560)*exp(x^2-39*x+400))*exp(20*exp(x^2-39*x+400)/exp(ex
p(x^3))))/exp(exp(x^3)),x, algorithm="maxima")

[Out]

e^(40*e^(x^2 - 39*x - e^(x^3) + 400)) + 2*e^(20*e^(x^2 - 39*x - e^(x^3) + 400))

Giac [F]

\[ \int e^{-e^{x^3}} \left (e^{20 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )+e^{40 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )\right ) \, dx=\int { -40 \, {\left ({\left (3 \, x^{2} e^{\left (x^{3} + x^{2} - 39 \, x + 400\right )} - {\left (2 \, x - 39\right )} e^{\left (x^{2} - 39 \, x + 400\right )}\right )} e^{\left (40 \, e^{\left (x^{2} - 39 \, x - e^{\left (x^{3}\right )} + 400\right )}\right )} + {\left (3 \, x^{2} e^{\left (x^{3} + x^{2} - 39 \, x + 400\right )} - {\left (2 \, x - 39\right )} e^{\left (x^{2} - 39 \, x + 400\right )}\right )} e^{\left (20 \, e^{\left (x^{2} - 39 \, x - e^{\left (x^{3}\right )} + 400\right )}\right )}\right )} e^{\left (-e^{\left (x^{3}\right )}\right )} \,d x } \]

[In]

integrate(((-120*x^2*exp(x^2-39*x+400)*exp(x^3)+(80*x-1560)*exp(x^2-39*x+400))*exp(20*exp(x^2-39*x+400)/exp(ex
p(x^3)))^2+(-120*x^2*exp(x^2-39*x+400)*exp(x^3)+(80*x-1560)*exp(x^2-39*x+400))*exp(20*exp(x^2-39*x+400)/exp(ex
p(x^3))))/exp(exp(x^3)),x, algorithm="giac")

[Out]

integrate(-40*((3*x^2*e^(x^3 + x^2 - 39*x + 400) - (2*x - 39)*e^(x^2 - 39*x + 400))*e^(40*e^(x^2 - 39*x - e^(x
^3) + 400)) + (3*x^2*e^(x^3 + x^2 - 39*x + 400) - (2*x - 39)*e^(x^2 - 39*x + 400))*e^(20*e^(x^2 - 39*x - e^(x^
3) + 400)))*e^(-e^(x^3)), x)

Mupad [B] (verification not implemented)

Time = 17.49 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int e^{-e^{x^3}} \left (e^{20 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )+e^{40 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )\right ) \, dx={\mathrm {e}}^{20\,{\mathrm {e}}^{-{\mathrm {e}}^{x^3}}\,{\mathrm {e}}^{-39\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{400}}\,\left ({\mathrm {e}}^{20\,{\mathrm {e}}^{-{\mathrm {e}}^{x^3}}\,{\mathrm {e}}^{-39\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{400}}+2\right ) \]

[In]

int(exp(-exp(x^3))*(exp(20*exp(-exp(x^3))*exp(x^2 - 39*x + 400))*(exp(x^2 - 39*x + 400)*(80*x - 1560) - 120*x^
2*exp(x^3)*exp(x^2 - 39*x + 400)) + exp(40*exp(-exp(x^3))*exp(x^2 - 39*x + 400))*(exp(x^2 - 39*x + 400)*(80*x
- 1560) - 120*x^2*exp(x^3)*exp(x^2 - 39*x + 400))),x)

[Out]

exp(20*exp(-exp(x^3))*exp(-39*x)*exp(x^2)*exp(400))*(exp(20*exp(-exp(x^3))*exp(-39*x)*exp(x^2)*exp(400)) + 2)