Integrand size = 125, antiderivative size = 26 \[ \int e^{-e^{x^3}} \left (e^{20 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )+e^{40 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )\right ) \, dx=\left (1+e^{20 e^{-e^{x^3}+(20-x)^2+x}}\right )^2 \]
Time = 1.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int e^{-e^{x^3}} \left (e^{20 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )+e^{40 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )\right ) \, dx=\left (1+e^{20 e^{400-e^{x^3}-39 x+x^2}}\right )^2 \]
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)*(-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)))/E^E^x^3,x]
Time = 1.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {7292, 27, 25, 7237}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int e^{-e^{x^3}} \left (e^{20 e^{-e^{x^3}+x^2-39 x+400}} \left (e^{x^2-39 x+400} (80 x-1560)-120 e^{x^3+x^2-39 x+400} x^2\right )+e^{40 e^{-e^{x^3}+x^2-39 x+400}} \left (e^{x^2-39 x+400} (80 x-1560)-120 e^{x^3+x^2-39 x+400} x^2\right )\right ) \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int 40 \left (e^{20 e^{-e^{x^3}+x^2-39 x+400}}+1\right ) \left (-3 e^{x^3} x^2+2 x-39\right ) \exp \left (-e^{x^3}+x^2+20 e^{-e^{x^3}+x^2-39 x+400}-39 x+400\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 40 \int -e^{x^2-39 x-e^{x^3}+20 e^{x^2-39 x-e^{x^3}+400}+400} \left (1+e^{20 e^{x^2-39 x-e^{x^3}+400}}\right ) \left (3 e^{x^3} x^2-2 x+39\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -40 \int e^{x^2-39 x-e^{x^3}+20 e^{x^2-39 x-e^{x^3}+400}+400} \left (1+e^{20 e^{x^2-39 x-e^{x^3}+400}}\right ) \left (3 e^{x^3} x^2-2 x+39\right )dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \left (e^{20 e^{-e^{x^3}+x^2-39 x+400}}+1\right )^2\) |
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)*(-1560 + 80*x)))/E^E^x^3,x]
3.8.40.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
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 | \({\mathrm e}^{40 \,{\mathrm e}^{x^{2}-39 x +400-{\mathrm e}^{x^{3}}}}+2 \,{\mathrm e}^{20 \,{\mathrm e}^{x^{2}-39 x +400} {\mathrm e}^{-{\mathrm e}^{x^{3}}}}\) | \(44\) |
int(((-120*x^2*exp(x^2-39*x+400)*exp(x^3)+(80*x-1560)*exp(x^2-39*x+400))*e xp(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)
Time = 0.37 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int e^{-e^{x^3}} \left (e^{20 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )+e^{40 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )\right ) \, dx=e^{\left (40 \, e^{\left (x^{2} - 39 \, x - e^{\left (x^{3}\right )} + 400\right )}\right )} + 2 \, e^{\left (20 \, e^{\left (x^{2} - 39 \, x - e^{\left (x^{3}\right )} + 400\right )}\right )} \]
integrate(((-120*x^2*exp(x^2-39*x+400)*exp(x^3)+(80*x-1560)*exp(x^2-39*x+4 00))*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, algorithm=\
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 \[ \int e^{-e^{x^3}} \left (e^{20 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )+e^{40 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )\right ) \, dx=e^{40 e^{x^{2} - 39 x + 400} e^{- e^{x^{3}}}} + 2 e^{20 e^{x^{2} - 39 x + 400} e^{- e^{x^{3}}}} \]
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(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+4 00)/exp(exp(x**3))))/exp(exp(x**3)),x)
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)))
Time = 0.55 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int e^{-e^{x^3}} \left (e^{20 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )+e^{40 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )\right ) \, dx=e^{\left (40 \, e^{\left (x^{2} - 39 \, x - e^{\left (x^{3}\right )} + 400\right )}\right )} + 2 \, e^{\left (20 \, e^{\left (x^{2} - 39 \, x - e^{\left (x^{3}\right )} + 400\right )}\right )} \]
integrate(((-120*x^2*exp(x^2-39*x+400)*exp(x^3)+(80*x-1560)*exp(x^2-39*x+4 00))*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, algorithm=\
\[ \int e^{-e^{x^3}} \left (e^{20 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )+e^{40 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )\right ) \, dx=\int { -40 \, {\left ({\left (3 \, x^{2} e^{\left (x^{3} + x^{2} - 39 \, x + 400\right )} - {\left (2 \, x - 39\right )} e^{\left (x^{2} - 39 \, x + 400\right )}\right )} e^{\left (40 \, e^{\left (x^{2} - 39 \, x - e^{\left (x^{3}\right )} + 400\right )}\right )} + {\left (3 \, x^{2} e^{\left (x^{3} + x^{2} - 39 \, x + 400\right )} - {\left (2 \, x - 39\right )} e^{\left (x^{2} - 39 \, x + 400\right )}\right )} e^{\left (20 \, e^{\left (x^{2} - 39 \, x - e^{\left (x^{3}\right )} + 400\right )}\right )}\right )} e^{\left (-e^{\left (x^{3}\right )}\right )} \,d x } \]
integrate(((-120*x^2*exp(x^2-39*x+400)*exp(x^3)+(80*x-1560)*exp(x^2-39*x+4 00))*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, algorithm=\
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)
Time = 17.49 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int e^{-e^{x^3}} \left (e^{20 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )+e^{40 e^{400-e^{x^3}-39 x+x^2}} \left (-120 e^{400-39 x+x^2+x^3} x^2+e^{400-39 x+x^2} (-1560+80 x)\right )\right ) \, dx={\mathrm {e}}^{20\,{\mathrm {e}}^{-{\mathrm {e}}^{x^3}}\,{\mathrm {e}}^{-39\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{400}}\,\left ({\mathrm {e}}^{20\,{\mathrm {e}}^{-{\mathrm {e}}^{x^3}}\,{\mathrm {e}}^{-39\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{400}}+2\right ) \]
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)) + ex p(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)