Integrand size = 152, antiderivative size = 25 \[ \int e^{81 x+e^{100 x-4 x^3} x-12 e^{75 x-3 x^3} x+54 e^{50 x-2 x^3} x-108 e^{25 x-x^3} x} \left (81+e^{50 x-2 x^3} \left (54+2700 x-324 x^3\right )+e^{100 x-4 x^3} \left (1+100 x-12 x^3\right )+e^{75 x-3 x^3} \left (-12-900 x+108 x^3\right )+e^{25 x-x^3} \left (-108-2700 x+324 x^3\right )\right ) \, dx=e^{\left (3-e^{-x+x \left (26-x^2\right )}\right )^4 x} \]
Time = 5.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int e^{81 x+e^{100 x-4 x^3} x-12 e^{75 x-3 x^3} x+54 e^{50 x-2 x^3} x-108 e^{25 x-x^3} x} \left (81+e^{50 x-2 x^3} \left (54+2700 x-324 x^3\right )+e^{100 x-4 x^3} \left (1+100 x-12 x^3\right )+e^{75 x-3 x^3} \left (-12-900 x+108 x^3\right )+e^{25 x-x^3} \left (-108-2700 x+324 x^3\right )\right ) \, dx=e^{e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x} \]
Integrate[E^(81*x + E^(100*x - 4*x^3)*x - 12*E^(75*x - 3*x^3)*x + 54*E^(50 *x - 2*x^3)*x - 108*E^(25*x - x^3)*x)*(81 + E^(50*x - 2*x^3)*(54 + 2700*x - 324*x^3) + E^(100*x - 4*x^3)*(1 + 100*x - 12*x^3) + E^(75*x - 3*x^3)*(-1 2 - 900*x + 108*x^3) + E^(25*x - x^3)*(-108 - 2700*x + 324*x^3)),x]
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.
| \(\displaystyle \int \left (e^{50 x-2 x^3} \left (-324 x^3+2700 x+54\right )+e^{100 x-4 x^3} \left (-12 x^3+100 x+1\right )+e^{75 x-3 x^3} \left (108 x^3-900 x-12\right )+e^{25 x-x^3} \left (324 x^3-2700 x-108\right )+81\right ) \exp \left (e^{100 x-4 x^3} x-12 e^{75 x-3 x^3} x+54 e^{50 x-2 x^3} x-108 e^{25 x-x^3} x+81 x\right ) \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \left (e^{25 x}-3 e^{x^3}\right )^3 \left (-e^{25 x} \left (12 x^3-100 x-1\right )-3 e^{x^3}\right ) \exp \left (e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x-4 x^3\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (108 \left (3 x^3-25 x-1\right ) \exp \left (e^{-4 x^3} x \left (e^{25 x}-3 e^{x^3}\right )^4-x^3+25 x\right )+12 \left (9 x^3-75 x-1\right ) \exp \left (e^{-4 x^3} x \left (e^{25 x}-3 e^{x^3}\right )^4-3 x^3+75 x\right )-\left (12 x^3-100 x-1\right ) \exp \left (e^{-4 x^3} x \left (e^{25 x}-3 e^{x^3}\right )^4-4 x^3+100 x\right )-54 \left (6 x^3-50 x-1\right ) \exp \left (e^{-4 x^3} x \left (e^{25 x}-3 e^{x^3}\right )^4-4 x^3+2 x \left (x^2+25\right )\right )+81 e^{e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \exp \left (e^{-4 x^3} x \left (e^{25 x}-3 e^{x^3}\right )^4-4 x^3+100 x\right )dx-12 \int \exp \left (e^{-4 x^3} x \left (e^{25 x}-3 e^{x^3}\right )^4-3 x^3+75 x\right )dx-108 \int \exp \left (e^{-4 x^3} x \left (e^{25 x}-3 e^{x^3}\right )^4-x^3+25 x\right )dx+100 \int \exp \left (e^{-4 x^3} x \left (e^{25 x}-3 e^{x^3}\right )^4-4 x^3+100 x\right ) xdx-900 \int \exp \left (e^{-4 x^3} x \left (e^{25 x}-3 e^{x^3}\right )^4-3 x^3+75 x\right ) xdx-2700 \int \exp \left (e^{-4 x^3} x \left (e^{25 x}-3 e^{x^3}\right )^4-x^3+25 x\right ) xdx-12 \int \exp \left (e^{-4 x^3} x \left (e^{25 x}-3 e^{x^3}\right )^4-4 x^3+100 x\right ) x^3dx+108 \int \exp \left (e^{-4 x^3} x \left (e^{25 x}-3 e^{x^3}\right )^4-3 x^3+75 x\right ) x^3dx+324 \int \exp \left (e^{-4 x^3} x \left (e^{25 x}-3 e^{x^3}\right )^4-x^3+25 x\right ) x^3dx+54 \int \exp \left (e^{-4 x^3} x \left (e^{25 x}-3 e^{x^3}\right )^4-4 x^3+2 x \left (x^2+25\right )\right )dx+2700 \int \exp \left (e^{-4 x^3} x \left (e^{25 x}-3 e^{x^3}\right )^4-4 x^3+2 x \left (x^2+25\right )\right ) xdx-324 \int \exp \left (e^{-4 x^3} x \left (e^{25 x}-3 e^{x^3}\right )^4-4 x^3+2 x \left (x^2+25\right )\right ) x^3dx+81 \int e^{e^{-4 x^3} \left (e^{25 x}-3 e^{x^3}\right )^4 x}dx\) |
Int[E^(81*x + E^(100*x - 4*x^3)*x - 12*E^(75*x - 3*x^3)*x + 54*E^(50*x - 2 *x^3)*x - 108*E^(25*x - x^3)*x)*(81 + E^(50*x - 2*x^3)*(54 + 2700*x - 324* x^3) + E^(100*x - 4*x^3)*(1 + 100*x - 12*x^3) + E^(75*x - 3*x^3)*(-12 - 90 0*x + 108*x^3) + E^(25*x - x^3)*(-108 - 2700*x + 324*x^3)),x]
3.19.60.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs. \(2(23)=46\).
Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.08
| method | result | size |
| risch | \({\mathrm e}^{x \left ({\mathrm e}^{-4 x \left (-5+x \right ) \left (5+x \right )}-12 \,{\mathrm e}^{-3 x \left (-5+x \right ) \left (5+x \right )}+54 \,{\mathrm e}^{-2 x \left (-5+x \right ) \left (5+x \right )}-108 \,{\mathrm e}^{-x \left (-5+x \right ) \left (5+x \right )}+81\right )}\) | \(52\) |
| parallelrisch | \({\mathrm e}^{x \left ({\mathrm e}^{-4 x^{3}+100 x}-12 \,{\mathrm e}^{-3 x^{3}+75 x}+54 \,{\mathrm e}^{-2 x^{3}+50 x}-108 \,{\mathrm e}^{-x^{3}+25 x}+81\right )}\) | \(58\) |
int(((-12*x^3+100*x+1)*exp(-x^3+25*x)^4+(108*x^3-900*x-12)*exp(-x^3+25*x)^ 3+(-324*x^3+2700*x+54)*exp(-x^3+25*x)^2+(324*x^3-2700*x-108)*exp(-x^3+25*x )+81)*exp(x*exp(-x^3+25*x)^4-12*x*exp(-x^3+25*x)^3+54*x*exp(-x^3+25*x)^2-1 08*x*exp(-x^3+25*x)+81*x),x,method=_RETURNVERBOSE)
exp(x*(exp(-4*x*(-5+x)*(5+x))-12*exp(-3*x*(-5+x)*(5+x))+54*exp(-2*x*(-5+x) *(5+x))-108*exp(-x*(-5+x)*(5+x))+81))
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int e^{81 x+e^{100 x-4 x^3} x-12 e^{75 x-3 x^3} x+54 e^{50 x-2 x^3} x-108 e^{25 x-x^3} x} \left (81+e^{50 x-2 x^3} \left (54+2700 x-324 x^3\right )+e^{100 x-4 x^3} \left (1+100 x-12 x^3\right )+e^{75 x-3 x^3} \left (-12-900 x+108 x^3\right )+e^{25 x-x^3} \left (-108-2700 x+324 x^3\right )\right ) \, dx=e^{\left (-108 \, x e^{\left (-x^{3} + 25 \, x\right )} + 54 \, x e^{\left (-2 \, x^{3} + 50 \, x\right )} - 12 \, x e^{\left (-3 \, x^{3} + 75 \, x\right )} + x e^{\left (-4 \, x^{3} + 100 \, x\right )} + 81 \, x\right )} \]
integrate(((-12*x^3+100*x+1)*exp(-x^3+25*x)^4+(108*x^3-900*x-12)*exp(-x^3+ 25*x)^3+(-324*x^3+2700*x+54)*exp(-x^3+25*x)^2+(324*x^3-2700*x-108)*exp(-x^ 3+25*x)+81)*exp(x*exp(-x^3+25*x)^4-12*x*exp(-x^3+25*x)^3+54*x*exp(-x^3+25* x)^2-108*x*exp(-x^3+25*x)+81*x),x, algorithm=\
e^(-108*x*e^(-x^3 + 25*x) + 54*x*e^(-2*x^3 + 50*x) - 12*x*e^(-3*x^3 + 75*x ) + x*e^(-4*x^3 + 100*x) + 81*x)
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (15) = 30\).
Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.16 \[ \int e^{81 x+e^{100 x-4 x^3} x-12 e^{75 x-3 x^3} x+54 e^{50 x-2 x^3} x-108 e^{25 x-x^3} x} \left (81+e^{50 x-2 x^3} \left (54+2700 x-324 x^3\right )+e^{100 x-4 x^3} \left (1+100 x-12 x^3\right )+e^{75 x-3 x^3} \left (-12-900 x+108 x^3\right )+e^{25 x-x^3} \left (-108-2700 x+324 x^3\right )\right ) \, dx=e^{x e^{- 4 x^{3} + 100 x} - 12 x e^{- 3 x^{3} + 75 x} + 54 x e^{- 2 x^{3} + 50 x} - 108 x e^{- x^{3} + 25 x} + 81 x} \]
integrate(((-12*x**3+100*x+1)*exp(-x**3+25*x)**4+(108*x**3-900*x-12)*exp(- x**3+25*x)**3+(-324*x**3+2700*x+54)*exp(-x**3+25*x)**2+(324*x**3-2700*x-10 8)*exp(-x**3+25*x)+81)*exp(x*exp(-x**3+25*x)**4-12*x*exp(-x**3+25*x)**3+54 *x*exp(-x**3+25*x)**2-108*x*exp(-x**3+25*x)+81*x),x)
exp(x*exp(-4*x**3 + 100*x) - 12*x*exp(-3*x**3 + 75*x) + 54*x*exp(-2*x**3 + 50*x) - 108*x*exp(-x**3 + 25*x) + 81*x)
\[ \int e^{81 x+e^{100 x-4 x^3} x-12 e^{75 x-3 x^3} x+54 e^{50 x-2 x^3} x-108 e^{25 x-x^3} x} \left (81+e^{50 x-2 x^3} \left (54+2700 x-324 x^3\right )+e^{100 x-4 x^3} \left (1+100 x-12 x^3\right )+e^{75 x-3 x^3} \left (-12-900 x+108 x^3\right )+e^{25 x-x^3} \left (-108-2700 x+324 x^3\right )\right ) \, dx=\int { {\left (108 \, {\left (3 \, x^{3} - 25 \, x - 1\right )} e^{\left (-x^{3} + 25 \, x\right )} - 54 \, {\left (6 \, x^{3} - 50 \, x - 1\right )} e^{\left (-2 \, x^{3} + 50 \, x\right )} + 12 \, {\left (9 \, x^{3} - 75 \, x - 1\right )} e^{\left (-3 \, x^{3} + 75 \, x\right )} - {\left (12 \, x^{3} - 100 \, x - 1\right )} e^{\left (-4 \, x^{3} + 100 \, x\right )} + 81\right )} e^{\left (-108 \, x e^{\left (-x^{3} + 25 \, x\right )} + 54 \, x e^{\left (-2 \, x^{3} + 50 \, x\right )} - 12 \, x e^{\left (-3 \, x^{3} + 75 \, x\right )} + x e^{\left (-4 \, x^{3} + 100 \, x\right )} + 81 \, x\right )} \,d x } \]
integrate(((-12*x^3+100*x+1)*exp(-x^3+25*x)^4+(108*x^3-900*x-12)*exp(-x^3+ 25*x)^3+(-324*x^3+2700*x+54)*exp(-x^3+25*x)^2+(324*x^3-2700*x-108)*exp(-x^ 3+25*x)+81)*exp(x*exp(-x^3+25*x)^4-12*x*exp(-x^3+25*x)^3+54*x*exp(-x^3+25* x)^2-108*x*exp(-x^3+25*x)+81*x),x, algorithm=\
integrate((108*(3*x^3 - 25*x - 1)*e^(-x^3 + 25*x) - 54*(6*x^3 - 50*x - 1)* e^(-2*x^3 + 50*x) + 12*(9*x^3 - 75*x - 1)*e^(-3*x^3 + 75*x) - (12*x^3 - 10 0*x - 1)*e^(-4*x^3 + 100*x) + 81)*e^(-108*x*e^(-x^3 + 25*x) + 54*x*e^(-2*x ^3 + 50*x) - 12*x*e^(-3*x^3 + 75*x) + x*e^(-4*x^3 + 100*x) + 81*x), x)
\[ \int e^{81 x+e^{100 x-4 x^3} x-12 e^{75 x-3 x^3} x+54 e^{50 x-2 x^3} x-108 e^{25 x-x^3} x} \left (81+e^{50 x-2 x^3} \left (54+2700 x-324 x^3\right )+e^{100 x-4 x^3} \left (1+100 x-12 x^3\right )+e^{75 x-3 x^3} \left (-12-900 x+108 x^3\right )+e^{25 x-x^3} \left (-108-2700 x+324 x^3\right )\right ) \, dx=\int { {\left (108 \, {\left (3 \, x^{3} - 25 \, x - 1\right )} e^{\left (-x^{3} + 25 \, x\right )} - 54 \, {\left (6 \, x^{3} - 50 \, x - 1\right )} e^{\left (-2 \, x^{3} + 50 \, x\right )} + 12 \, {\left (9 \, x^{3} - 75 \, x - 1\right )} e^{\left (-3 \, x^{3} + 75 \, x\right )} - {\left (12 \, x^{3} - 100 \, x - 1\right )} e^{\left (-4 \, x^{3} + 100 \, x\right )} + 81\right )} e^{\left (-108 \, x e^{\left (-x^{3} + 25 \, x\right )} + 54 \, x e^{\left (-2 \, x^{3} + 50 \, x\right )} - 12 \, x e^{\left (-3 \, x^{3} + 75 \, x\right )} + x e^{\left (-4 \, x^{3} + 100 \, x\right )} + 81 \, x\right )} \,d x } \]
integrate(((-12*x^3+100*x+1)*exp(-x^3+25*x)^4+(108*x^3-900*x-12)*exp(-x^3+ 25*x)^3+(-324*x^3+2700*x+54)*exp(-x^3+25*x)^2+(324*x^3-2700*x-108)*exp(-x^ 3+25*x)+81)*exp(x*exp(-x^3+25*x)^4-12*x*exp(-x^3+25*x)^3+54*x*exp(-x^3+25* x)^2-108*x*exp(-x^3+25*x)+81*x),x, algorithm=\
integrate((108*(3*x^3 - 25*x - 1)*e^(-x^3 + 25*x) - 54*(6*x^3 - 50*x - 1)* e^(-2*x^3 + 50*x) + 12*(9*x^3 - 75*x - 1)*e^(-3*x^3 + 75*x) - (12*x^3 - 10 0*x - 1)*e^(-4*x^3 + 100*x) + 81)*e^(-108*x*e^(-x^3 + 25*x) + 54*x*e^(-2*x ^3 + 50*x) - 12*x*e^(-3*x^3 + 75*x) + x*e^(-4*x^3 + 100*x) + 81*x), x)
Time = 10.66 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.40 \[ \int e^{81 x+e^{100 x-4 x^3} x-12 e^{75 x-3 x^3} x+54 e^{50 x-2 x^3} x-108 e^{25 x-x^3} x} \left (81+e^{50 x-2 x^3} \left (54+2700 x-324 x^3\right )+e^{100 x-4 x^3} \left (1+100 x-12 x^3\right )+e^{75 x-3 x^3} \left (-12-900 x+108 x^3\right )+e^{25 x-x^3} \left (-108-2700 x+324 x^3\right )\right ) \, dx={\mathrm {e}}^{-12\,x\,{\mathrm {e}}^{75\,x}\,{\mathrm {e}}^{-3\,x^3}}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{100\,x}\,{\mathrm {e}}^{-4\,x^3}}\,{\mathrm {e}}^{54\,x\,{\mathrm {e}}^{50\,x}\,{\mathrm {e}}^{-2\,x^3}}\,{\mathrm {e}}^{-108\,x\,{\mathrm {e}}^{25\,x}\,{\mathrm {e}}^{-x^3}}\,{\mathrm {e}}^{81\,x} \]
int(exp(81*x - 108*x*exp(25*x - x^3) + 54*x*exp(50*x - 2*x^3) - 12*x*exp(7 5*x - 3*x^3) + x*exp(100*x - 4*x^3))*(exp(100*x - 4*x^3)*(100*x - 12*x^3 + 1) - exp(75*x - 3*x^3)*(900*x - 108*x^3 + 12) + exp(50*x - 2*x^3)*(2700*x - 324*x^3 + 54) - exp(25*x - x^3)*(2700*x - 324*x^3 + 108) + 81),x)