Integrand size = 95, antiderivative size = 24 \[ \int \frac {1}{9} e^{e^{\frac {1}{9} \left (4 e^{16} x-8 e^8 x^2+4 x^3\right )} (3+x)+\frac {1}{9} \left (4 e^{16} x-8 e^8 x^2+4 x^3\right )} \left (9+36 x^2+12 x^3+e^{16} (12+4 x)+e^8 \left (-48 x-16 x^2\right )\right ) \, dx=1+e^{e^{\frac {4}{9} x \left (-e^8+x\right )^2} (3+x)} \]
Time = 5.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {1}{9} e^{e^{\frac {1}{9} \left (4 e^{16} x-8 e^8 x^2+4 x^3\right )} (3+x)+\frac {1}{9} \left (4 e^{16} x-8 e^8 x^2+4 x^3\right )} \left (9+36 x^2+12 x^3+e^{16} (12+4 x)+e^8 \left (-48 x-16 x^2\right )\right ) \, dx=e^{e^{\frac {4}{9} \left (e^8-x\right )^2 x} (3+x)} \]
Integrate[(E^(E^((4*E^16*x - 8*E^8*x^2 + 4*x^3)/9)*(3 + x) + (4*E^16*x - 8 *E^8*x^2 + 4*x^3)/9)*(9 + 36*x^2 + 12*x^3 + E^16*(12 + 4*x) + E^8*(-48*x - 16*x^2)))/9,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.
\(\displaystyle \int \frac {1}{9} \left (12 x^3+36 x^2+e^8 \left (-16 x^2-48 x\right )+e^{16} (4 x+12)+9\right ) \exp \left (e^{\frac {1}{9} \left (4 x^3-8 e^8 x^2+4 e^{16} x\right )} (x+3)+\frac {1}{9} \left (4 x^3-8 e^8 x^2+4 e^{16} x\right )\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \int \exp \left (e^{\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )} (x+3)+\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )\right ) \left (12 x^3+36 x^2+4 e^{16} (x+3)-16 e^8 \left (x^2+3 x\right )+9\right )dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \frac {1}{9} \int \exp \left (e^{\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )} (x+3)+\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )\right ) \left (12 x^3+4 \left (9-4 e^8\right ) x^2-4 e^8 \left (12-e^8\right ) x+3 \left (3+4 e^{16}\right )\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{9} \int \left (12 \exp \left (e^{\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )} (x+3)+\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )\right ) x^3-4 \exp \left (e^{\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )} (x+3)+\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )\right ) \left (-9+4 e^8\right ) x^2+4 \exp \left (e^{\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )} (x+3)+\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )+8\right ) \left (-12+e^8\right ) x+3 \exp \left (e^{\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )} (x+3)+\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )\right ) \left (3+4 e^{16}\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{9} \left (3 \left (3+4 e^{16}\right ) \int \exp \left (e^{\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )} (x+3)+\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )\right )dx-4 \left (12-e^8\right ) \int \exp \left (e^{\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )} (x+3)+\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )+8\right ) xdx+4 \left (9-4 e^8\right ) \int \exp \left (e^{\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )} (x+3)+\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )\right ) x^2dx+12 \int \exp \left (e^{\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )} (x+3)+\frac {4}{9} \left (x^3-2 e^8 x^2+e^{16} x\right )\right ) x^3dx\right )\) |
Int[(E^(E^((4*E^16*x - 8*E^8*x^2 + 4*x^3)/9)*(3 + x) + (4*E^16*x - 8*E^8*x ^2 + 4*x^3)/9)*(9 + 36*x^2 + 12*x^3 + E^16*(12 + 4*x) + E^8*(-48*x - 16*x^ 2)))/9,x]
3.15.67.3.1 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]]
Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
method | result | size |
risch | \({\mathrm e}^{\left (3+x \right ) {\mathrm e}^{\frac {4 x \left (-2 x \,{\mathrm e}^{8}+x^{2}+{\mathrm e}^{16}\right )}{9}}}\) | \(21\) |
norman | \({\mathrm e}^{\left (3+x \right ) {\mathrm e}^{\frac {4 x \,{\mathrm e}^{16}}{9}-\frac {8 x^{2} {\mathrm e}^{8}}{9}+\frac {4 x^{3}}{9}}}\) | \(27\) |
parallelrisch | \({\mathrm e}^{\left (3+x \right ) {\mathrm e}^{\frac {4 x \,{\mathrm e}^{16}}{9}-\frac {8 x^{2} {\mathrm e}^{8}}{9}+\frac {4 x^{3}}{9}}}\) | \(27\) |
int(1/9*((4*x+12)*exp(8)^2+(-16*x^2-48*x)*exp(8)+12*x^3+36*x^2+9)*exp(4/9* x*exp(8)^2-8/9*x^2*exp(8)+4/9*x^3)*exp((3+x)*exp(4/9*x*exp(8)^2-8/9*x^2*ex p(8)+4/9*x^3)),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{9} e^{e^{\frac {1}{9} \left (4 e^{16} x-8 e^8 x^2+4 x^3\right )} (3+x)+\frac {1}{9} \left (4 e^{16} x-8 e^8 x^2+4 x^3\right )} \left (9+36 x^2+12 x^3+e^{16} (12+4 x)+e^8 \left (-48 x-16 x^2\right )\right ) \, dx=e^{\left ({\left (x + 3\right )} e^{\left (\frac {4}{9} \, x^{3} - \frac {8}{9} \, x^{2} e^{8} + \frac {4}{9} \, x e^{16}\right )}\right )} \]
integrate(1/9*((4*x+12)*exp(8)^2+(-16*x^2-48*x)*exp(8)+12*x^3+36*x^2+9)*ex p(4/9*x*exp(8)^2-8/9*x^2*exp(8)+4/9*x^3)*exp((3+x)*exp(4/9*x*exp(8)^2-8/9* x^2*exp(8)+4/9*x^3)),x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {1}{9} e^{e^{\frac {1}{9} \left (4 e^{16} x-8 e^8 x^2+4 x^3\right )} (3+x)+\frac {1}{9} \left (4 e^{16} x-8 e^8 x^2+4 x^3\right )} \left (9+36 x^2+12 x^3+e^{16} (12+4 x)+e^8 \left (-48 x-16 x^2\right )\right ) \, dx=e^{\left (x + 3\right ) e^{\frac {4 x^{3}}{9} - \frac {8 x^{2} e^{8}}{9} + \frac {4 x e^{16}}{9}}} \]
integrate(1/9*((4*x+12)*exp(8)**2+(-16*x**2-48*x)*exp(8)+12*x**3+36*x**2+9 )*exp(4/9*x*exp(8)**2-8/9*x**2*exp(8)+4/9*x**3)*exp((3+x)*exp(4/9*x*exp(8) **2-8/9*x**2*exp(8)+4/9*x**3)),x)
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).
Time = 0.53 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {1}{9} e^{e^{\frac {1}{9} \left (4 e^{16} x-8 e^8 x^2+4 x^3\right )} (3+x)+\frac {1}{9} \left (4 e^{16} x-8 e^8 x^2+4 x^3\right )} \left (9+36 x^2+12 x^3+e^{16} (12+4 x)+e^8 \left (-48 x-16 x^2\right )\right ) \, dx=e^{\left (x e^{\left (\frac {4}{9} \, x^{3} - \frac {8}{9} \, x^{2} e^{8} + \frac {4}{9} \, x e^{16}\right )} + 3 \, e^{\left (\frac {4}{9} \, x^{3} - \frac {8}{9} \, x^{2} e^{8} + \frac {4}{9} \, x e^{16}\right )}\right )} \]
integrate(1/9*((4*x+12)*exp(8)^2+(-16*x^2-48*x)*exp(8)+12*x^3+36*x^2+9)*ex p(4/9*x*exp(8)^2-8/9*x^2*exp(8)+4/9*x^3)*exp((3+x)*exp(4/9*x*exp(8)^2-8/9* x^2*exp(8)+4/9*x^3)),x, algorithm=\
\[ \int \frac {1}{9} e^{e^{\frac {1}{9} \left (4 e^{16} x-8 e^8 x^2+4 x^3\right )} (3+x)+\frac {1}{9} \left (4 e^{16} x-8 e^8 x^2+4 x^3\right )} \left (9+36 x^2+12 x^3+e^{16} (12+4 x)+e^8 \left (-48 x-16 x^2\right )\right ) \, dx=\int { \frac {1}{9} \, {\left (12 \, x^{3} + 36 \, x^{2} + 4 \, {\left (x + 3\right )} e^{16} - 16 \, {\left (x^{2} + 3 \, x\right )} e^{8} + 9\right )} e^{\left (\frac {4}{9} \, x^{3} - \frac {8}{9} \, x^{2} e^{8} + \frac {4}{9} \, x e^{16} + {\left (x + 3\right )} e^{\left (\frac {4}{9} \, x^{3} - \frac {8}{9} \, x^{2} e^{8} + \frac {4}{9} \, x e^{16}\right )}\right )} \,d x } \]
integrate(1/9*((4*x+12)*exp(8)^2+(-16*x^2-48*x)*exp(8)+12*x^3+36*x^2+9)*ex p(4/9*x*exp(8)^2-8/9*x^2*exp(8)+4/9*x^3)*exp((3+x)*exp(4/9*x*exp(8)^2-8/9* x^2*exp(8)+4/9*x^3)),x, algorithm=\
integrate(1/9*(12*x^3 + 36*x^2 + 4*(x + 3)*e^16 - 16*(x^2 + 3*x)*e^8 + 9)* e^(4/9*x^3 - 8/9*x^2*e^8 + 4/9*x*e^16 + (x + 3)*e^(4/9*x^3 - 8/9*x^2*e^8 + 4/9*x*e^16)), x)
Time = 13.08 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.96 \[ \int \frac {1}{9} e^{e^{\frac {1}{9} \left (4 e^{16} x-8 e^8 x^2+4 x^3\right )} (3+x)+\frac {1}{9} \left (4 e^{16} x-8 e^8 x^2+4 x^3\right )} \left (9+36 x^2+12 x^3+e^{16} (12+4 x)+e^8 \left (-48 x-16 x^2\right )\right ) \, dx={\mathrm {e}}^{3\,{\mathrm {e}}^{-\frac {8\,x^2\,{\mathrm {e}}^8}{9}}\,{\mathrm {e}}^{\frac {4\,x^3}{9}}\,{\mathrm {e}}^{\frac {4\,x\,{\mathrm {e}}^{16}}{9}}}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-\frac {8\,x^2\,{\mathrm {e}}^8}{9}}\,{\mathrm {e}}^{\frac {4\,x^3}{9}}\,{\mathrm {e}}^{\frac {4\,x\,{\mathrm {e}}^{16}}{9}}} \]
int((exp(exp((4*x*exp(16))/9 - (8*x^2*exp(8))/9 + (4*x^3)/9)*(x + 3))*exp( (4*x*exp(16))/9 - (8*x^2*exp(8))/9 + (4*x^3)/9)*(36*x^2 - exp(8)*(48*x + 1 6*x^2) + 12*x^3 + exp(16)*(4*x + 12) + 9))/9,x)