Integrand size = 121, antiderivative size = 28 \[ \int \frac {e^{-\frac {e^3 \left (-3 x+x^3\right )}{1+e^3 \left (36 x-24 x^2+4 x^3\right )}} \left (e^3 \left (3-3 x^2\right )+e^6 \left (72 x^2-96 x^3+24 x^4\right )\right )}{1+e^3 \left (72 x-48 x^2+8 x^3\right )+e^6 \left (1296 x^2-1728 x^3+864 x^4-192 x^5+16 x^6\right )} \, dx=4+e^{-\frac {-3+x^2}{(6-2 x)^2+\frac {1}{e^3 x}}} \]
Time = 2.76 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\frac {e^3 \left (-3 x+x^3\right )}{1+e^3 \left (36 x-24 x^2+4 x^3\right )}} \left (e^3 \left (3-3 x^2\right )+e^6 \left (72 x^2-96 x^3+24 x^4\right )\right )}{1+e^3 \left (72 x-48 x^2+8 x^3\right )+e^6 \left (1296 x^2-1728 x^3+864 x^4-192 x^5+16 x^6\right )} \, dx=e^{-\frac {e^3 x \left (-3+x^2\right )}{1+4 e^3 (-3+x)^2 x}} \]
Integrate[(E^3*(3 - 3*x^2) + E^6*(72*x^2 - 96*x^3 + 24*x^4))/(E^((E^3*(-3* x + x^3))/(1 + E^3*(36*x - 24*x^2 + 4*x^3)))*(1 + E^3*(72*x - 48*x^2 + 8*x ^3) + E^6*(1296*x^2 - 1728*x^3 + 864*x^4 - 192*x^5 + 16*x^6))),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.
\(\displaystyle \int \frac {\left (e^3 \left (3-3 x^2\right )+e^6 \left (24 x^4-96 x^3+72 x^2\right )\right ) \exp \left (-\frac {e^3 \left (x^3-3 x\right )}{e^3 \left (4 x^3-24 x^2+36 x\right )+1}\right )}{e^3 \left (8 x^3-48 x^2+72 x\right )+e^6 \left (16 x^6-192 x^5+864 x^4-1728 x^3+1296 x^2\right )+1} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \frac {\left (e^3 \left (3-3 x^2\right )+e^6 \left (24 x^4-96 x^3+72 x^2\right )\right ) \exp \left (-\frac {e^3 \left (x^3-3 x\right )}{e^3 \left (4 x^3-24 x^2+36 x\right )+1}\right )}{\left (4 e^3 x^3-24 e^3 x^2+36 e^3 x+1\right )^2}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (24 e^6 x^4-96 e^6 x^3-3 e^3 \left (1-24 e^3\right ) x^2+3 e^3\right ) \exp \left (-\frac {e^3 x \left (x^2-3\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}\right )}{\left (4 e^3 x^3-24 e^3 x^2+36 e^3 x+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {6 (x+2) \exp \left (3-\frac {e^3 x \left (x^2-3\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}+\frac {3 \left (-\left (\left (1-48 e^3\right ) x^2\right )-2 \left (1+72 e^3\right ) x-3\right ) \exp \left (3-\frac {e^3 x \left (x^2-3\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}\right )}{\left (4 e^3 x^3-24 e^3 x^2+36 e^3 x+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -9 \int \frac {\exp \left (3-\frac {e^3 x \left (x^2-3\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}\right )}{\left (4 e^3 x^3-24 e^3 x^2+36 e^3 x+1\right )^2}dx-6 \left (1+72 e^3\right ) \int \frac {\exp \left (3-\frac {e^3 x \left (x^2-3\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}\right ) x}{\left (4 e^3 x^3-24 e^3 x^2+36 e^3 x+1\right )^2}dx-3 \left (1-48 e^3\right ) \int \frac {\exp \left (3-\frac {e^3 x \left (x^2-3\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}\right ) x^2}{\left (4 e^3 x^3-24 e^3 x^2+36 e^3 x+1\right )^2}dx+12 \int \frac {\exp \left (3-\frac {e^3 x \left (x^2-3\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}dx+6 \int \frac {\exp \left (3-\frac {e^3 x \left (x^2-3\right )}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}\right ) x}{4 e^3 x^3-24 e^3 x^2+36 e^3 x+1}dx\) |
Int[(E^3*(3 - 3*x^2) + E^6*(72*x^2 - 96*x^3 + 24*x^4))/(E^((E^3*(-3*x + x^ 3))/(1 + E^3*(36*x - 24*x^2 + 4*x^3)))*(1 + E^3*(72*x - 48*x^2 + 8*x^3) + E^6*(1296*x^2 - 1728*x^3 + 864*x^4 - 192*x^5 + 16*x^6))),x]
3.6.99.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 0.68 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25
method | result | size |
norman | \({\mathrm e}^{-\frac {\left (x^{3}-3 x \right ) {\mathrm e}^{3}}{\left (4 x^{3}-24 x^{2}+36 x \right ) {\mathrm e}^{3}+1}}\) | \(35\) |
risch | \({\mathrm e}^{-\frac {x \left (x^{2}-3\right ) {\mathrm e}^{3}}{4 x^{3} {\mathrm e}^{3}-24 x^{2} {\mathrm e}^{3}+36 x \,{\mathrm e}^{3}+1}}\) | \(35\) |
gosper | \({\mathrm e}^{-\frac {x \left (x^{2}-3\right ) {\mathrm e}^{3}}{4 x^{3} {\mathrm e}^{3}-24 x^{2} {\mathrm e}^{3}+36 x \,{\mathrm e}^{3}+1}}\) | \(36\) |
parallelrisch | \({\mathrm e}^{-\frac {\left (x^{3}-3 x \right ) {\mathrm e}^{3}}{4 x^{3} {\mathrm e}^{3}-24 x^{2} {\mathrm e}^{3}+36 x \,{\mathrm e}^{3}+1}}\) | \(37\) |
int(((24*x^4-96*x^3+72*x^2)*exp(3)^2+(-3*x^2+3)*exp(3))/((16*x^6-192*x^5+8 64*x^4-1728*x^3+1296*x^2)*exp(3)^2+(8*x^3-48*x^2+72*x)*exp(3)+1)/exp((x^3- 3*x)*exp(3)/((4*x^3-24*x^2+36*x)*exp(3)+1)),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {e^{-\frac {e^3 \left (-3 x+x^3\right )}{1+e^3 \left (36 x-24 x^2+4 x^3\right )}} \left (e^3 \left (3-3 x^2\right )+e^6 \left (72 x^2-96 x^3+24 x^4\right )\right )}{1+e^3 \left (72 x-48 x^2+8 x^3\right )+e^6 \left (1296 x^2-1728 x^3+864 x^4-192 x^5+16 x^6\right )} \, dx=e^{\left (-\frac {{\left (x^{3} - 3 \, x\right )} e^{3}}{4 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} e^{3} + 1}\right )} \]
integrate(((24*x^4-96*x^3+72*x^2)*exp(3)^2+(-3*x^2+3)*exp(3))/((16*x^6-192 *x^5+864*x^4-1728*x^3+1296*x^2)*exp(3)^2+(8*x^3-48*x^2+72*x)*exp(3)+1)/exp ((x^3-3*x)*exp(3)/((4*x^3-24*x^2+36*x)*exp(3)+1)),x, algorithm=\
Time = 0.82 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-\frac {e^3 \left (-3 x+x^3\right )}{1+e^3 \left (36 x-24 x^2+4 x^3\right )}} \left (e^3 \left (3-3 x^2\right )+e^6 \left (72 x^2-96 x^3+24 x^4\right )\right )}{1+e^3 \left (72 x-48 x^2+8 x^3\right )+e^6 \left (1296 x^2-1728 x^3+864 x^4-192 x^5+16 x^6\right )} \, dx=e^{- \frac {\left (x^{3} - 3 x\right ) e^{3}}{\left (4 x^{3} - 24 x^{2} + 36 x\right ) e^{3} + 1}} \]
integrate(((24*x**4-96*x**3+72*x**2)*exp(3)**2+(-3*x**2+3)*exp(3))/((16*x* *6-192*x**5+864*x**4-1728*x**3+1296*x**2)*exp(3)**2+(8*x**3-48*x**2+72*x)* exp(3)+1)/exp((x**3-3*x)*exp(3)/((4*x**3-24*x**2+36*x)*exp(3)+1)),x)
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (26) = 52\).
Time = 0.39 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.07 \[ \int \frac {e^{-\frac {e^3 \left (-3 x+x^3\right )}{1+e^3 \left (36 x-24 x^2+4 x^3\right )}} \left (e^3 \left (3-3 x^2\right )+e^6 \left (72 x^2-96 x^3+24 x^4\right )\right )}{1+e^3 \left (72 x-48 x^2+8 x^3\right )+e^6 \left (1296 x^2-1728 x^3+864 x^4-192 x^5+16 x^6\right )} \, dx=e^{\left (-\frac {6 \, x^{2} e^{3}}{4 \, x^{3} e^{3} - 24 \, x^{2} e^{3} + 36 \, x e^{3} + 1} + \frac {12 \, x e^{3}}{4 \, x^{3} e^{3} - 24 \, x^{2} e^{3} + 36 \, x e^{3} + 1} + \frac {1}{4 \, {\left (4 \, x^{3} e^{3} - 24 \, x^{2} e^{3} + 36 \, x e^{3} + 1\right )}} - \frac {1}{4}\right )} \]
integrate(((24*x^4-96*x^3+72*x^2)*exp(3)^2+(-3*x^2+3)*exp(3))/((16*x^6-192 *x^5+864*x^4-1728*x^3+1296*x^2)*exp(3)^2+(8*x^3-48*x^2+72*x)*exp(3)+1)/exp ((x^3-3*x)*exp(3)/((4*x^3-24*x^2+36*x)*exp(3)+1)),x, algorithm=\
e^(-6*x^2*e^3/(4*x^3*e^3 - 24*x^2*e^3 + 36*x*e^3 + 1) + 12*x*e^3/(4*x^3*e^ 3 - 24*x^2*e^3 + 36*x*e^3 + 1) + 1/4/(4*x^3*e^3 - 24*x^2*e^3 + 36*x*e^3 + 1) - 1/4)
\[ \int \frac {e^{-\frac {e^3 \left (-3 x+x^3\right )}{1+e^3 \left (36 x-24 x^2+4 x^3\right )}} \left (e^3 \left (3-3 x^2\right )+e^6 \left (72 x^2-96 x^3+24 x^4\right )\right )}{1+e^3 \left (72 x-48 x^2+8 x^3\right )+e^6 \left (1296 x^2-1728 x^3+864 x^4-192 x^5+16 x^6\right )} \, dx=\int { \frac {3 \, {\left (8 \, {\left (x^{4} - 4 \, x^{3} + 3 \, x^{2}\right )} e^{6} - {\left (x^{2} - 1\right )} e^{3}\right )} e^{\left (-\frac {{\left (x^{3} - 3 \, x\right )} e^{3}}{4 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} e^{3} + 1}\right )}}{16 \, {\left (x^{6} - 12 \, x^{5} + 54 \, x^{4} - 108 \, x^{3} + 81 \, x^{2}\right )} e^{6} + 8 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} e^{3} + 1} \,d x } \]
integrate(((24*x^4-96*x^3+72*x^2)*exp(3)^2+(-3*x^2+3)*exp(3))/((16*x^6-192 *x^5+864*x^4-1728*x^3+1296*x^2)*exp(3)^2+(8*x^3-48*x^2+72*x)*exp(3)+1)/exp ((x^3-3*x)*exp(3)/((4*x^3-24*x^2+36*x)*exp(3)+1)),x, algorithm=\
integrate(3*(8*(x^4 - 4*x^3 + 3*x^2)*e^6 - (x^2 - 1)*e^3)*e^(-(x^3 - 3*x)* e^3/(4*(x^3 - 6*x^2 + 9*x)*e^3 + 1))/(16*(x^6 - 12*x^5 + 54*x^4 - 108*x^3 + 81*x^2)*e^6 + 8*(x^3 - 6*x^2 + 9*x)*e^3 + 1), x)
Time = 13.49 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {e^{-\frac {e^3 \left (-3 x+x^3\right )}{1+e^3 \left (36 x-24 x^2+4 x^3\right )}} \left (e^3 \left (3-3 x^2\right )+e^6 \left (72 x^2-96 x^3+24 x^4\right )\right )}{1+e^3 \left (72 x-48 x^2+8 x^3\right )+e^6 \left (1296 x^2-1728 x^3+864 x^4-192 x^5+16 x^6\right )} \, dx={\mathrm {e}}^{\frac {3\,x\,{\mathrm {e}}^3-x^3\,{\mathrm {e}}^3}{4\,{\mathrm {e}}^3\,x^3-24\,{\mathrm {e}}^3\,x^2+36\,{\mathrm {e}}^3\,x+1}} \]