Integrand size = 219, antiderivative size = 31 \[ \int \frac {e^e \left (-432 x^2-576 x^3-180 x^4-256 x^6-512 x^7-384 x^8-128 x^9-16 x^{10}+e^{x^2} \left (288 x^2+384 x^3-72 x^4-192 x^5-48 x^6\right )+e^{2 x^2} \left (-48 x^2-64 x^3+44 x^4+64 x^5+16 x^6\right )\right )}{81-12 e^{3 x^2}+e^{4 x^2}-288 x^4-288 x^5-72 x^6+256 x^8+512 x^9+384 x^{10}+128 x^{11}+16 x^{12}+e^{2 x^2} \left (54-32 x^4-32 x^5-8 x^6\right )+e^{x^2} \left (-108+192 x^4+192 x^5+48 x^6\right )} \, dx=\frac {e^e}{x-\frac {\left (3-e^{x^2}\right )^2}{x^3 (4+2 x)^2}} \]
Time = 9.80 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.55 \[ \int \frac {e^e \left (-432 x^2-576 x^3-180 x^4-256 x^6-512 x^7-384 x^8-128 x^9-16 x^{10}+e^{x^2} \left (288 x^2+384 x^3-72 x^4-192 x^5-48 x^6\right )+e^{2 x^2} \left (-48 x^2-64 x^3+44 x^4+64 x^5+16 x^6\right )\right )}{81-12 e^{3 x^2}+e^{4 x^2}-288 x^4-288 x^5-72 x^6+256 x^8+512 x^9+384 x^{10}+128 x^{11}+16 x^{12}+e^{2 x^2} \left (54-32 x^4-32 x^5-8 x^6\right )+e^{x^2} \left (-108+192 x^4+192 x^5+48 x^6\right )} \, dx=\frac {4 e^e x^3 (2+x)^2}{-9+6 e^{x^2}-e^{2 x^2}+16 x^4+16 x^5+4 x^6} \]
Integrate[(E^E*(-432*x^2 - 576*x^3 - 180*x^4 - 256*x^6 - 512*x^7 - 384*x^8 - 128*x^9 - 16*x^10 + E^x^2*(288*x^2 + 384*x^3 - 72*x^4 - 192*x^5 - 48*x^ 6) + E^(2*x^2)*(-48*x^2 - 64*x^3 + 44*x^4 + 64*x^5 + 16*x^6)))/(81 - 12*E^ (3*x^2) + E^(4*x^2) - 288*x^4 - 288*x^5 - 72*x^6 + 256*x^8 + 512*x^9 + 384 *x^10 + 128*x^11 + 16*x^12 + E^(2*x^2)*(54 - 32*x^4 - 32*x^5 - 8*x^6) + E^ x^2*(-108 + 192*x^4 + 192*x^5 + 48*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.
| \(\displaystyle \int \frac {e^e \left (-16 x^{10}-128 x^9-384 x^8-512 x^7-256 x^6-180 x^4-576 x^3-432 x^2+e^{x^2} \left (-48 x^6-192 x^5-72 x^4+384 x^3+288 x^2\right )+e^{2 x^2} \left (16 x^6+64 x^5+44 x^4-64 x^3-48 x^2\right )\right )}{16 x^{12}+128 x^{11}+384 x^{10}+512 x^9+256 x^8-72 x^6-288 x^5-288 x^4-12 e^{3 x^2}+e^{4 x^2}+e^{2 x^2} \left (-8 x^6-32 x^5-32 x^4+54\right )+e^{x^2} \left (48 x^6+192 x^5+192 x^4-108\right )+81} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle e^e \int -\frac {4 \left (4 x^{10}+32 x^9+96 x^8+128 x^7+64 x^6+45 x^4+144 x^3+108 x^2+e^{2 x^2} \left (-4 x^6-16 x^5-11 x^4+16 x^3+12 x^2\right )-6 e^{x^2} \left (-2 x^6-8 x^5-3 x^4+16 x^3+12 x^2\right )\right )}{16 x^{12}+128 x^{11}+384 x^{10}+512 x^9+256 x^8-72 x^6-288 x^5-288 x^4-12 e^{3 x^2}+e^{4 x^2}-12 e^{x^2} \left (-4 x^6-16 x^5-16 x^4+9\right )+2 e^{2 x^2} \left (-4 x^6-16 x^5-16 x^4+27\right )+81}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -4 e^e \int \frac {4 x^{10}+32 x^9+96 x^8+128 x^7+64 x^6+45 x^4+144 x^3+108 x^2+e^{2 x^2} \left (-4 x^6-16 x^5-11 x^4+16 x^3+12 x^2\right )-6 e^{x^2} \left (-2 x^6-8 x^5-3 x^4+16 x^3+12 x^2\right )}{16 x^{12}+128 x^{11}+384 x^{10}+512 x^9+256 x^8-72 x^6-288 x^5-288 x^4-12 e^{3 x^2}+e^{4 x^2}-12 e^{x^2} \left (-4 x^6-16 x^5-16 x^4+9\right )+2 e^{2 x^2} \left (-4 x^6-16 x^5-16 x^4+27\right )+81}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 e^e \int \frac {x^2 (x+2) \left (4 x^7+24 x^6+48 x^5+32 x^4+45 x+e^{2 x^2} \left (-4 x^3-8 x^2+5 x+6\right )+6 e^{x^2} \left (2 x^3+4 x^2-5 x-6\right )+54\right )}{\left (-4 x^6-16 x^5-16 x^4-6 e^{x^2}+e^{2 x^2}+9\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 e^e \int \left (-\frac {\left (2 x^4+8 x^3+5 x^2-13 x-14\right ) x^2}{2 \left (2 x^3+4 x^2+e^{x^2}-3\right )^2}-\frac {\left (2 x^4+8 x^3+5 x^2-7 x-2\right ) x^2}{2 \left (2 x^3+4 x^2-e^{x^2}+3\right )^2}-\frac {x^3+2 x^2-x-1}{2 \left (-2 x^3-4 x^2+e^{x^2}-3\right )}+\frac {x^3+2 x^2-x-1}{2 \left (2 x^3+4 x^2+e^{x^2}-3\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 e^e \left (\frac {1}{2} \int \frac {1}{-2 x^3-4 x^2+e^{x^2}-3}dx+\int \frac {x^2}{\left (2 x^3+4 x^2-e^{x^2}+3\right )^2}dx+\frac {7}{2} \int \frac {x^3}{\left (2 x^3+4 x^2-e^{x^2}+3\right )^2}dx-\frac {1}{2} \int \frac {x}{2 x^3+4 x^2-e^{x^2}+3}dx+\int \frac {x^2}{2 x^3+4 x^2-e^{x^2}+3}dx+\frac {1}{2} \int \frac {x^3}{2 x^3+4 x^2-e^{x^2}+3}dx+7 \int \frac {x^2}{\left (2 x^3+4 x^2+e^{x^2}-3\right )^2}dx+\frac {13}{2} \int \frac {x^3}{\left (2 x^3+4 x^2+e^{x^2}-3\right )^2}dx-\frac {1}{2} \int \frac {1}{2 x^3+4 x^2+e^{x^2}-3}dx-\frac {1}{2} \int \frac {x}{2 x^3+4 x^2+e^{x^2}-3}dx+\int \frac {x^2}{2 x^3+4 x^2+e^{x^2}-3}dx+\frac {1}{2} \int \frac {x^3}{2 x^3+4 x^2+e^{x^2}-3}dx-\int \frac {x^6}{\left (2 x^3+4 x^2-e^{x^2}+3\right )^2}dx-\int \frac {x^6}{\left (2 x^3+4 x^2+e^{x^2}-3\right )^2}dx-4 \int \frac {x^5}{\left (2 x^3+4 x^2-e^{x^2}+3\right )^2}dx-4 \int \frac {x^5}{\left (2 x^3+4 x^2+e^{x^2}-3\right )^2}dx-\frac {5}{2} \int \frac {x^4}{\left (2 x^3+4 x^2-e^{x^2}+3\right )^2}dx-\frac {5}{2} \int \frac {x^4}{\left (2 x^3+4 x^2+e^{x^2}-3\right )^2}dx\right )\) |
Int[(E^E*(-432*x^2 - 576*x^3 - 180*x^4 - 256*x^6 - 512*x^7 - 384*x^8 - 128 *x^9 - 16*x^10 + E^x^2*(288*x^2 + 384*x^3 - 72*x^4 - 192*x^5 - 48*x^6) + E ^(2*x^2)*(-48*x^2 - 64*x^3 + 44*x^4 + 64*x^5 + 16*x^6)))/(81 - 12*E^(3*x^2 ) + E^(4*x^2) - 288*x^4 - 288*x^5 - 72*x^6 + 256*x^8 + 512*x^9 + 384*x^10 + 128*x^11 + 16*x^12 + E^(2*x^2)*(54 - 32*x^4 - 32*x^5 - 8*x^6) + E^x^2*(- 108 + 192*x^4 + 192*x^5 + 48*x^6)),x]
3.4.47.3.1 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_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.45 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.61
| method | result | size |
| risch | \(\frac {4 \,{\mathrm e}^{{\mathrm e}} \left (x^{2}+4 x +4\right ) x^{3}}{4 x^{6}+16 x^{5}+16 x^{4}-{\mathrm e}^{2 x^{2}}+6 \,{\mathrm e}^{x^{2}}-9}\) | \(50\) |
| parallelrisch | \(-\frac {{\mathrm e}^{{\mathrm e}} \left (-4 x^{5}-16 x^{4}-16 x^{3}\right )}{4 x^{6}+16 x^{5}+16 x^{4}-{\mathrm e}^{2 x^{2}}+6 \,{\mathrm e}^{x^{2}}-9}\) | \(55\) |
int(((16*x^6+64*x^5+44*x^4-64*x^3-48*x^2)*exp(x^2)^2+(-48*x^6-192*x^5-72*x ^4+384*x^3+288*x^2)*exp(x^2)-16*x^10-128*x^9-384*x^8-512*x^7-256*x^6-180*x ^4-576*x^3-432*x^2)*exp(exp(1))/(exp(x^2)^4-12*exp(x^2)^3+(-8*x^6-32*x^5-3 2*x^4+54)*exp(x^2)^2+(48*x^6+192*x^5+192*x^4-108)*exp(x^2)+16*x^12+128*x^1 1+384*x^10+512*x^9+256*x^8-72*x^6-288*x^5-288*x^4+81),x,method=_RETURNVERB OSE)
Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int \frac {e^e \left (-432 x^2-576 x^3-180 x^4-256 x^6-512 x^7-384 x^8-128 x^9-16 x^{10}+e^{x^2} \left (288 x^2+384 x^3-72 x^4-192 x^5-48 x^6\right )+e^{2 x^2} \left (-48 x^2-64 x^3+44 x^4+64 x^5+16 x^6\right )\right )}{81-12 e^{3 x^2}+e^{4 x^2}-288 x^4-288 x^5-72 x^6+256 x^8+512 x^9+384 x^{10}+128 x^{11}+16 x^{12}+e^{2 x^2} \left (54-32 x^4-32 x^5-8 x^6\right )+e^{x^2} \left (-108+192 x^4+192 x^5+48 x^6\right )} \, dx=\frac {4 \, {\left (x^{5} + 4 \, x^{4} + 4 \, x^{3}\right )} e^{e}}{4 \, x^{6} + 16 \, x^{5} + 16 \, x^{4} - e^{\left (2 \, x^{2}\right )} + 6 \, e^{\left (x^{2}\right )} - 9} \]
integrate(((16*x^6+64*x^5+44*x^4-64*x^3-48*x^2)*exp(x^2)^2+(-48*x^6-192*x^ 5-72*x^4+384*x^3+288*x^2)*exp(x^2)-16*x^10-128*x^9-384*x^8-512*x^7-256*x^6 -180*x^4-576*x^3-432*x^2)*exp(exp(1))/(exp(x^2)^4-12*exp(x^2)^3+(-8*x^6-32 *x^5-32*x^4+54)*exp(x^2)^2+(48*x^6+192*x^5+192*x^4-108)*exp(x^2)+16*x^12+1 28*x^11+384*x^10+512*x^9+256*x^8-72*x^6-288*x^5-288*x^4+81),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (24) = 48\).
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.97 \[ \int \frac {e^e \left (-432 x^2-576 x^3-180 x^4-256 x^6-512 x^7-384 x^8-128 x^9-16 x^{10}+e^{x^2} \left (288 x^2+384 x^3-72 x^4-192 x^5-48 x^6\right )+e^{2 x^2} \left (-48 x^2-64 x^3+44 x^4+64 x^5+16 x^6\right )\right )}{81-12 e^{3 x^2}+e^{4 x^2}-288 x^4-288 x^5-72 x^6+256 x^8+512 x^9+384 x^{10}+128 x^{11}+16 x^{12}+e^{2 x^2} \left (54-32 x^4-32 x^5-8 x^6\right )+e^{x^2} \left (-108+192 x^4+192 x^5+48 x^6\right )} \, dx=\frac {- 4 x^{5} e^{e} - 16 x^{4} e^{e} - 16 x^{3} e^{e}}{- 4 x^{6} - 16 x^{5} - 16 x^{4} + e^{2 x^{2}} - 6 e^{x^{2}} + 9} \]
integrate(((16*x**6+64*x**5+44*x**4-64*x**3-48*x**2)*exp(x**2)**2+(-48*x** 6-192*x**5-72*x**4+384*x**3+288*x**2)*exp(x**2)-16*x**10-128*x**9-384*x**8 -512*x**7-256*x**6-180*x**4-576*x**3-432*x**2)*exp(exp(1))/(exp(x**2)**4-1 2*exp(x**2)**3+(-8*x**6-32*x**5-32*x**4+54)*exp(x**2)**2+(48*x**6+192*x**5 +192*x**4-108)*exp(x**2)+16*x**12+128*x**11+384*x**10+512*x**9+256*x**8-72 *x**6-288*x**5-288*x**4+81),x)
(-4*x**5*exp(E) - 16*x**4*exp(E) - 16*x**3*exp(E))/(-4*x**6 - 16*x**5 - 16 *x**4 + exp(2*x**2) - 6*exp(x**2) + 9)
Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int \frac {e^e \left (-432 x^2-576 x^3-180 x^4-256 x^6-512 x^7-384 x^8-128 x^9-16 x^{10}+e^{x^2} \left (288 x^2+384 x^3-72 x^4-192 x^5-48 x^6\right )+e^{2 x^2} \left (-48 x^2-64 x^3+44 x^4+64 x^5+16 x^6\right )\right )}{81-12 e^{3 x^2}+e^{4 x^2}-288 x^4-288 x^5-72 x^6+256 x^8+512 x^9+384 x^{10}+128 x^{11}+16 x^{12}+e^{2 x^2} \left (54-32 x^4-32 x^5-8 x^6\right )+e^{x^2} \left (-108+192 x^4+192 x^5+48 x^6\right )} \, dx=\frac {4 \, {\left (x^{5} + 4 \, x^{4} + 4 \, x^{3}\right )} e^{e}}{4 \, x^{6} + 16 \, x^{5} + 16 \, x^{4} - e^{\left (2 \, x^{2}\right )} + 6 \, e^{\left (x^{2}\right )} - 9} \]
integrate(((16*x^6+64*x^5+44*x^4-64*x^3-48*x^2)*exp(x^2)^2+(-48*x^6-192*x^ 5-72*x^4+384*x^3+288*x^2)*exp(x^2)-16*x^10-128*x^9-384*x^8-512*x^7-256*x^6 -180*x^4-576*x^3-432*x^2)*exp(exp(1))/(exp(x^2)^4-12*exp(x^2)^3+(-8*x^6-32 *x^5-32*x^4+54)*exp(x^2)^2+(48*x^6+192*x^5+192*x^4-108)*exp(x^2)+16*x^12+1 28*x^11+384*x^10+512*x^9+256*x^8-72*x^6-288*x^5-288*x^4+81),x, algorithm=\
Time = 0.49 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int \frac {e^e \left (-432 x^2-576 x^3-180 x^4-256 x^6-512 x^7-384 x^8-128 x^9-16 x^{10}+e^{x^2} \left (288 x^2+384 x^3-72 x^4-192 x^5-48 x^6\right )+e^{2 x^2} \left (-48 x^2-64 x^3+44 x^4+64 x^5+16 x^6\right )\right )}{81-12 e^{3 x^2}+e^{4 x^2}-288 x^4-288 x^5-72 x^6+256 x^8+512 x^9+384 x^{10}+128 x^{11}+16 x^{12}+e^{2 x^2} \left (54-32 x^4-32 x^5-8 x^6\right )+e^{x^2} \left (-108+192 x^4+192 x^5+48 x^6\right )} \, dx=\frac {4 \, {\left (x^{5} + 4 \, x^{4} + 4 \, x^{3}\right )} e^{e}}{4 \, x^{6} + 16 \, x^{5} + 16 \, x^{4} - e^{\left (2 \, x^{2}\right )} + 6 \, e^{\left (x^{2}\right )} - 9} \]
integrate(((16*x^6+64*x^5+44*x^4-64*x^3-48*x^2)*exp(x^2)^2+(-48*x^6-192*x^ 5-72*x^4+384*x^3+288*x^2)*exp(x^2)-16*x^10-128*x^9-384*x^8-512*x^7-256*x^6 -180*x^4-576*x^3-432*x^2)*exp(exp(1))/(exp(x^2)^4-12*exp(x^2)^3+(-8*x^6-32 *x^5-32*x^4+54)*exp(x^2)^2+(48*x^6+192*x^5+192*x^4-108)*exp(x^2)+16*x^12+1 28*x^11+384*x^10+512*x^9+256*x^8-72*x^6-288*x^5-288*x^4+81),x, algorithm=\
Time = 8.83 (sec) , antiderivative size = 172, normalized size of antiderivative = 5.55 \[ \int \frac {e^e \left (-432 x^2-576 x^3-180 x^4-256 x^6-512 x^7-384 x^8-128 x^9-16 x^{10}+e^{x^2} \left (288 x^2+384 x^3-72 x^4-192 x^5-48 x^6\right )+e^{2 x^2} \left (-48 x^2-64 x^3+44 x^4+64 x^5+16 x^6\right )\right )}{81-12 e^{3 x^2}+e^{4 x^2}-288 x^4-288 x^5-72 x^6+256 x^8+512 x^9+384 x^{10}+128 x^{11}+16 x^{12}+e^{2 x^2} \left (54-32 x^4-32 x^5-8 x^6\right )+e^{x^2} \left (-108+192 x^4+192 x^5+48 x^6\right )} \, dx=\frac {4\,\left (4\,{\mathrm {e}}^{\mathrm {e}}\,x^{17}+48\,{\mathrm {e}}^{\mathrm {e}}\,x^{16}+228\,{\mathrm {e}}^{\mathrm {e}}\,x^{15}+504\,{\mathrm {e}}^{\mathrm {e}}\,x^{14}+329\,{\mathrm {e}}^{\mathrm {e}}\,x^{13}-736\,{\mathrm {e}}^{\mathrm {e}}\,x^{12}-1569\,{\mathrm {e}}^{\mathrm {e}}\,x^{11}-744\,{\mathrm {e}}^{\mathrm {e}}\,x^{10}+568\,{\mathrm {e}}^{\mathrm {e}}\,x^9+608\,{\mathrm {e}}^{\mathrm {e}}\,x^8+112\,{\mathrm {e}}^{\mathrm {e}}\,x^7\right )}{\left (6\,{\mathrm {e}}^{x^2}-{\mathrm {e}}^{2\,x^2}+16\,x^4+16\,x^5+4\,x^6-9\right )\,\left (4\,x^{12}+32\,x^{11}+84\,x^{10}+40\,x^9-167\,x^8-228\,x^7+11\,x^6+124\,x^5+28\,x^4\right )} \]
int(-(exp(exp(1))*(exp(x^2)*(72*x^4 - 384*x^3 - 288*x^2 + 192*x^5 + 48*x^6 ) - exp(2*x^2)*(44*x^4 - 64*x^3 - 48*x^2 + 64*x^5 + 16*x^6) + 432*x^2 + 57 6*x^3 + 180*x^4 + 256*x^6 + 512*x^7 + 384*x^8 + 128*x^9 + 16*x^10))/(exp(4 *x^2) - 12*exp(3*x^2) + exp(x^2)*(192*x^4 + 192*x^5 + 48*x^6 - 108) - 288* x^4 - 288*x^5 - 72*x^6 + 256*x^8 + 512*x^9 + 384*x^10 + 128*x^11 + 16*x^12 - exp(2*x^2)*(32*x^4 + 32*x^5 + 8*x^6 - 54) + 81),x)
(4*(112*x^7*exp(exp(1)) + 608*x^8*exp(exp(1)) + 568*x^9*exp(exp(1)) - 744* x^10*exp(exp(1)) - 1569*x^11*exp(exp(1)) - 736*x^12*exp(exp(1)) + 329*x^13 *exp(exp(1)) + 504*x^14*exp(exp(1)) + 228*x^15*exp(exp(1)) + 48*x^16*exp(e xp(1)) + 4*x^17*exp(exp(1))))/((6*exp(x^2) - exp(2*x^2) + 16*x^4 + 16*x^5 + 4*x^6 - 9)*(28*x^4 + 124*x^5 + 11*x^6 - 228*x^7 - 167*x^8 + 40*x^9 + 84* x^10 + 32*x^11 + 4*x^12))