Integrand size = 183, antiderivative size = 32 \[ \int \frac {e^{-32 x^2} \left (4394 x-140608 x^3-2028 x^5+64896 x^7+1326 x^9-42432 x^{11}-328 x^{13}+10496 x^{15}+102 x^{17}-3264 x^{19}-12 x^{21}+384 x^{23}+2 x^{25}-64 x^{27}+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (-5408+173056 x^2-1664 x^4-53248 x^6+2880 x^8+30720 x^{10}-640 x^{12}-4096 x^{14}+224 x^{16}+1024 x^{18}\right )\right )}{2197-1014 x^4+663 x^8-164 x^{12}+51 x^{16}-6 x^{20}+x^{24}} \, dx=\left (-e^{-16 x^2} x+\frac {4}{3+\frac {1}{4} \left (1-x^4\right )^2}\right )^2 \] Output:
(4/(3+1/2*(-x^4+1)*(1/2-1/2*x^4))-x/exp(16*x^2))^2
Time = 2.93 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34 \[ \int \frac {e^{-32 x^2} \left (4394 x-140608 x^3-2028 x^5+64896 x^7+1326 x^9-42432 x^{11}-328 x^{13}+10496 x^{15}+102 x^{17}-3264 x^{19}-12 x^{21}+384 x^{23}+2 x^{25}-64 x^{27}+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (-5408+173056 x^2-1664 x^4-53248 x^6+2880 x^8+30720 x^{10}-640 x^{12}-4096 x^{14}+224 x^{16}+1024 x^{18}\right )\right )}{2197-1014 x^4+663 x^8-164 x^{12}+51 x^{16}-6 x^{20}+x^{24}} \, dx=\frac {e^{-32 x^2} \left (-16 e^{16 x^2}+13 x-2 x^5+x^9\right )^2}{\left (13-2 x^4+x^8\right )^2} \] Input:
Integrate[(4394*x - 140608*x^3 - 2028*x^5 + 64896*x^7 + 1326*x^9 - 42432*x ^11 - 328*x^13 + 10496*x^15 + 102*x^17 - 3264*x^19 - 12*x^21 + 384*x^23 + 2*x^25 - 64*x^27 + E^(32*x^2)*(4096*x^3 - 4096*x^7) + E^(16*x^2)*(-5408 + 173056*x^2 - 1664*x^4 - 53248*x^6 + 2880*x^8 + 30720*x^10 - 640*x^12 - 409 6*x^14 + 224*x^16 + 1024*x^18))/(E^(32*x^2)*(2197 - 1014*x^4 + 663*x^8 - 1 64*x^12 + 51*x^16 - 6*x^20 + x^24)),x]
Output:
(-16*E^(16*x^2) + 13*x - 2*x^5 + x^9)^2/(E^(32*x^2)*(13 - 2*x^4 + x^8)^2)
Time = 11.66 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {2461, 7239, 2009}
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.
| \(\displaystyle \int \frac {e^{-32 x^2} \left (-64 x^{27}+2 x^{25}+384 x^{23}-12 x^{21}-3264 x^{19}+102 x^{17}+10496 x^{15}-328 x^{13}-42432 x^{11}+1326 x^9+64896 x^7-2028 x^5-140608 x^3+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (1024 x^{18}+224 x^{16}-4096 x^{14}-640 x^{12}+30720 x^{10}+2880 x^8-53248 x^6-1664 x^4+173056 x^2-5408\right )+4394 x\right )}{x^{24}-6 x^{20}+51 x^{16}-164 x^{12}+663 x^8-1014 x^4+2197} \, dx\) |
| \(\Big \downarrow \) 2461 |
| \(\displaystyle \int \left (\frac {i e^{-32 x^2} \left (-64 x^{27}+2 x^{25}+384 x^{23}-12 x^{21}-3264 x^{19}+102 x^{17}+10496 x^{15}-328 x^{13}-42432 x^{11}+1326 x^9+64896 x^7-2028 x^5-140608 x^3+4394 x+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (1024 x^{18}+224 x^{16}-4096 x^{14}-640 x^{12}+30720 x^{10}+2880 x^8-53248 x^6-1664 x^4+173056 x^2-5408\right )\right )}{768 \sqrt {3} \left (-2 x^4+4 i \sqrt {3}+2\right )}+\frac {i e^{-32 x^2} \left (-64 x^{27}+2 x^{25}+384 x^{23}-12 x^{21}-3264 x^{19}+102 x^{17}+10496 x^{15}-328 x^{13}-42432 x^{11}+1326 x^9+64896 x^7-2028 x^5-140608 x^3+4394 x+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (1024 x^{18}+224 x^{16}-4096 x^{14}-640 x^{12}+30720 x^{10}+2880 x^8-53248 x^6-1664 x^4+173056 x^2-5408\right )\right )}{768 \sqrt {3} \left (2 x^4+4 i \sqrt {3}-2\right )}-\frac {e^{-32 x^2} \left (-64 x^{27}+2 x^{25}+384 x^{23}-12 x^{21}-3264 x^{19}+102 x^{17}+10496 x^{15}-328 x^{13}-42432 x^{11}+1326 x^9+64896 x^7-2028 x^5-140608 x^3+4394 x+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (1024 x^{18}+224 x^{16}-4096 x^{14}-640 x^{12}+30720 x^{10}+2880 x^8-53248 x^6-1664 x^4+173056 x^2-5408\right )\right )}{192 \left (-2 x^4+4 i \sqrt {3}+2\right )^2}-\frac {e^{-32 x^2} \left (-64 x^{27}+2 x^{25}+384 x^{23}-12 x^{21}-3264 x^{19}+102 x^{17}+10496 x^{15}-328 x^{13}-42432 x^{11}+1326 x^9+64896 x^7-2028 x^5-140608 x^3+4394 x+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (1024 x^{18}+224 x^{16}-4096 x^{14}-640 x^{12}+30720 x^{10}+2880 x^8-53248 x^6-1664 x^4+173056 x^2-5408\right )\right )}{192 \left (2 x^4+4 i \sqrt {3}-2\right )^2}-\frac {i e^{-32 x^2} \left (-64 x^{27}+2 x^{25}+384 x^{23}-12 x^{21}-3264 x^{19}+102 x^{17}+10496 x^{15}-328 x^{13}-42432 x^{11}+1326 x^9+64896 x^7-2028 x^5-140608 x^3+4394 x+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (1024 x^{18}+224 x^{16}-4096 x^{14}-640 x^{12}+30720 x^{10}+2880 x^8-53248 x^6-1664 x^4+173056 x^2-5408\right )\right )}{24 \sqrt {3} \left (-2 x^4+4 i \sqrt {3}+2\right )^3}-\frac {i e^{-32 x^2} \left (-64 x^{27}+2 x^{25}+384 x^{23}-12 x^{21}-3264 x^{19}+102 x^{17}+10496 x^{15}-328 x^{13}-42432 x^{11}+1326 x^9+64896 x^7-2028 x^5-140608 x^3+4394 x+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (1024 x^{18}+224 x^{16}-4096 x^{14}-640 x^{12}+30720 x^{10}+2880 x^8-53248 x^6-1664 x^4+173056 x^2-5408\right )\right )}{24 \sqrt {3} \left (2 x^4+4 i \sqrt {3}-2\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \left (e^{-32 x^2} \left (2-64 x^2\right ) x-\frac {4096 \left (x^4-1\right ) x^3}{\left (x^8-2 x^4+13\right )^3}+\frac {32 e^{-16 x^2} \left (32 x^{10}+7 x^8-64 x^6-6 x^4+416 x^2-13\right )}{\left (x^8-2 x^4+13\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle e^{-32 x^2} x^2+\frac {256}{\left (x^8-2 x^4+13\right )^2}-\frac {32 e^{-16 x^2} \left (x^{10}-2 x^6+13 x^2\right )}{\left (x^8-2 x^4+13\right )^2 x}\) |
Input:
Int[(4394*x - 140608*x^3 - 2028*x^5 + 64896*x^7 + 1326*x^9 - 42432*x^11 - 328*x^13 + 10496*x^15 + 102*x^17 - 3264*x^19 - 12*x^21 + 384*x^23 + 2*x^25 - 64*x^27 + E^(32*x^2)*(4096*x^3 - 4096*x^7) + E^(16*x^2)*(-5408 + 173056 *x^2 - 1664*x^4 - 53248*x^6 + 2880*x^8 + 30720*x^10 - 640*x^12 - 4096*x^14 + 224*x^16 + 1024*x^18))/(E^(32*x^2)*(2197 - 1014*x^4 + 663*x^8 - 164*x^1 2 + 51*x^16 - 6*x^20 + x^24)),x]
Output:
x^2/E^(32*x^2) + 256/(13 - 2*x^4 + x^8)^2 - (32*(13*x^2 - 2*x^6 + x^10))/( E^(16*x^2)*x*(13 - 2*x^4 + x^8)^2)
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px /. x -> Sqrt[x]]}, Int[ExpandIntegrand[u, (Qx /. x -> x^2)^p, x], x] /; !SumQ[NonfreeFactors[ Qx, x]]] /; PolyQ[Px, x^2] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 20.95 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53
| method | result | size |
| parts | \(\frac {256}{\left (x^{8}-2 x^{4}+13\right )^{2}}+x^{2} {\mathrm e}^{-32 x^{2}}-\frac {32 x \,{\mathrm e}^{-16 x^{2}}}{x^{8}-2 x^{4}+13}\) | \(49\) |
| risch | \(\frac {256}{x^{16}-4 x^{12}+30 x^{8}-52 x^{4}+169}-\frac {32 x \,{\mathrm e}^{-16 x^{2}}}{x^{8}-2 x^{4}+13}+x^{2} {\mathrm e}^{-32 x^{2}}\) | \(57\) |
| parallelrisch | \(\frac {\left (30 x^{18}-120 x^{14}+900 x^{10}-960 \,{\mathrm e}^{16 x^{2}} x^{9}-1560 x^{6}+1920 \,{\mathrm e}^{16 x^{2}} x^{5}+5070 x^{2}-12480 x \,{\mathrm e}^{16 x^{2}}+7680 \,{\mathrm e}^{32 x^{2}}\right ) {\mathrm e}^{-32 x^{2}}}{30 x^{16}-120 x^{12}+900 x^{8}-1560 x^{4}+5070}\) | \(100\) |
| orering | \(\text {Expression too large to display}\) | \(3360\) |
Input:
int(((-4096*x^7+4096*x^3)*exp(16*x^2)^2+(1024*x^18+224*x^16-4096*x^14-640* x^12+30720*x^10+2880*x^8-53248*x^6-1664*x^4+173056*x^2-5408)*exp(16*x^2)-6 4*x^27+2*x^25+384*x^23-12*x^21-3264*x^19+102*x^17+10496*x^15-328*x^13-4243 2*x^11+1326*x^9+64896*x^7-2028*x^5-140608*x^3+4394*x)/(x^24-6*x^20+51*x^16 -164*x^12+663*x^8-1014*x^4+2197)/exp(16*x^2)^2,x,method=_RETURNVERBOSE)
Output:
256/(x^8-2*x^4+13)^2+x^2/exp(x^2)^32-32*x/(x^8-2*x^4+13)/exp(16*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (24) = 48\).
Time = 0.10 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.53 \[ \int \frac {e^{-32 x^2} \left (4394 x-140608 x^3-2028 x^5+64896 x^7+1326 x^9-42432 x^{11}-328 x^{13}+10496 x^{15}+102 x^{17}-3264 x^{19}-12 x^{21}+384 x^{23}+2 x^{25}-64 x^{27}+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (-5408+173056 x^2-1664 x^4-53248 x^6+2880 x^8+30720 x^{10}-640 x^{12}-4096 x^{14}+224 x^{16}+1024 x^{18}\right )\right )}{2197-1014 x^4+663 x^8-164 x^{12}+51 x^{16}-6 x^{20}+x^{24}} \, dx=\frac {{\left (x^{18} - 4 \, x^{14} + 30 \, x^{10} - 52 \, x^{6} + 169 \, x^{2} - 32 \, {\left (x^{9} - 2 \, x^{5} + 13 \, x\right )} e^{\left (16 \, x^{2}\right )} + 256 \, e^{\left (32 \, x^{2}\right )}\right )} e^{\left (-32 \, x^{2}\right )}}{x^{16} - 4 \, x^{12} + 30 \, x^{8} - 52 \, x^{4} + 169} \] Input:
integrate(((-4096*x^7+4096*x^3)*exp(16*x^2)^2+(1024*x^18+224*x^16-4096*x^1 4-640*x^12+30720*x^10+2880*x^8-53248*x^6-1664*x^4+173056*x^2-5408)*exp(16* x^2)-64*x^27+2*x^25+384*x^23-12*x^21-3264*x^19+102*x^17+10496*x^15-328*x^1 3-42432*x^11+1326*x^9+64896*x^7-2028*x^5-140608*x^3+4394*x)/(x^24-6*x^20+5 1*x^16-164*x^12+663*x^8-1014*x^4+2197)/exp(16*x^2)^2,x, algorithm="fricas" )
Output:
(x^18 - 4*x^14 + 30*x^10 - 52*x^6 + 169*x^2 - 32*(x^9 - 2*x^5 + 13*x)*e^(1 6*x^2) + 256*e^(32*x^2))*e^(-32*x^2)/(x^16 - 4*x^12 + 30*x^8 - 52*x^4 + 16 9)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (22) = 44\).
Time = 0.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.91 \[ \int \frac {e^{-32 x^2} \left (4394 x-140608 x^3-2028 x^5+64896 x^7+1326 x^9-42432 x^{11}-328 x^{13}+10496 x^{15}+102 x^{17}-3264 x^{19}-12 x^{21}+384 x^{23}+2 x^{25}-64 x^{27}+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (-5408+173056 x^2-1664 x^4-53248 x^6+2880 x^8+30720 x^{10}-640 x^{12}-4096 x^{14}+224 x^{16}+1024 x^{18}\right )\right )}{2197-1014 x^4+663 x^8-164 x^{12}+51 x^{16}-6 x^{20}+x^{24}} \, dx=\frac {- 32 x e^{- 16 x^{2}} + \left (x^{10} - 2 x^{6} + 13 x^{2}\right ) e^{- 32 x^{2}}}{x^{8} - 2 x^{4} + 13} + \frac {256}{x^{16} - 4 x^{12} + 30 x^{8} - 52 x^{4} + 169} \] Input:
integrate(((-4096*x**7+4096*x**3)*exp(16*x**2)**2+(1024*x**18+224*x**16-40 96*x**14-640*x**12+30720*x**10+2880*x**8-53248*x**6-1664*x**4+173056*x**2- 5408)*exp(16*x**2)-64*x**27+2*x**25+384*x**23-12*x**21-3264*x**19+102*x**1 7+10496*x**15-328*x**13-42432*x**11+1326*x**9+64896*x**7-2028*x**5-140608* x**3+4394*x)/(x**24-6*x**20+51*x**16-164*x**12+663*x**8-1014*x**4+2197)/ex p(16*x**2)**2,x)
Output:
(-32*x*exp(-16*x**2) + (x**10 - 2*x**6 + 13*x**2)*exp(-32*x**2))/(x**8 - 2 *x**4 + 13) + 256/(x**16 - 4*x**12 + 30*x**8 - 52*x**4 + 169)
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (24) = 48\).
Time = 0.10 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.44 \[ \int \frac {e^{-32 x^2} \left (4394 x-140608 x^3-2028 x^5+64896 x^7+1326 x^9-42432 x^{11}-328 x^{13}+10496 x^{15}+102 x^{17}-3264 x^{19}-12 x^{21}+384 x^{23}+2 x^{25}-64 x^{27}+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (-5408+173056 x^2-1664 x^4-53248 x^6+2880 x^8+30720 x^{10}-640 x^{12}-4096 x^{14}+224 x^{16}+1024 x^{18}\right )\right )}{2197-1014 x^4+663 x^8-164 x^{12}+51 x^{16}-6 x^{20}+x^{24}} \, dx=-\frac {32 \, {\left (x^{9} - 2 \, x^{5} + 13 \, x\right )} e^{\left (-16 \, x^{2}\right )} - {\left (x^{18} - 4 \, x^{14} + 30 \, x^{10} - 52 \, x^{6} + 169 \, x^{2}\right )} e^{\left (-32 \, x^{2}\right )} - 256}{x^{16} - 4 \, x^{12} + 30 \, x^{8} - 52 \, x^{4} + 169} \] Input:
integrate(((-4096*x^7+4096*x^3)*exp(16*x^2)^2+(1024*x^18+224*x^16-4096*x^1 4-640*x^12+30720*x^10+2880*x^8-53248*x^6-1664*x^4+173056*x^2-5408)*exp(16* x^2)-64*x^27+2*x^25+384*x^23-12*x^21-3264*x^19+102*x^17+10496*x^15-328*x^1 3-42432*x^11+1326*x^9+64896*x^7-2028*x^5-140608*x^3+4394*x)/(x^24-6*x^20+5 1*x^16-164*x^12+663*x^8-1014*x^4+2197)/exp(16*x^2)^2,x, algorithm="maxima" )
Output:
-(32*(x^9 - 2*x^5 + 13*x)*e^(-16*x^2) - (x^18 - 4*x^14 + 30*x^10 - 52*x^6 + 169*x^2)*e^(-32*x^2) - 256)/(x^16 - 4*x^12 + 30*x^8 - 52*x^4 + 169)
Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (24) = 48\).
Time = 0.18 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.69 \[ \int \frac {e^{-32 x^2} \left (4394 x-140608 x^3-2028 x^5+64896 x^7+1326 x^9-42432 x^{11}-328 x^{13}+10496 x^{15}+102 x^{17}-3264 x^{19}-12 x^{21}+384 x^{23}+2 x^{25}-64 x^{27}+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (-5408+173056 x^2-1664 x^4-53248 x^6+2880 x^8+30720 x^{10}-640 x^{12}-4096 x^{14}+224 x^{16}+1024 x^{18}\right )\right )}{2197-1014 x^4+663 x^8-164 x^{12}+51 x^{16}-6 x^{20}+x^{24}} \, dx=\frac {x^{18} - 4 \, x^{14} + 30 \, x^{10} - 32 \, x^{9} e^{\left (16 \, x^{2}\right )} - 52 \, x^{6} + 64 \, x^{5} e^{\left (16 \, x^{2}\right )} + 169 \, x^{2} - 416 \, x e^{\left (16 \, x^{2}\right )} + 256 \, e^{\left (32 \, x^{2}\right )}}{x^{16} e^{\left (32 \, x^{2}\right )} - 4 \, x^{12} e^{\left (32 \, x^{2}\right )} + 30 \, x^{8} e^{\left (32 \, x^{2}\right )} - 52 \, x^{4} e^{\left (32 \, x^{2}\right )} + 169 \, e^{\left (32 \, x^{2}\right )}} \] Input:
integrate(((-4096*x^7+4096*x^3)*exp(16*x^2)^2+(1024*x^18+224*x^16-4096*x^1 4-640*x^12+30720*x^10+2880*x^8-53248*x^6-1664*x^4+173056*x^2-5408)*exp(16* x^2)-64*x^27+2*x^25+384*x^23-12*x^21-3264*x^19+102*x^17+10496*x^15-328*x^1 3-42432*x^11+1326*x^9+64896*x^7-2028*x^5-140608*x^3+4394*x)/(x^24-6*x^20+5 1*x^16-164*x^12+663*x^8-1014*x^4+2197)/exp(16*x^2)^2,x, algorithm="giac")
Output:
(x^18 - 4*x^14 + 30*x^10 - 32*x^9*e^(16*x^2) - 52*x^6 + 64*x^5*e^(16*x^2) + 169*x^2 - 416*x*e^(16*x^2) + 256*e^(32*x^2))/(x^16*e^(32*x^2) - 4*x^12*e ^(32*x^2) + 30*x^8*e^(32*x^2) - 52*x^4*e^(32*x^2) + 169*e^(32*x^2))
Time = 3.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {e^{-32 x^2} \left (4394 x-140608 x^3-2028 x^5+64896 x^7+1326 x^9-42432 x^{11}-328 x^{13}+10496 x^{15}+102 x^{17}-3264 x^{19}-12 x^{21}+384 x^{23}+2 x^{25}-64 x^{27}+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (-5408+173056 x^2-1664 x^4-53248 x^6+2880 x^8+30720 x^{10}-640 x^{12}-4096 x^{14}+224 x^{16}+1024 x^{18}\right )\right )}{2197-1014 x^4+663 x^8-164 x^{12}+51 x^{16}-6 x^{20}+x^{24}} \, dx=\frac {{\mathrm {e}}^{-32\,x^2}\,{\left (13\,x-16\,{\mathrm {e}}^{16\,x^2}-2\,x^5+x^9\right )}^2}{{\left (x^8-2\,x^4+13\right )}^2} \] Input:
int((exp(-32*x^2)*(4394*x + exp(16*x^2)*(173056*x^2 - 1664*x^4 - 53248*x^6 + 2880*x^8 + 30720*x^10 - 640*x^12 - 4096*x^14 + 224*x^16 + 1024*x^18 - 5 408) + exp(32*x^2)*(4096*x^3 - 4096*x^7) - 140608*x^3 - 2028*x^5 + 64896*x ^7 + 1326*x^9 - 42432*x^11 - 328*x^13 + 10496*x^15 + 102*x^17 - 3264*x^19 - 12*x^21 + 384*x^23 + 2*x^25 - 64*x^27))/(663*x^8 - 1014*x^4 - 164*x^12 + 51*x^16 - 6*x^20 + x^24 + 2197),x)
Output:
(exp(-32*x^2)*(13*x - 16*exp(16*x^2) - 2*x^5 + x^9)^2)/(x^8 - 2*x^4 + 13)^ 2
Time = 0.39 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.09 \[ \int \frac {e^{-32 x^2} \left (4394 x-140608 x^3-2028 x^5+64896 x^7+1326 x^9-42432 x^{11}-328 x^{13}+10496 x^{15}+102 x^{17}-3264 x^{19}-12 x^{21}+384 x^{23}+2 x^{25}-64 x^{27}+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (-5408+173056 x^2-1664 x^4-53248 x^6+2880 x^8+30720 x^{10}-640 x^{12}-4096 x^{14}+224 x^{16}+1024 x^{18}\right )\right )}{2197-1014 x^4+663 x^8-164 x^{12}+51 x^{16}-6 x^{20}+x^{24}} \, dx=\frac {256 e^{32 x^{2}}-32 e^{16 x^{2}} x^{9}+64 e^{16 x^{2}} x^{5}-416 e^{16 x^{2}} x +x^{18}-4 x^{14}+30 x^{10}-52 x^{6}+169 x^{2}}{e^{32 x^{2}} \left (x^{16}-4 x^{12}+30 x^{8}-52 x^{4}+169\right )} \] Input:
int(((-4096*x^7+4096*x^3)*exp(16*x^2)^2+(1024*x^18+224*x^16-4096*x^14-640* x^12+30720*x^10+2880*x^8-53248*x^6-1664*x^4+173056*x^2-5408)*exp(16*x^2)-6 4*x^27+2*x^25+384*x^23-12*x^21-3264*x^19+102*x^17+10496*x^15-328*x^13-4243 2*x^11+1326*x^9+64896*x^7-2028*x^5-140608*x^3+4394*x)/(x^24-6*x^20+51*x^16 -164*x^12+663*x^8-1014*x^4+2197)/exp(16*x^2)^2,x)
Output:
(256*e**(32*x**2) - 32*e**(16*x**2)*x**9 + 64*e**(16*x**2)*x**5 - 416*e**( 16*x**2)*x + x**18 - 4*x**14 + 30*x**10 - 52*x**6 + 169*x**2)/(e**(32*x**2 )*(x**16 - 4*x**12 + 30*x**8 - 52*x**4 + 169))