\(\int \frac {e^{-32 x^2} (4394 x-140608 x^3-2028 x^5+64896 x^7+1326 x^9-42432 x^{11}-328 x^{13}+10496 x^{15}+102 x^{17}-3264 x^{19}-12 x^{21}+384 x^{23}+2 x^{25}-64 x^{27}+e^{32 x^2} (4096 x^3-4096 x^7)+e^{16 x^2} (-5408+173056 x^2-1664 x^4-53248 x^6+2880 x^8+30720 x^{10}-640 x^{12}-4096 x^{14}+224 x^{16}+1024 x^{18}))}{2197-1014 x^4+663 x^8-164 x^{12}+51 x^{16}-6 x^{20}+x^{24}} \, dx\) [6559]

Optimal result

Integrand size = 183, antiderivative size = 32 \[ \int \frac {e^{-32 x^2} \left (4394 x-140608 x^3-2028 x^5+64896 x^7+1326 x^9-42432 x^{11}-328 x^{13}+10496 x^{15}+102 x^{17}-3264 x^{19}-12 x^{21}+384 x^{23}+2 x^{25}-64 x^{27}+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (-5408+173056 x^2-1664 x^4-53248 x^6+2880 x^8+30720 x^{10}-640 x^{12}-4096 x^{14}+224 x^{16}+1024 x^{18}\right )\right )}{2197-1014 x^4+663 x^8-164 x^{12}+51 x^{16}-6 x^{20}+x^{24}} \, dx=\left (-e^{-16 x^2} x+\frac {4}{3+\frac {1}{4} \left (1-x^4\right )^2}\right )^2 \]

[Out]

(4/(3+1/2*(-x^4+1)*(1/2-1/2*x^4))-x/exp(16*x^2))^2

Rubi [F(-1)]

Timed out. \[ \int \frac {e^{-32 x^2} \left (4394 x-140608 x^3-2028 x^5+64896 x^7+1326 x^9-42432 x^{11}-328 x^{13}+10496 x^{15}+102 x^{17}-3264 x^{19}-12 x^{21}+384 x^{23}+2 x^{25}-64 x^{27}+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (-5408+173056 x^2-1664 x^4-53248 x^6+2880 x^8+30720 x^{10}-640 x^{12}-4096 x^{14}+224 x^{16}+1024 x^{18}\right )\right )}{2197-1014 x^4+663 x^8-164 x^{12}+51 x^{16}-6 x^{20}+x^{24}} \, dx=\text {\$Aborted} \]

[In]

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 - 3
264*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 + 1730
56*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^12 + 51*x^16 - 6*x^20 + x^24)),x]

[Out]

$Aborted

Rubi steps Aborted

Mathematica [A] (verified)

Time = 4.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34 \[ \int \frac {e^{-32 x^2} \left (4394 x-140608 x^3-2028 x^5+64896 x^7+1326 x^9-42432 x^{11}-328 x^{13}+10496 x^{15}+102 x^{17}-3264 x^{19}-12 x^{21}+384 x^{23}+2 x^{25}-64 x^{27}+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (-5408+173056 x^2-1664 x^4-53248 x^6+2880 x^8+30720 x^{10}-640 x^{12}-4096 x^{14}+224 x^{16}+1024 x^{18}\right )\right )}{2197-1014 x^4+663 x^8-164 x^{12}+51 x^{16}-6 x^{20}+x^{24}} \, dx=\frac {e^{-32 x^2} \left (-16 e^{16 x^2}+13 x-2 x^5+x^9\right )^2}{\left (13-2 x^4+x^8\right )^2} \]

[In]

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 - 4096*x^14 + 224*x^16 + 1024*x^18))/(E
^(32*x^2)*(2197 - 1014*x^4 + 663*x^8 - 164*x^12 + 51*x^16 - 6*x^20 + x^24)),x]

[Out]

(-16*E^(16*x^2) + 13*x - 2*x^5 + x^9)^2/(E^(32*x^2)*(13 - 2*x^4 + x^8)^2)

Maple [A] (verified)

Time = 1.10 (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	\(\frac {\left (4 x^{18}-16 x^{14}+120 x^{10}-128 \,{\mathrm e}^{16 x^{2}} x^{9}-208 x^{6}+256 \,{\mathrm e}^{16 x^{2}} x^{5}+676 x^{2}-1664 x \,{\mathrm e}^{16 x^{2}}+1024 \,{\mathrm e}^{32 x^{2}}\right ) {\mathrm e}^{-32 x^{2}}}{4 x^{16}-16 x^{12}+120 x^{8}-208 x^{4}+676}\)	\(100\)

[In]

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^6-1
664*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^13-42
432*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)/e
xp(16*x^2)^2,x,method=_RETURNVERBOSE)

[Out]

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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (24) = 48\).

Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.53 \[ \int \frac {e^{-32 x^2} \left (4394 x-140608 x^3-2028 x^5+64896 x^7+1326 x^9-42432 x^{11}-328 x^{13}+10496 x^{15}+102 x^{17}-3264 x^{19}-12 x^{21}+384 x^{23}+2 x^{25}-64 x^{27}+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (-5408+173056 x^2-1664 x^4-53248 x^6+2880 x^8+30720 x^{10}-640 x^{12}-4096 x^{14}+224 x^{16}+1024 x^{18}\right )\right )}{2197-1014 x^4+663 x^8-164 x^{12}+51 x^{16}-6 x^{20}+x^{24}} \, dx=\frac {{\left (x^{18} - 4 \, x^{14} + 30 \, x^{10} - 52 \, x^{6} + 169 \, x^{2} - 32 \, {\left (x^{9} - 2 \, x^{5} + 13 \, x\right )} e^{\left (16 \, x^{2}\right )} + 256 \, e^{\left (32 \, x^{2}\right )}\right )} e^{\left (-32 \, x^{2}\right )}}{x^{16} - 4 \, x^{12} + 30 \, x^{8} - 52 \, x^{4} + 169} \]

[In]

integrate(((-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)-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)/(x^24-6*x^20+51*x^16-164*x^12+663*x^8-1014*x^4+2
197)/exp(16*x^2)^2,x, algorithm="fricas")

[Out]

(x^18 - 4*x^14 + 30*x^10 - 52*x^6 + 169*x^2 - 32*(x^9 - 2*x^5 + 13*x)*e^(16*x^2) + 256*e^(32*x^2))*e^(-32*x^2)
/(x^16 - 4*x^12 + 30*x^8 - 52*x^4 + 169)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (22) = 44\).

Time = 0.22 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.91 \[ \int \frac {e^{-32 x^2} \left (4394 x-140608 x^3-2028 x^5+64896 x^7+1326 x^9-42432 x^{11}-328 x^{13}+10496 x^{15}+102 x^{17}-3264 x^{19}-12 x^{21}+384 x^{23}+2 x^{25}-64 x^{27}+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (-5408+173056 x^2-1664 x^4-53248 x^6+2880 x^8+30720 x^{10}-640 x^{12}-4096 x^{14}+224 x^{16}+1024 x^{18}\right )\right )}{2197-1014 x^4+663 x^8-164 x^{12}+51 x^{16}-6 x^{20}+x^{24}} \, dx=\frac {- 32 x e^{- 16 x^{2}} + \left (x^{10} - 2 x^{6} + 13 x^{2}\right ) e^{- 32 x^{2}}}{x^{8} - 2 x^{4} + 13} + \frac {256}{x^{16} - 4 x^{12} + 30 x^{8} - 52 x^{4} + 169} \]

[In]

integrate(((-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)-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)/(x**24-6*x**20+51*x**
16-164*x**12+663*x**8-1014*x**4+2197)/exp(16*x**2)**2,x)

[Out]

(-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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (24) = 48\).

Time = 0.25 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.44 \[ \int \frac {e^{-32 x^2} \left (4394 x-140608 x^3-2028 x^5+64896 x^7+1326 x^9-42432 x^{11}-328 x^{13}+10496 x^{15}+102 x^{17}-3264 x^{19}-12 x^{21}+384 x^{23}+2 x^{25}-64 x^{27}+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (-5408+173056 x^2-1664 x^4-53248 x^6+2880 x^8+30720 x^{10}-640 x^{12}-4096 x^{14}+224 x^{16}+1024 x^{18}\right )\right )}{2197-1014 x^4+663 x^8-164 x^{12}+51 x^{16}-6 x^{20}+x^{24}} \, dx=-\frac {32 \, {\left (x^{9} - 2 \, x^{5} + 13 \, x\right )} e^{\left (-16 \, x^{2}\right )} - {\left (x^{18} - 4 \, x^{14} + 30 \, x^{10} - 52 \, x^{6} + 169 \, x^{2}\right )} e^{\left (-32 \, x^{2}\right )} - 256}{x^{16} - 4 \, x^{12} + 30 \, x^{8} - 52 \, x^{4} + 169} \]

[In]

integrate(((-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)-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)/(x^24-6*x^20+51*x^16-164*x^12+663*x^8-1014*x^4+2
197)/exp(16*x^2)^2,x, algorithm="maxima")

[Out]

-(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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (24) = 48\).

Time = 0.36 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.44 \[ \int \frac {e^{-32 x^2} \left (4394 x-140608 x^3-2028 x^5+64896 x^7+1326 x^9-42432 x^{11}-328 x^{13}+10496 x^{15}+102 x^{17}-3264 x^{19}-12 x^{21}+384 x^{23}+2 x^{25}-64 x^{27}+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (-5408+173056 x^2-1664 x^4-53248 x^6+2880 x^8+30720 x^{10}-640 x^{12}-4096 x^{14}+224 x^{16}+1024 x^{18}\right )\right )}{2197-1014 x^4+663 x^8-164 x^{12}+51 x^{16}-6 x^{20}+x^{24}} \, dx=\frac {x^{18} e^{\left (-32 \, x^{2}\right )} - 4 \, x^{14} e^{\left (-32 \, x^{2}\right )} + 30 \, x^{10} e^{\left (-32 \, x^{2}\right )} - 32 \, x^{9} e^{\left (-16 \, x^{2}\right )} - 52 \, x^{6} e^{\left (-32 \, x^{2}\right )} + 64 \, x^{5} e^{\left (-16 \, x^{2}\right )} + 169 \, x^{2} e^{\left (-32 \, x^{2}\right )} - 416 \, x e^{\left (-16 \, x^{2}\right )} + 256}{x^{16} - 4 \, x^{12} + 30 \, x^{8} - 52 \, x^{4} + 169} \]

[In]

integrate(((-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)-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)/(x^24-6*x^20+51*x^16-164*x^12+663*x^8-1014*x^4+2
197)/exp(16*x^2)^2,x, algorithm="giac")

[Out]

(x^18*e^(-32*x^2) - 4*x^14*e^(-32*x^2) + 30*x^10*e^(-32*x^2) - 32*x^9*e^(-16*x^2) - 52*x^6*e^(-32*x^2) + 64*x^
5*e^(-16*x^2) + 169*x^2*e^(-32*x^2) - 416*x*e^(-16*x^2) + 256)/(x^16 - 4*x^12 + 30*x^8 - 52*x^4 + 169)

Mupad [B] (verification not implemented)

Time = 12.43 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {e^{-32 x^2} \left (4394 x-140608 x^3-2028 x^5+64896 x^7+1326 x^9-42432 x^{11}-328 x^{13}+10496 x^{15}+102 x^{17}-3264 x^{19}-12 x^{21}+384 x^{23}+2 x^{25}-64 x^{27}+e^{32 x^2} \left (4096 x^3-4096 x^7\right )+e^{16 x^2} \left (-5408+173056 x^2-1664 x^4-53248 x^6+2880 x^8+30720 x^{10}-640 x^{12}-4096 x^{14}+224 x^{16}+1024 x^{18}\right )\right )}{2197-1014 x^4+663 x^8-164 x^{12}+51 x^{16}-6 x^{20}+x^{24}} \, dx=\frac {{\mathrm {e}}^{-32\,x^2}\,{\left (13\,x-16\,{\mathrm {e}}^{16\,x^2}-2\,x^5+x^9\right )}^2}{{\left (x^8-2\,x^4+13\right )}^2} \]

[In]

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 - 5408) + 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)

[Out]

(exp(-32*x^2)*(13*x - 16*exp(16*x^2) - 2*x^5 + x^9)^2)/(x^8 - 2*x^4 + 13)^2