∫ e−32x2 (4394x−140608x3−2028x5+64896x7+1326x9−42432x11−328x13+10496x15+102x17−3264x19−12x21+384x23+2x25−64x27+e32x2 (4096x3−4096x7)+e16x2 (−5408+173056x2−1664x4−53248x6+2880x8+30720x10−640x12−4096x14+224x16+1024x18))______________________________________________________________________________________________________________________________________________________________________________________ 2197−1014x4+663x8−164x12+51x16−6x20+x24 dx

3.66.59 $\int \frac {e^{-32 x^2} (4394 x-140608 x^3-2028 x^5+64896 x^7+1326 x^9-42432 x^{11}-328 x^{13}+10496 x^{15}+102 x^{17}-3264 x^{19}-12 x^{21}+384 x^{23}+2 x^{25}-64 x^{27}+e^{32 x^2} (4096 x^3-4096 x^7)+e^{16 x^2} (-5408+173056 x^2-1664 x^4-53248 x^6+2880 x^8+30720 x^{10}-640 x^{12}-4096 x^{14}+224 x^{16}+1024 x^{18}))}{2197-1014 x^4+663 x^8-164 x^{12}+51 x^{16}-6 x^{20}+x^{24}} \, dx$

Optimal. Leaf size=32 \[ \left (-e^{-16 x^2} x+\frac {4}{3+\frac {1}{4} \left (1-x^4\right )^2}\right )^2 \]

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Rubi [F] time = 180.00, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.000, Rules used = {} \begin {gather*} \text {\$Aborted} \end {gather*}

Veriﬁcation is not applicable to the result.

[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

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Mathematica [A] time = 0.12, size = 43, normalized size = 1.34 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully veriﬁed.

[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)

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fricas [B] time = 0.90, size = 81, normalized size = 2.53 \begin {gather*} \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} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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giac [B] time = 0.23, size = 110, normalized size = 3.44 \begin {gather*} \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} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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maple [A] time = 0.24, size = 57, normalized size = 1.78


method	result	size

risch	$\frac {256}{x^{16}-4 x^{12}+30 x^{8}-52 x^{4}+169}-\frac {32 x \,{\mathrm e}^{-16 x^{2}}}{x^{8}-2 x^{4}+13}+x^{2} {\mathrm e}^{-32 x^{2}}$	$57$

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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^16-4*x^12+30*x^8-52*x^4+169)-32*x/(x^8-2*x^4+13)*exp(-16*x^2)+x^2*exp(-32*x^2)

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maxima [B] time = 0.44, size = 78, normalized size = 2.44 \begin {gather*} -\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} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B] time = 4.30, size = 41, normalized size = 1.28 \begin {gather*} \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} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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

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sympy [B] time = 0.41, size = 61, normalized size = 1.91 \begin {gather*} \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} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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