∫ ee(−432x2−576x3−180x4−256x6−512x7−384x8−128x9−16x10+ex2 (288x2+384x3−72x4−192x5−48x6)+e2x2 (−48x2−64x3+44x4+64x5+16x6))_______________________________________________________________________________________________________ 81−12e3x2+e4x2−288x4−288x5−72x6+256x8+512x9+384x10+128x11+16x12+e2x2(54−32x4−32x5−8x6)+ex2(−108+192x4+192x5+48x6) dx

3.4.47 \(\int \frac {e^e (-432 x^2-576 x^3-180 x^4-256 x^6-512 x^7-384 x^8-128 x^9-16 x^{10}+e^{x^2} (288 x^2+384 x^3-72 x^4-192 x^5-48 x^6)+e^{2 x^2} (-48 x^2-64 x^3+44 x^4+64 x^5+16 x^6))}{81-12 e^{3 x^2}+e^{4 x^2}-288 x^4-288 x^5-72 x^6+256 x^8+512 x^9+384 x^{10}+128 x^{11}+16 x^{12}+e^{2 x^2} (54-32 x^4-32 x^5-8 x^6)+e^{x^2} (-108+192 x^4+192 x^5+48 x^6)} \, dx\)

Optimal. Leaf size=31 \[ \frac {e^e}{x-\frac {\left (3-e^{x^2}\right )^2}{x^3 (4+2 x)^2}} \]

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Rubi [F] time = 5.19, 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*} \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 \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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 + 38

4*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]

[Out]

-2*E^E*Defer[Int][(-3 + E^x^2 - 4*x^2 - 2*x^3)^(-1), x] - 4*E^E*Defer[Int][x^2/(3 - E^x^2 + 4*x^2 + 2*x^3)^2,

x] - 14*E^E*Defer[Int][x^3/(3 - E^x^2 + 4*x^2 + 2*x^3)^2, x] + 10*E^E*Defer[Int][x^4/(3 - E^x^2 + 4*x^2 + 2*x^

3)^2, x] + 16*E^E*Defer[Int][x^5/(3 - E^x^2 + 4*x^2 + 2*x^3)^2, x] + 4*E^E*Defer[Int][x^6/(3 - E^x^2 + 4*x^2 +

2*x^3)^2, x] + 2*E^E*Defer[Int][x/(3 - E^x^2 + 4*x^2 + 2*x^3), x] - 4*E^E*Defer[Int][x^2/(3 - E^x^2 + 4*x^2 +

2*x^3), x] - 2*E^E*Defer[Int][x^3/(3 - E^x^2 + 4*x^2 + 2*x^3), x] - 28*E^E*Defer[Int][x^2/(-3 + E^x^2 + 4*x^2

+ 2*x^3)^2, x] - 26*E^E*Defer[Int][x^3/(-3 + E^x^2 + 4*x^2 + 2*x^3)^2, x] + 10*E^E*Defer[Int][x^4/(-3 + E^x^2

+ 4*x^2 + 2*x^3)^2, x] + 16*E^E*Defer[Int][x^5/(-3 + E^x^2 + 4*x^2 + 2*x^3)^2, x] + 4*E^E*Defer[Int][x^6/(-3

+ E^x^2 + 4*x^2 + 2*x^3)^2, x] + 2*E^E*Defer[Int][(-3 + E^x^2 + 4*x^2 + 2*x^3)^(-1), x] + 2*E^E*Defer[Int][x/(

-3 + E^x^2 + 4*x^2 + 2*x^3), x] - 4*E^E*Defer[Int][x^2/(-3 + E^x^2 + 4*x^2 + 2*x^3), x] - 2*E^E*Defer[Int][x^3

/(-3 + E^x^2 + 4*x^2 + 2*x^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^e \int \frac {-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 )}{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\\ &=e^e \int \frac {4 x^2 (2+x) \left (-54-45 x-32 x^4-48 x^5-24 x^6-4 x^7-e^{2 x^2} \left (6+5 x-8 x^2-4 x^3\right )-6 e^{x^2} \left (-6-5 x+4 x^2+2 x^3\right )\right )}{\left (9-6 e^{x^2}+e^{2 x^2}-16 x^4-16 x^5-4 x^6\right )^2} \, dx\\ &=\left (4 e^e\right ) \int \frac {x^2 (2+x) \left (-54-45 x-32 x^4-48 x^5-24 x^6-4 x^7-e^{2 x^2} \left (6+5 x-8 x^2-4 x^3\right )-6 e^{x^2} \left (-6-5 x+4 x^2+2 x^3\right )\right )}{\left (9-6 e^{x^2}+e^{2 x^2}-16 x^4-16 x^5-4 x^6\right )^2} \, dx\\ &=\left (4 e^e\right ) \int \left (\frac {-1-x+2 x^2+x^3}{2 \left (-3+e^{x^2}-4 x^2-2 x^3\right )}-\frac {-1-x+2 x^2+x^3}{2 \left (-3+e^{x^2}+4 x^2+2 x^3\right )}+\frac {x^2 \left (-14-13 x+5 x^2+8 x^3+2 x^4\right )}{2 \left (-3+e^{x^2}+4 x^2+2 x^3\right )^2}+\frac {x^2 \left (-2-7 x+5 x^2+8 x^3+2 x^4\right )}{2 \left (3-e^{x^2}+4 x^2+2 x^3\right )^2}\right ) \, dx\\ &=\left (2 e^e\right ) \int \frac {-1-x+2 x^2+x^3}{-3+e^{x^2}-4 x^2-2 x^3} \, dx-\left (2 e^e\right ) \int \frac {-1-x+2 x^2+x^3}{-3+e^{x^2}+4 x^2+2 x^3} \, dx+\left (2 e^e\right ) \int \frac {x^2 \left (-14-13 x+5 x^2+8 x^3+2 x^4\right )}{\left (-3+e^{x^2}+4 x^2+2 x^3\right )^2} \, dx+\left (2 e^e\right ) \int \frac {x^2 \left (-2-7 x+5 x^2+8 x^3+2 x^4\right )}{\left (3-e^{x^2}+4 x^2+2 x^3\right )^2} \, dx\\ &=\left (2 e^e\right ) \int \left (-\frac {2 x^2}{\left (3-e^{x^2}+4 x^2+2 x^3\right )^2}-\frac {7 x^3}{\left (3-e^{x^2}+4 x^2+2 x^3\right )^2}+\frac {5 x^4}{\left (3-e^{x^2}+4 x^2+2 x^3\right )^2}+\frac {8 x^5}{\left (3-e^{x^2}+4 x^2+2 x^3\right )^2}+\frac {2 x^6}{\left (3-e^{x^2}+4 x^2+2 x^3\right )^2}\right ) \, dx+\left (2 e^e\right ) \int \left (-\frac {1}{-3+e^{x^2}-4 x^2-2 x^3}+\frac {x}{3-e^{x^2}+4 x^2+2 x^3}-\frac {2 x^2}{3-e^{x^2}+4 x^2+2 x^3}-\frac {x^3}{3-e^{x^2}+4 x^2+2 x^3}\right ) \, dx+\left (2 e^e\right ) \int \left (-\frac {14 x^2}{\left (-3+e^{x^2}+4 x^2+2 x^3\right )^2}-\frac {13 x^3}{\left (-3+e^{x^2}+4 x^2+2 x^3\right )^2}+\frac {5 x^4}{\left (-3+e^{x^2}+4 x^2+2 x^3\right )^2}+\frac {8 x^5}{\left (-3+e^{x^2}+4 x^2+2 x^3\right )^2}+\frac {2 x^6}{\left (-3+e^{x^2}+4 x^2+2 x^3\right )^2}\right ) \, dx-\left (2 e^e\right ) \int \left (-\frac {1}{-3+e^{x^2}+4 x^2+2 x^3}-\frac {x}{-3+e^{x^2}+4 x^2+2 x^3}+\frac {2 x^2}{-3+e^{x^2}+4 x^2+2 x^3}+\frac {x^3}{-3+e^{x^2}+4 x^2+2 x^3}\right ) \, dx\\ &=-\left (\left (2 e^e\right ) \int \frac {1}{-3+e^{x^2}-4 x^2-2 x^3} \, dx\right )+\left (2 e^e\right ) \int \frac {x}{3-e^{x^2}+4 x^2+2 x^3} \, dx-\left (2 e^e\right ) \int \frac {x^3}{3-e^{x^2}+4 x^2+2 x^3} \, dx+\left (2 e^e\right ) \int \frac {1}{-3+e^{x^2}+4 x^2+2 x^3} \, dx+\left (2 e^e\right ) \int \frac {x}{-3+e^{x^2}+4 x^2+2 x^3} \, dx-\left (2 e^e\right ) \int \frac {x^3}{-3+e^{x^2}+4 x^2+2 x^3} \, dx-\left (4 e^e\right ) \int \frac {x^2}{\left (3-e^{x^2}+4 x^2+2 x^3\right )^2} \, dx+\left (4 e^e\right ) \int \frac {x^6}{\left (3-e^{x^2}+4 x^2+2 x^3\right )^2} \, dx-\left (4 e^e\right ) \int \frac {x^2}{3-e^{x^2}+4 x^2+2 x^3} \, dx+\left (4 e^e\right ) \int \frac {x^6}{\left (-3+e^{x^2}+4 x^2+2 x^3\right )^2} \, dx-\left (4 e^e\right ) \int \frac {x^2}{-3+e^{x^2}+4 x^2+2 x^3} \, dx+\left (10 e^e\right ) \int \frac {x^4}{\left (3-e^{x^2}+4 x^2+2 x^3\right )^2} \, dx+\left (10 e^e\right ) \int \frac {x^4}{\left (-3+e^{x^2}+4 x^2+2 x^3\right )^2} \, dx-\left (14 e^e\right ) \int \frac {x^3}{\left (3-e^{x^2}+4 x^2+2 x^3\right )^2} \, dx+\left (16 e^e\right ) \int \frac {x^5}{\left (3-e^{x^2}+4 x^2+2 x^3\right )^2} \, dx+\left (16 e^e\right ) \int \frac {x^5}{\left (-3+e^{x^2}+4 x^2+2 x^3\right )^2} \, dx-\left (26 e^e\right ) \int \frac {x^3}{\left (-3+e^{x^2}+4 x^2+2 x^3\right )^2} \, dx-\left (28 e^e\right ) \int \frac {x^2}{\left (-3+e^{x^2}+4 x^2+2 x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.07, size = 48, normalized size = 1.55 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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]

[Out]

(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)

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fricas [A] time = 0.97, size = 52, normalized size = 1.68 \begin {gather*} \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} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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)-1

6*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+128*x^11+384*x^10+512*x^9+256*x^8-7

2*x^6-288*x^5-288*x^4+81),x, algorithm="fricas")

[Out]

4*(x^5 + 4*x^4 + 4*x^3)*e^e/(4*x^6 + 16*x^5 + 16*x^4 - e^(2*x^2) + 6*e^(x^2) - 9)

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giac [A] time = 3.03, size = 52, normalized size = 1.68 \begin {gather*} \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} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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)-1

6*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+128*x^11+384*x^10+512*x^9+256*x^8-7

2*x^6-288*x^5-288*x^4+81),x, algorithm="giac")

[Out]

4*(x^5 + 4*x^4 + 4*x^3)*e^e/(4*x^6 + 16*x^5 + 16*x^4 - e^(2*x^2) + 6*e^(x^2) - 9)

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maple [A] time = 0.08, size = 50, normalized size = 1.61


method	result	size

risch	\(\frac {4 \,{\mathrm e}^{{\mathrm e}} \left (x^{2}+4 x +4\right ) x^{3}}{4 x^{6}+16 x^{5}+16 x^{4}-{\mathrm e}^{2 x^{2}}+6 \,{\mathrm e}^{x^{2}}-9}\)	\(50\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

-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,method=_RETURNVERBOSE)

[Out]

4*exp(exp(1))*(x^2+4*x+4)*x^3/(4*x^6+16*x^5+16*x^4-exp(2*x^2)+6*exp(x^2)-9)

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maxima [A] time = 0.56, size = 52, normalized size = 1.68 \begin {gather*} \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} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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)-1

6*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+128*x^11+384*x^10+512*x^9+256*x^8-7

2*x^6-288*x^5-288*x^4+81),x, algorithm="maxima")

[Out]

4*(x^5 + 4*x^4 + 4*x^3)*e^e/(4*x^6 + 16*x^5 + 16*x^4 - e^(2*x^2) + 6*e^(x^2) - 9)

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mupad [B] time = 0.68, size = 172, normalized size = 5.55 \begin {gather*} \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 )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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 + 576*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)

[Out]

(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

(exp(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 + 1

1*x^6 - 228*x^7 - 167*x^8 + 40*x^9 + 84*x^10 + 32*x^11 + 4*x^12))

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sympy [B] time = 0.27, size = 61, normalized size = 1.97 \begin {gather*} \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} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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+

128*x**11+384*x**10+512*x**9+256*x**8-72*x**6-288*x**5-288*x**4+81),x)

[Out]

(-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) +

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