3.86.68 \(\int \frac {e^{\frac {4+8 x+4 x^2}{x^2+e^{x^2} (2 x-2 x^2)+e^{2 x^2} (1-2 x+x^2)}} (8+8 x+e^{x^2} (-16+16 x^2-16 x^3-16 x^4))}{-x^3+e^{2 x^2} (-3 x+6 x^2-3 x^3)+e^{3 x^2} (-1+3 x-3 x^2+x^3)+e^{x^2} (-3 x^2+3 x^3)} \, dx\)

Optimal. Leaf size=26 \[ -16+e^{\frac {4 (1+x)^2}{\left (e^{x^2} (-1+x)-x\right )^2}} \]

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Rubi [F]  time = 10.75, 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 {\exp \left (\frac {4+8 x+4 x^2}{x^2+e^{x^2} \left (2 x-2 x^2\right )+e^{2 x^2} \left (1-2 x+x^2\right )}\right ) \left (8+8 x+e^{x^2} \left (-16+16 x^2-16 x^3-16 x^4\right )\right )}{-x^3+e^{2 x^2} \left (-3 x+6 x^2-3 x^3\right )+e^{3 x^2} \left (-1+3 x-3 x^2+x^3\right )+e^{x^2} \left (-3 x^2+3 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((4 + 8*x + 4*x^2)/(x^2 + E^x^2*(2*x - 2*x^2) + E^(2*x^2)*(1 - 2*x + x^2)))*(8 + 8*x + E^x^2*(-16 + 16*
x^2 - 16*x^3 - 16*x^4)))/(-x^3 + E^(2*x^2)*(-3*x + 6*x^2 - 3*x^3) + E^(3*x^2)*(-1 + 3*x - 3*x^2 + x^3) + E^x^2
*(-3*x^2 + 3*x^3)),x]

[Out]

-24*Defer[Int][E^((4*(1 + x)^2)/(-(E^x^2*(-1 + x)) + x)^2)/(-E^x^2 - x + E^x^2*x)^3, x] - 32*Defer[Int][E^((4*
(1 + x)^2)/(-(E^x^2*(-1 + x)) + x)^2)/((-1 + x)*(-E^x^2 - x + E^x^2*x)^3), x] - 8*Defer[Int][(E^((4*(1 + x)^2)
/(-(E^x^2*(-1 + x)) + x)^2)*x)/(-E^x^2 - x + E^x^2*x)^3, x] - 16*Defer[Int][(E^((4*(1 + x)^2)/(-(E^x^2*(-1 + x
)) + x)^2)*x^2)/(-E^x^2 - x + E^x^2*x)^3, x] - 32*Defer[Int][(E^((4*(1 + x)^2)/(-(E^x^2*(-1 + x)) + x)^2)*x^3)
/(-E^x^2 - x + E^x^2*x)^3, x] - 16*Defer[Int][(E^((4*(1 + x)^2)/(-(E^x^2*(-1 + x)) + x)^2)*x^4)/(-E^x^2 - x +
E^x^2*x)^3, x] - 16*Defer[Int][E^((4*(1 + x)^2)/(-(E^x^2*(-1 + x)) + x)^2)/(-E^x^2 - x + E^x^2*x)^2, x] - 32*D
efer[Int][E^((4*(1 + x)^2)/(-(E^x^2*(-1 + x)) + x)^2)/((-1 + x)*(-E^x^2 - x + E^x^2*x)^2), x] - 16*Defer[Int][
(E^((4*(1 + x)^2)/(-(E^x^2*(-1 + x)) + x)^2)*x)/(-E^x^2 - x + E^x^2*x)^2, x] - 32*Defer[Int][(E^((4*(1 + x)^2)
/(-(E^x^2*(-1 + x)) + x)^2)*x^2)/(-E^x^2 - x + E^x^2*x)^2, x] - 16*Defer[Int][(E^((4*(1 + x)^2)/(-(E^x^2*(-1 +
x)) + x)^2)*x^3)/(-E^x^2 - x + E^x^2*x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} (1+x) \left (1-2 e^{x^2} \left (1-x+x^3\right )\right )}{\left (e^{x^2} (-1+x)-x\right )^3} \, dx\\ &=8 \int \frac {e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} (1+x) \left (1-2 e^{x^2} \left (1-x+x^3\right )\right )}{\left (e^{x^2} (-1+x)-x\right )^3} \, dx\\ &=8 \int \left (-\frac {e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} (1+x)^2 \left (1-2 x^2+2 x^3\right )}{(-1+x) \left (-e^{x^2}-x+e^{x^2} x\right )^3}-\frac {2 e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} \left (1-x^2+x^3+x^4\right )}{(-1+x) \left (-e^{x^2}-x+e^{x^2} x\right )^2}\right ) \, dx\\ &=-\left (8 \int \frac {e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} (1+x)^2 \left (1-2 x^2+2 x^3\right )}{(-1+x) \left (-e^{x^2}-x+e^{x^2} x\right )^3} \, dx\right )-16 \int \frac {e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} \left (1-x^2+x^3+x^4\right )}{(-1+x) \left (-e^{x^2}-x+e^{x^2} x\right )^2} \, dx\\ &=-\left (8 \int \left (\frac {3 e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}}}{\left (-e^{x^2}-x+e^{x^2} x\right )^3}+\frac {4 e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}}}{(-1+x) \left (-e^{x^2}-x+e^{x^2} x\right )^3}+\frac {e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} x}{\left (-e^{x^2}-x+e^{x^2} x\right )^3}+\frac {2 e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} x^2}{\left (-e^{x^2}-x+e^{x^2} x\right )^3}+\frac {4 e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} x^3}{\left (-e^{x^2}-x+e^{x^2} x\right )^3}+\frac {2 e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} x^4}{\left (-e^{x^2}-x+e^{x^2} x\right )^3}\right ) \, dx\right )-16 \int \left (\frac {e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}}}{\left (-e^{x^2}-x+e^{x^2} x\right )^2}+\frac {2 e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}}}{(-1+x) \left (-e^{x^2}-x+e^{x^2} x\right )^2}+\frac {e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} x}{\left (-e^{x^2}-x+e^{x^2} x\right )^2}+\frac {2 e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} x^2}{\left (-e^{x^2}-x+e^{x^2} x\right )^2}+\frac {e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} x^3}{\left (-e^{x^2}-x+e^{x^2} x\right )^2}\right ) \, dx\\ &=-\left (8 \int \frac {e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} x}{\left (-e^{x^2}-x+e^{x^2} x\right )^3} \, dx\right )-16 \int \frac {e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} x^2}{\left (-e^{x^2}-x+e^{x^2} x\right )^3} \, dx-16 \int \frac {e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} x^4}{\left (-e^{x^2}-x+e^{x^2} x\right )^3} \, dx-16 \int \frac {e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}}}{\left (-e^{x^2}-x+e^{x^2} x\right )^2} \, dx-16 \int \frac {e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} x}{\left (-e^{x^2}-x+e^{x^2} x\right )^2} \, dx-16 \int \frac {e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} x^3}{\left (-e^{x^2}-x+e^{x^2} x\right )^2} \, dx-24 \int \frac {e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}}}{\left (-e^{x^2}-x+e^{x^2} x\right )^3} \, dx-32 \int \frac {e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}}}{(-1+x) \left (-e^{x^2}-x+e^{x^2} x\right )^3} \, dx-32 \int \frac {e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} x^3}{\left (-e^{x^2}-x+e^{x^2} x\right )^3} \, dx-32 \int \frac {e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}}}{(-1+x) \left (-e^{x^2}-x+e^{x^2} x\right )^2} \, dx-32 \int \frac {e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} x^2}{\left (-e^{x^2}-x+e^{x^2} x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.78, size = 23, normalized size = 0.88 \begin {gather*} e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((4 + 8*x + 4*x^2)/(x^2 + E^x^2*(2*x - 2*x^2) + E^(2*x^2)*(1 - 2*x + x^2)))*(8 + 8*x + E^x^2*(-16
+ 16*x^2 - 16*x^3 - 16*x^4)))/(-x^3 + E^(2*x^2)*(-3*x + 6*x^2 - 3*x^3) + E^(3*x^2)*(-1 + 3*x - 3*x^2 + x^3) +
E^x^2*(-3*x^2 + 3*x^3)),x]

[Out]

E^((4*(1 + x)^2)/(-(E^x^2*(-1 + x)) + x)^2)

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fricas [A]  time = 1.11, size = 45, normalized size = 1.73 \begin {gather*} e^{\left (\frac {4 \, {\left (x^{2} + 2 \, x + 1\right )}}{x^{2} + {\left (x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x^{2}\right )} - 2 \, {\left (x^{2} - x\right )} e^{\left (x^{2}\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^4-16*x^3+16*x^2-16)*exp(x^2)+8*x+8)*exp((4*x^2+8*x+4)/((x^2-2*x+1)*exp(x^2)^2+(-2*x^2+2*x)*e
xp(x^2)+x^2))/((x^3-3*x^2+3*x-1)*exp(x^2)^3+(-3*x^3+6*x^2-3*x)*exp(x^2)^2+(3*x^3-3*x^2)*exp(x^2)-x^3),x, algor
ithm="fricas")

[Out]

e^(4*(x^2 + 2*x + 1)/(x^2 + (x^2 - 2*x + 1)*e^(2*x^2) - 2*(x^2 - x)*e^(x^2)))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {8 \, {\left (2 \, {\left (x^{4} + x^{3} - x^{2} + 1\right )} e^{\left (x^{2}\right )} - x - 1\right )} e^{\left (\frac {4 \, {\left (x^{2} + 2 \, x + 1\right )}}{x^{2} + {\left (x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x^{2}\right )} - 2 \, {\left (x^{2} - x\right )} e^{\left (x^{2}\right )}}\right )}}{x^{3} - {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )} e^{\left (3 \, x^{2}\right )} + 3 \, {\left (x^{3} - 2 \, x^{2} + x\right )} e^{\left (2 \, x^{2}\right )} - 3 \, {\left (x^{3} - x^{2}\right )} e^{\left (x^{2}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^4-16*x^3+16*x^2-16)*exp(x^2)+8*x+8)*exp((4*x^2+8*x+4)/((x^2-2*x+1)*exp(x^2)^2+(-2*x^2+2*x)*e
xp(x^2)+x^2))/((x^3-3*x^2+3*x-1)*exp(x^2)^3+(-3*x^3+6*x^2-3*x)*exp(x^2)^2+(3*x^3-3*x^2)*exp(x^2)-x^3),x, algor
ithm="giac")

[Out]

integrate(8*(2*(x^4 + x^3 - x^2 + 1)*e^(x^2) - x - 1)*e^(4*(x^2 + 2*x + 1)/(x^2 + (x^2 - 2*x + 1)*e^(2*x^2) -
2*(x^2 - x)*e^(x^2)))/(x^3 - (x^3 - 3*x^2 + 3*x - 1)*e^(3*x^2) + 3*(x^3 - 2*x^2 + x)*e^(2*x^2) - 3*(x^3 - x^2)
*e^(x^2)), x)

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maple [B]  time = 0.17, size = 56, normalized size = 2.15

method	result	size
risch	\({\mathrm e}^{\frac {4 \left (x +1\right )^{2}}{-2 x^{2} {\mathrm e}^{x^{2}}+{\mathrm e}^{2 x^{2}} x^{2}+2 \,{\mathrm e}^{x^{2}} x -2 x \,{\mathrm e}^{2 x^{2}}+x^{2}+{\mathrm e}^{2 x^{2}}}}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-16*x^4-16*x^3+16*x^2-16)*exp(x^2)+8*x+8)*exp((4*x^2+8*x+4)/((x^2-2*x+1)*exp(x^2)^2+(-2*x^2+2*x)*exp(x^2
)+x^2))/((x^3-3*x^2+3*x-1)*exp(x^2)^3+(-3*x^3+6*x^2-3*x)*exp(x^2)^2+(3*x^3-3*x^2)*exp(x^2)-x^3),x,method=_RETU
RNVERBOSE)

[Out]

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

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maxima [B]  time = 1.01, size = 267, normalized size = 10.27 \begin {gather*} e^{\left (\frac {4 \, e^{\left (2 \, x^{2}\right )}}{x^{2} + {\left (x^{2} - 2 \, x + 1\right )} e^{\left (4 \, x^{2}\right )} - 2 \, {\left (2 \, x^{2} - 3 \, x + 1\right )} e^{\left (3 \, x^{2}\right )} + {\left (6 \, x^{2} - 6 \, x + 1\right )} e^{\left (2 \, x^{2}\right )} - 2 \, {\left (2 \, x^{2} - x\right )} e^{\left (x^{2}\right )}} - \frac {8 \, e^{\left (x^{2}\right )}}{x^{2} - {\left (x^{2} - 2 \, x + 1\right )} e^{\left (3 \, x^{2}\right )} + {\left (3 \, x^{2} - 4 \, x + 1\right )} e^{\left (2 \, x^{2}\right )} - {\left (3 \, x^{2} - 2 \, x\right )} e^{\left (x^{2}\right )}} + \frac {8 \, e^{\left (x^{2}\right )}}{{\left (x - 1\right )} e^{\left (3 \, x^{2}\right )} - {\left (3 \, x - 2\right )} e^{\left (2 \, x^{2}\right )} + {\left (3 \, x - 1\right )} e^{\left (x^{2}\right )} - x} + \frac {4}{x^{2} + {\left (x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x^{2}\right )} - 2 \, {\left (x^{2} - x\right )} e^{\left (x^{2}\right )}} + \frac {8}{{\left (x - 1\right )} e^{\left (2 \, x^{2}\right )} - {\left (2 \, x - 1\right )} e^{\left (x^{2}\right )} + x} + \frac {4}{e^{\left (2 \, x^{2}\right )} - 2 \, e^{\left (x^{2}\right )} + 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^4-16*x^3+16*x^2-16)*exp(x^2)+8*x+8)*exp((4*x^2+8*x+4)/((x^2-2*x+1)*exp(x^2)^2+(-2*x^2+2*x)*e
xp(x^2)+x^2))/((x^3-3*x^2+3*x-1)*exp(x^2)^3+(-3*x^3+6*x^2-3*x)*exp(x^2)^2+(3*x^3-3*x^2)*exp(x^2)-x^3),x, algor
ithm="maxima")

[Out]

e^(4*e^(2*x^2)/(x^2 + (x^2 - 2*x + 1)*e^(4*x^2) - 2*(2*x^2 - 3*x + 1)*e^(3*x^2) + (6*x^2 - 6*x + 1)*e^(2*x^2)
- 2*(2*x^2 - x)*e^(x^2)) - 8*e^(x^2)/(x^2 - (x^2 - 2*x + 1)*e^(3*x^2) + (3*x^2 - 4*x + 1)*e^(2*x^2) - (3*x^2 -
2*x)*e^(x^2)) + 8*e^(x^2)/((x - 1)*e^(3*x^2) - (3*x - 2)*e^(2*x^2) + (3*x - 1)*e^(x^2) - x) + 4/(x^2 + (x^2 -
2*x + 1)*e^(2*x^2) - 2*(x^2 - x)*e^(x^2)) + 8/((x - 1)*e^(2*x^2) - (2*x - 1)*e^(x^2) + x) + 4/(e^(2*x^2) - 2*
e^(x^2) + 1))

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mupad [B]  time = 5.42, size = 155, normalized size = 5.96 \begin {gather*} {\mathrm {e}}^{\frac {8\,x}{{\mathrm {e}}^{2\,x^2}+2\,x\,{\mathrm {e}}^{x^2}-2\,x\,{\mathrm {e}}^{2\,x^2}-2\,x^2\,{\mathrm {e}}^{x^2}+x^2\,{\mathrm {e}}^{2\,x^2}+x^2}}\,{\mathrm {e}}^{\frac {4\,x^2}{{\mathrm {e}}^{2\,x^2}+2\,x\,{\mathrm {e}}^{x^2}-2\,x\,{\mathrm {e}}^{2\,x^2}-2\,x^2\,{\mathrm {e}}^{x^2}+x^2\,{\mathrm {e}}^{2\,x^2}+x^2}}\,{\mathrm {e}}^{\frac {4}{{\mathrm {e}}^{2\,x^2}+2\,x\,{\mathrm {e}}^{x^2}-2\,x\,{\mathrm {e}}^{2\,x^2}-2\,x^2\,{\mathrm {e}}^{x^2}+x^2\,{\mathrm {e}}^{2\,x^2}+x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((8*x + 4*x^2 + 4)/(exp(x^2)*(2*x - 2*x^2) + exp(2*x^2)*(x^2 - 2*x + 1) + x^2))*(8*x - exp(x^2)*(16*x
^3 - 16*x^2 + 16*x^4 + 16) + 8))/(exp(x^2)*(3*x^2 - 3*x^3) - exp(3*x^2)*(3*x - 3*x^2 + x^3 - 1) + exp(2*x^2)*(
3*x - 6*x^2 + 3*x^3) + x^3),x)

[Out]

exp((8*x)/(exp(2*x^2) + 2*x*exp(x^2) - 2*x*exp(2*x^2) - 2*x^2*exp(x^2) + x^2*exp(2*x^2) + x^2))*exp((4*x^2)/(e
xp(2*x^2) + 2*x*exp(x^2) - 2*x*exp(2*x^2) - 2*x^2*exp(x^2) + x^2*exp(2*x^2) + x^2))*exp(4/(exp(2*x^2) + 2*x*ex
p(x^2) - 2*x*exp(2*x^2) - 2*x^2*exp(x^2) + x^2*exp(2*x^2) + x^2))

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sympy [B]  time = 0.95, size = 42, normalized size = 1.62 \begin {gather*} e^{\frac {4 x^{2} + 8 x + 4}{x^{2} + \left (- 2 x^{2} + 2 x\right ) e^{x^{2}} + \left (x^{2} - 2 x + 1\right ) e^{2 x^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x**4-16*x**3+16*x**2-16)*exp(x**2)+8*x+8)*exp((4*x**2+8*x+4)/((x**2-2*x+1)*exp(x**2)**2+(-2*x*
*2+2*x)*exp(x**2)+x**2))/((x**3-3*x**2+3*x-1)*exp(x**2)**3+(-3*x**3+6*x**2-3*x)*exp(x**2)**2+(3*x**3-3*x**2)*e
xp(x**2)-x**3),x)

[Out]

exp((4*x**2 + 8*x + 4)/(x**2 + (-2*x**2 + 2*x)*exp(x**2) + (x**2 - 2*x + 1)*exp(2*x**2)))

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