3.83.86 \(\int \frac {2 e^{6-3 e^x+3 e^{2 x-x^2}} x-6 e^{4-2 e^x+2 e^{2 x-x^2}} x^2-e^4 x^3-2 x^4+e^{-e^x+e^{2 x-x^2}} (3 e^6 x^2+6 e^2 x^3+2 e^{6+x} x^3+e^{6+2 x-x^2} (-4 x^3+4 x^4))}{4 e^{6-3 e^x+3 e^{2 x-x^2}}-12 e^{4-2 e^x+2 e^{2 x-x^2}} x+12 e^{2-e^x+e^{2 x-x^2}} x^2-4 x^3} \, dx\)

Optimal. Leaf size=38 \[ \frac {1}{4} x \left (x+\frac {x^2}{\left (-e^{-e^x+e^{(2-x) x}}+\frac {x}{e^2}\right )^2}\right ) \]

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Rubi [F]  time = 22.71, 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 {2 e^{6-3 e^x+3 e^{2 x-x^2}} x-6 e^{4-2 e^x+2 e^{2 x-x^2}} x^2-e^4 x^3-2 x^4+e^{-e^x+e^{2 x-x^2}} \left (3 e^6 x^2+6 e^2 x^3+2 e^{6+x} x^3+e^{6+2 x-x^2} \left (-4 x^3+4 x^4\right )\right )}{4 e^{6-3 e^x+3 e^{2 x-x^2}}-12 e^{4-2 e^x+2 e^{2 x-x^2}} x+12 e^{2-e^x+e^{2 x-x^2}} x^2-4 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*E^(6 - 3*E^x + 3*E^(2*x - x^2))*x - 6*E^(4 - 2*E^x + 2*E^(2*x - x^2))*x^2 - E^4*x^3 - 2*x^4 + E^(-E^x +
E^(2*x - x^2))*(3*E^6*x^2 + 6*E^2*x^3 + 2*E^(6 + x)*x^3 + E^(6 + 2*x - x^2)*(-4*x^3 + 4*x^4)))/(4*E^(6 - 3*E^
x + 3*E^(2*x - x^2)) - 12*E^(4 - 2*E^x + 2*E^(2*x - x^2))*x + 12*E^(2 - E^x + E^(2*x - x^2))*x^2 - 4*x^3),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{3 e^x} \left (2 e^{6-3 e^x+3 e^{2 x-x^2}} x-6 e^{4-2 e^x+2 e^{2 x-x^2}} x^2-e^4 x^3-2 x^4+e^{-e^x+e^{2 x-x^2}} \left (3 e^6 x^2+6 e^2 x^3+2 e^{6+x} x^3+e^{6+2 x-x^2} \left (-4 x^3+4 x^4\right )\right )\right )}{4 \left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx\\ &=\frac {1}{4} \int \frac {e^{3 e^x} \left (2 e^{6-3 e^x+3 e^{2 x-x^2}} x-6 e^{4-2 e^x+2 e^{2 x-x^2}} x^2-e^4 x^3-2 x^4+e^{-e^x+e^{2 x-x^2}} \left (3 e^6 x^2+6 e^2 x^3+2 e^{6+x} x^3+e^{6+2 x-x^2} \left (-4 x^3+4 x^4\right )\right )\right )}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx\\ &=\frac {1}{4} \int \left (\frac {2 e^{6+3 e^{-((-2+x) x)}} x}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3}+\frac {3 e^{6+2 e^x+e^{-((-2+x) x)}} x^2}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3}-\frac {6 e^{4+e^x+2 e^{-((-2+x) x)}} x^2}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3}-\frac {e^{4+3 e^x} x^3}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3}+\frac {6 e^{2+2 e^x+e^{-((-2+x) x)}} x^3}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3}+\frac {2 e^{6+2 e^x+e^{-((-2+x) x)}+x} x^3}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3}+\frac {4 e^{6+2 e^x+e^{-((-2+x) x)}+2 x-x^2} (-1+x) x^3}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3}+\frac {2 e^{3 e^x} x^4}{\left (-e^{2+e^{-((-2+x) x)}}+e^{e^x} x\right )^3}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {e^{4+3 e^x} x^3}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx\right )+\frac {1}{2} \int \frac {e^{6+3 e^{-((-2+x) x)}} x}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx+\frac {1}{2} \int \frac {e^{6+2 e^x+e^{-((-2+x) x)}+x} x^3}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx+\frac {1}{2} \int \frac {e^{3 e^x} x^4}{\left (-e^{2+e^{-((-2+x) x)}}+e^{e^x} x\right )^3} \, dx+\frac {3}{4} \int \frac {e^{6+2 e^x+e^{-((-2+x) x)}} x^2}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx-\frac {3}{2} \int \frac {e^{4+e^x+2 e^{-((-2+x) x)}} x^2}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx+\frac {3}{2} \int \frac {e^{2+2 e^x+e^{-((-2+x) x)}} x^3}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx+\int \frac {e^{6+2 e^x+e^{-((-2+x) x)}+2 x-x^2} (-1+x) x^3}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx\\ &=-\left (\frac {1}{4} \int \frac {e^{4+3 e^x} x^3}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx\right )+\frac {1}{2} \int \frac {e^{6+3 e^{-((-2+x) x)}} x}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx+\frac {1}{2} \int \frac {e^{6+2 e^x+e^{-((-2+x) x)}+x} x^3}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx+\frac {1}{2} \int \frac {e^{3 e^x} x^4}{\left (-e^{2+e^{-((-2+x) x)}}+e^{e^x} x\right )^3} \, dx+\frac {3}{4} \int \frac {e^{6+2 e^x+e^{-((-2+x) x)}} x^2}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx-\frac {3}{2} \int \frac {e^{4+e^x+2 e^{-((-2+x) x)}} x^2}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx+\frac {3}{2} \int \frac {e^{2+2 e^x+e^{-((-2+x) x)}} x^3}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx+\int \left (\frac {e^{6+2 e^x+e^{-((-2+x) x)}+2 x-x^2} x^3}{\left (-e^{2+e^{-((-2+x) x)}}+e^{e^x} x\right )^3}-\frac {e^{6+2 e^x+e^{-((-2+x) x)}+2 x-x^2} x^4}{\left (-e^{2+e^{-((-2+x) x)}}+e^{e^x} x\right )^3}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {e^{4+3 e^x} x^3}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx\right )+\frac {1}{2} \int \frac {e^{6+3 e^{-((-2+x) x)}} x}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx+\frac {1}{2} \int \frac {e^{6+2 e^x+e^{-((-2+x) x)}+x} x^3}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx+\frac {1}{2} \int \frac {e^{3 e^x} x^4}{\left (-e^{2+e^{-((-2+x) x)}}+e^{e^x} x\right )^3} \, dx+\frac {3}{4} \int \frac {e^{6+2 e^x+e^{-((-2+x) x)}} x^2}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx-\frac {3}{2} \int \frac {e^{4+e^x+2 e^{-((-2+x) x)}} x^2}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx+\frac {3}{2} \int \frac {e^{2+2 e^x+e^{-((-2+x) x)}} x^3}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^3} \, dx+\int \frac {e^{6+2 e^x+e^{-((-2+x) x)}+2 x-x^2} x^3}{\left (-e^{2+e^{-((-2+x) x)}}+e^{e^x} x\right )^3} \, dx-\int \frac {e^{6+2 e^x+e^{-((-2+x) x)}+2 x-x^2} x^4}{\left (-e^{2+e^{-((-2+x) x)}}+e^{e^x} x\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.49, size = 44, normalized size = 1.16 \begin {gather*} \frac {1}{4} \left (x^2+\frac {e^{4+2 e^x} x^3}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*E^(6 - 3*E^x + 3*E^(2*x - x^2))*x - 6*E^(4 - 2*E^x + 2*E^(2*x - x^2))*x^2 - E^4*x^3 - 2*x^4 + E^(
-E^x + E^(2*x - x^2))*(3*E^6*x^2 + 6*E^2*x^3 + 2*E^(6 + x)*x^3 + E^(6 + 2*x - x^2)*(-4*x^3 + 4*x^4)))/(4*E^(6
- 3*E^x + 3*E^(2*x - x^2)) - 12*E^(4 - 2*E^x + 2*E^(2*x - x^2))*x + 12*E^(2 - E^x + E^(2*x - x^2))*x^2 - 4*x^3
),x]

[Out]

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

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fricas [B]  time = 0.59, size = 136, normalized size = 3.58 \begin {gather*} \frac {x^{4} + x^{3} e^{4} - 2 \, x^{3} e^{\left ({\left (2 \, e^{6} + e^{\left (-x^{2} + 2 \, x + 6\right )} - e^{\left (x + 6\right )}\right )} e^{\left (-6\right )}\right )} + x^{2} e^{\left (2 \, {\left (2 \, e^{6} + e^{\left (-x^{2} + 2 \, x + 6\right )} - e^{\left (x + 6\right )}\right )} e^{\left (-6\right )}\right )}}{4 \, {\left (x^{2} - 2 \, x e^{\left ({\left (2 \, e^{6} + e^{\left (-x^{2} + 2 \, x + 6\right )} - e^{\left (x + 6\right )}\right )} e^{\left (-6\right )}\right )} + e^{\left (2 \, {\left (2 \, e^{6} + e^{\left (-x^{2} + 2 \, x + 6\right )} - e^{\left (x + 6\right )}\right )} e^{\left (-6\right )}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(2)^3*exp(-exp(x)+exp(-x^2+2*x))^3-6*x^2*exp(2)^2*exp(-exp(x)+exp(-x^2+2*x))^2+(2*x^3*exp(2)
^3*exp(x)+(4*x^4-4*x^3)*exp(2)^3*exp(-x^2+2*x)+3*x^2*exp(2)^3+6*x^3*exp(2))*exp(-exp(x)+exp(-x^2+2*x))-x^3*exp
(2)^2-2*x^4)/(4*exp(2)^3*exp(-exp(x)+exp(-x^2+2*x))^3-12*x*exp(2)^2*exp(-exp(x)+exp(-x^2+2*x))^2+12*x^2*exp(2)
*exp(-exp(x)+exp(-x^2+2*x))-4*x^3),x, algorithm="fricas")

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(2)^3*exp(-exp(x)+exp(-x^2+2*x))^3-6*x^2*exp(2)^2*exp(-exp(x)+exp(-x^2+2*x))^2+(2*x^3*exp(2)
^3*exp(x)+(4*x^4-4*x^3)*exp(2)^3*exp(-x^2+2*x)+3*x^2*exp(2)^3+6*x^3*exp(2))*exp(-exp(x)+exp(-x^2+2*x))-x^3*exp
(2)^2-2*x^4)/(4*exp(2)^3*exp(-exp(x)+exp(-x^2+2*x))^3-12*x*exp(2)^2*exp(-exp(x)+exp(-x^2+2*x))^2+12*x^2*exp(2)
*exp(-exp(x)+exp(-x^2+2*x))-4*x^3),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.09, size = 34, normalized size = 0.89

method	result	size
risch	\(\frac {x^{2}}{4}+\frac {x^{3} {\mathrm e}^{4}}{4 \left ({\mathrm e}^{-{\mathrm e}^{x}+{\mathrm e}^{-\left (x -2\right ) x}+2}-x \right )^{2}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*exp(2)^3*exp(-exp(x)+exp(-x^2+2*x))^3-6*x^2*exp(2)^2*exp(-exp(x)+exp(-x^2+2*x))^2+(2*x^3*exp(2)^3*exp
(x)+(4*x^4-4*x^3)*exp(2)^3*exp(-x^2+2*x)+3*x^2*exp(2)^3+6*x^3*exp(2))*exp(-exp(x)+exp(-x^2+2*x))-x^3*exp(2)^2-
2*x^4)/(4*exp(2)^3*exp(-exp(x)+exp(-x^2+2*x))^3-12*x*exp(2)^2*exp(-exp(x)+exp(-x^2+2*x))^2+12*x^2*exp(2)*exp(-
exp(x)+exp(-x^2+2*x))-4*x^3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.52, size = 105, normalized size = 2.76 \begin {gather*} -\frac {2 \, x^{3} e^{\left (e^{\left (-x^{2} + 2 \, x\right )} + e^{x} + 2\right )} - x^{2} e^{\left (2 \, e^{\left (-x^{2} + 2 \, x\right )} + 4\right )} - {\left (x^{4} + x^{3} e^{4}\right )} e^{\left (2 \, e^{x}\right )}}{4 \, {\left (x^{2} e^{\left (2 \, e^{x}\right )} - 2 \, x e^{\left (e^{\left (-x^{2} + 2 \, x\right )} + e^{x} + 2\right )} + e^{\left (2 \, e^{\left (-x^{2} + 2 \, x\right )} + 4\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(2)^3*exp(-exp(x)+exp(-x^2+2*x))^3-6*x^2*exp(2)^2*exp(-exp(x)+exp(-x^2+2*x))^2+(2*x^3*exp(2)
^3*exp(x)+(4*x^4-4*x^3)*exp(2)^3*exp(-x^2+2*x)+3*x^2*exp(2)^3+6*x^3*exp(2))*exp(-exp(x)+exp(-x^2+2*x))-x^3*exp
(2)^2-2*x^4)/(4*exp(2)^3*exp(-exp(x)+exp(-x^2+2*x))^3-12*x*exp(2)^2*exp(-exp(x)+exp(-x^2+2*x))^2+12*x^2*exp(2)
*exp(-exp(x)+exp(-x^2+2*x))-4*x^3),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.41, size = 102, normalized size = 2.68 \begin {gather*} \frac {x^2\,\left (x+{\mathrm {e}}^{2\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-x^2}-2\,{\mathrm {e}}^x}-2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-x^2}-{\mathrm {e}}^x-2}+x^2\,{\mathrm {e}}^{-4}\right )}{4\,\left ({\mathrm {e}}^{2\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-x^2}-2\,{\mathrm {e}}^x}-2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-x^2}-{\mathrm {e}}^x-2}+x^2\,{\mathrm {e}}^{-4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*exp(4) - exp(exp(2*x - x^2) - exp(x))*(6*x^3*exp(2) + 3*x^2*exp(6) - exp(6)*exp(2*x - x^2)*(4*x^3 - 4
*x^4) + 2*x^3*exp(6)*exp(x)) + 2*x^4 - 2*x*exp(3*exp(2*x - x^2) - 3*exp(x))*exp(6) + 6*x^2*exp(2*exp(2*x - x^2
) - 2*exp(x))*exp(4))/(4*x^3 - 4*exp(3*exp(2*x - x^2) - 3*exp(x))*exp(6) + 12*x*exp(2*exp(2*x - x^2) - 2*exp(x
))*exp(4) - 12*x^2*exp(2)*exp(exp(2*x - x^2) - exp(x))),x)

[Out]

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

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sympy [B]  time = 0.37, size = 58, normalized size = 1.53 \begin {gather*} \frac {x^{3} e^{4}}{4 x^{2} - 8 x e^{2} e^{- e^{x} + e^{- x^{2} + 2 x}} + 4 e^{4} e^{- 2 e^{x} + 2 e^{- x^{2} + 2 x}}} + \frac {x^{2}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(2)**3*exp(-exp(x)+exp(-x**2+2*x))**3-6*x**2*exp(2)**2*exp(-exp(x)+exp(-x**2+2*x))**2+(2*x**
3*exp(2)**3*exp(x)+(4*x**4-4*x**3)*exp(2)**3*exp(-x**2+2*x)+3*x**2*exp(2)**3+6*x**3*exp(2))*exp(-exp(x)+exp(-x
**2+2*x))-x**3*exp(2)**2-2*x**4)/(4*exp(2)**3*exp(-exp(x)+exp(-x**2+2*x))**3-12*x*exp(2)**2*exp(-exp(x)+exp(-x
**2+2*x))**2+12*x**2*exp(2)*exp(-exp(x)+exp(-x**2+2*x))-4*x**3),x)

[Out]

x**3*exp(4)/(4*x**2 - 8*x*exp(2)*exp(-exp(x) + exp(-x**2 + 2*x)) + 4*exp(4)*exp(-2*exp(x) + 2*exp(-x**2 + 2*x)
)) + x**2/4

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