3.7.34 \(\int \frac {e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} (e^{4 e^x}+e^4 x^2-3 e^2 x^3+4 x^4+e^{2 e^x} (-2 e^2 x+2 x^2+2 e^x x^3)+e^{\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} (-9 e^{4 e^x} x^2-9 e^4 x^4+36 e^2 x^5-36 x^6+e^{2 e^x} (18 e^2 x^3-36 x^4)))}{e^{4 e^x}+e^4 x^2-4 e^2 x^3+4 x^4+e^{2 e^x} (-2 e^2 x+4 x^2)} \, dx\)

Optimal. Leaf size=35 \[ e^{-\frac {x}{-e^2+\frac {e^{2 e^x}}{x}+2 x}} x-3 x^3 \]

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Rubi [F]  time = 50.32, 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^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (e^{4 e^x}+e^4 x^2-3 e^2 x^3+4 x^4+e^{2 e^x} \left (-2 e^2 x+2 x^2+2 e^x x^3\right )+e^{\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (-9 e^{4 e^x} x^2-9 e^4 x^4+36 e^2 x^5-36 x^6+e^{2 e^x} \left (18 e^2 x^3-36 x^4\right )\right )\right )}{e^{4 e^x}+e^4 x^2-4 e^2 x^3+4 x^4+e^{2 e^x} \left (-2 e^2 x+4 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(4*E^x) + E^4*x^2 - 3*E^2*x^3 + 4*x^4 + E^(2*E^x)*(-2*E^2*x + 2*x^2 + 2*E^x*x^3) + E^(x^2/(E^(2*E^x) -
E^2*x + 2*x^2))*(-9*E^(4*E^x)*x^2 - 9*E^4*x^4 + 36*E^2*x^5 - 36*x^6 + E^(2*E^x)*(18*E^2*x^3 - 36*x^4)))/(E^(x^
2/(E^(2*E^x) - E^2*x + 2*x^2))*(E^(4*E^x) + E^4*x^2 - 4*E^2*x^3 + 4*x^4 + E^(2*E^x)*(-2*E^2*x + 4*x^2))),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (e^{4 e^x}+e^4 x^2-3 e^2 x^3+4 x^4-9 e^{\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^2 \left (e^{2 e^x}-e^2 x+2 x^2\right )^2+2 e^{2 e^x} x \left (-e^2+x+e^x x^2\right )\right )}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx\\ &=\int \left (-9 x^2+\frac {e^{4-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^2}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2}-\frac {3 e^{2-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2}+\frac {e^{4 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}}}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2}+\frac {4 e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^4}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2}+\frac {2 e^{2 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x \left (-e^2+x+e^x x^2\right )}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2}\right ) \, dx\\ &=-3 x^3+2 \int \frac {e^{2 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x \left (-e^2+x+e^x x^2\right )}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx-3 \int \frac {e^{2-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+4 \int \frac {e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^4}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx+\int \frac {e^{4-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^2}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+\int \frac {e^{4 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}}}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx\\ &=-3 x^3+2 \int \left (\frac {e^{2 e^x+x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2}+\frac {e^{2 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x \left (-e^2+x\right )}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2}\right ) \, dx-3 \int \frac {e^{2-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+4 \int \frac {e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^4}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx+\int \frac {e^{4-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^2}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+\int \frac {e^{4 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}}}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx\\ &=-3 x^3+2 \int \frac {e^{2 e^x+x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx+2 \int \frac {e^{2 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x \left (-e^2+x\right )}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx-3 \int \frac {e^{2-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+4 \int \frac {e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^4}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx+\int \frac {e^{4-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^2}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+\int \frac {e^{4 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}}}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx\\ &=-3 x^3+2 \int \frac {e^{2 e^x+x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx+2 \int \left (-\frac {e^{2+2 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2}+\frac {e^{2 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^2}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2}\right ) \, dx-3 \int \frac {e^{2-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+4 \int \frac {e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^4}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx+\int \frac {e^{4-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^2}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+\int \frac {e^{4 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}}}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx\\ &=-3 x^3-2 \int \frac {e^{2+2 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+2 \int \frac {e^{2 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^2}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx+2 \int \frac {e^{2 e^x+x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx-3 \int \frac {e^{2-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^3}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+4 \int \frac {e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^4}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx+\int \frac {e^{4-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x^2}{\left (-e^{2 e^x}+e^2 x-2 x^2\right )^2} \, dx+\int \frac {e^{4 e^x-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}}}{\left (e^{2 e^x}-e^2 x+2 x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.42, size = 36, normalized size = 1.03 \begin {gather*} e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x-3 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4*E^x) + E^4*x^2 - 3*E^2*x^3 + 4*x^4 + E^(2*E^x)*(-2*E^2*x + 2*x^2 + 2*E^x*x^3) + E^(x^2/(E^(2*E
^x) - E^2*x + 2*x^2))*(-9*E^(4*E^x)*x^2 - 9*E^4*x^4 + 36*E^2*x^5 - 36*x^6 + E^(2*E^x)*(18*E^2*x^3 - 36*x^4)))/
(E^(x^2/(E^(2*E^x) - E^2*x + 2*x^2))*(E^(4*E^x) + E^4*x^2 - 4*E^2*x^3 + 4*x^4 + E^(2*E^x)*(-2*E^2*x + 4*x^2)))
,x]

[Out]

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

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fricas [A]  time = 0.68, size = 58, normalized size = 1.66 \begin {gather*} -{\left (3 \, x^{3} e^{\left (\frac {x^{2}}{2 \, x^{2} - x e^{2} + e^{\left (2 \, e^{x}\right )}}\right )} - x\right )} e^{\left (-\frac {x^{2}}{2 \, x^{2} - x e^{2} + e^{\left (2 \, e^{x}\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x^2*exp(exp(x))^4+(18*x^3*exp(2)-36*x^4)*exp(exp(x))^2-9*x^4*exp(2)^2+36*exp(2)*x^5-36*x^6)*exp
(x^2/(exp(exp(x))^2-exp(2)*x+2*x^2))+exp(exp(x))^4+(2*exp(x)*x^3-2*exp(2)*x+2*x^2)*exp(exp(x))^2+x^2*exp(2)^2-
3*x^3*exp(2)+4*x^4)/(exp(exp(x))^4+(-2*exp(2)*x+4*x^2)*exp(exp(x))^2+x^2*exp(2)^2-4*x^3*exp(2)+4*x^4)/exp(x^2/
(exp(exp(x))^2-exp(2)*x+2*x^2)),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 2.72, size = 58, normalized size = 1.66 \begin {gather*} -{\left (3 \, x^{3} e^{\left (\frac {x^{2}}{2 \, x^{2} - x e^{2} + e^{\left (2 \, e^{x}\right )}}\right )} - x\right )} e^{\left (-\frac {x^{2}}{2 \, x^{2} - x e^{2} + e^{\left (2 \, e^{x}\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x^2*exp(exp(x))^4+(18*x^3*exp(2)-36*x^4)*exp(exp(x))^2-9*x^4*exp(2)^2+36*exp(2)*x^5-36*x^6)*exp
(x^2/(exp(exp(x))^2-exp(2)*x+2*x^2))+exp(exp(x))^4+(2*exp(x)*x^3-2*exp(2)*x+2*x^2)*exp(exp(x))^2+x^2*exp(2)^2-
3*x^3*exp(2)+4*x^4)/(exp(exp(x))^4+(-2*exp(2)*x+4*x^2)*exp(exp(x))^2+x^2*exp(2)^2-4*x^3*exp(2)+4*x^4)/exp(x^2/
(exp(exp(x))^2-exp(2)*x+2*x^2)),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.13, size = 33, normalized size = 0.94

method	result	size
risch	\(-3 x^{3}+x \,{\mathrm e}^{\frac {x^{2}}{{\mathrm e}^{2} x -2 x^{2}-{\mathrm e}^{2 \,{\mathrm e}^{x}}}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-9*x^2*exp(exp(x))^4+(18*x^3*exp(2)-36*x^4)*exp(exp(x))^2-9*x^4*exp(2)^2+36*exp(2)*x^5-36*x^6)*exp(x^2/(
exp(exp(x))^2-exp(2)*x+2*x^2))+exp(exp(x))^4+(2*exp(x)*x^3-2*exp(2)*x+2*x^2)*exp(exp(x))^2+x^2*exp(2)^2-3*x^3*
exp(2)+4*x^4)/(exp(exp(x))^4+(-2*exp(2)*x+4*x^2)*exp(exp(x))^2+x^2*exp(2)^2-4*x^3*exp(2)+4*x^4)/exp(x^2/(exp(e
xp(x))^2-exp(2)*x+2*x^2)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -3 \, x^{3} + \int \frac {{\left (4 \, x^{4} - 3 \, x^{3} e^{2} + x^{2} e^{4} + 2 \, {\left (x^{3} e^{x} + x^{2} - x e^{2}\right )} e^{\left (2 \, e^{x}\right )} + e^{\left (4 \, e^{x}\right )}\right )} e^{\left (-\frac {x e^{2}}{2 \, {\left (2 \, x^{2} - x e^{2} + e^{\left (2 \, e^{x}\right )}\right )}} + \frac {e^{\left (2 \, e^{x}\right )}}{2 \, {\left (2 \, x^{2} - x e^{2} + e^{\left (2 \, e^{x}\right )}\right )}}\right )}}{4 \, x^{4} e^{\frac {1}{2}} - 4 \, x^{3} e^{\frac {5}{2}} + x^{2} e^{\frac {9}{2}} + 2 \, {\left (2 \, x^{2} e^{\frac {1}{2}} - x e^{\frac {5}{2}}\right )} e^{\left (2 \, e^{x}\right )} + e^{\left (4 \, e^{x} + \frac {1}{2}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x^2*exp(exp(x))^4+(18*x^3*exp(2)-36*x^4)*exp(exp(x))^2-9*x^4*exp(2)^2+36*exp(2)*x^5-36*x^6)*exp
(x^2/(exp(exp(x))^2-exp(2)*x+2*x^2))+exp(exp(x))^4+(2*exp(x)*x^3-2*exp(2)*x+2*x^2)*exp(exp(x))^2+x^2*exp(2)^2-
3*x^3*exp(2)+4*x^4)/(exp(exp(x))^4+(-2*exp(2)*x+4*x^2)*exp(exp(x))^2+x^2*exp(2)^2-4*x^3*exp(2)+4*x^4)/exp(x^2/
(exp(exp(x))^2-exp(2)*x+2*x^2)),x, algorithm="maxima")

[Out]

-3*x^3 + integrate((4*x^4 - 3*x^3*e^2 + x^2*e^4 + 2*(x^3*e^x + x^2 - x*e^2)*e^(2*e^x) + e^(4*e^x))*e^(-1/2*x*e
^2/(2*x^2 - x*e^2 + e^(2*e^x)) + 1/2*e^(2*e^x)/(2*x^2 - x*e^2 + e^(2*e^x)))/(4*x^4*e^(1/2) - 4*x^3*e^(5/2) + x
^2*e^(9/2) + 2*(2*x^2*e^(1/2) - x*e^(5/2))*e^(2*e^x) + e^(4*e^x + 1/2)), x)

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mupad [B]  time = 1.22, size = 32, normalized size = 0.91 \begin {gather*} x\,{\mathrm {e}}^{-\frac {x^2}{{\mathrm {e}}^{2\,{\mathrm {e}}^x}-x\,{\mathrm {e}}^2+2\,x^2}}-3\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x^2/(exp(2*exp(x)) - x*exp(2) + 2*x^2))*(exp(4*exp(x)) - 3*x^3*exp(2) + x^2*exp(4) - exp(x^2/(exp(2*
exp(x)) - x*exp(2) + 2*x^2))*(9*x^4*exp(4) - 36*x^5*exp(2) - exp(2*exp(x))*(18*x^3*exp(2) - 36*x^4) + 36*x^6 +
9*x^2*exp(4*exp(x))) + 4*x^4 + exp(2*exp(x))*(2*x^3*exp(x) - 2*x*exp(2) + 2*x^2)))/(exp(4*exp(x)) - 4*x^3*exp
(2) + x^2*exp(4) - exp(2*exp(x))*(2*x*exp(2) - 4*x^2) + 4*x^4),x)

[Out]

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

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sympy [A]  time = 3.00, size = 27, normalized size = 0.77 \begin {gather*} - 3 x^{3} + x e^{- \frac {x^{2}}{2 x^{2} - x e^{2} + e^{2 e^{x}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x**2*exp(exp(x))**4+(18*x**3*exp(2)-36*x**4)*exp(exp(x))**2-9*x**4*exp(2)**2+36*exp(2)*x**5-36*
x**6)*exp(x**2/(exp(exp(x))**2-exp(2)*x+2*x**2))+exp(exp(x))**4+(2*exp(x)*x**3-2*exp(2)*x+2*x**2)*exp(exp(x))*
*2+x**2*exp(2)**2-3*x**3*exp(2)+4*x**4)/(exp(exp(x))**4+(-2*exp(2)*x+4*x**2)*exp(exp(x))**2+x**2*exp(2)**2-4*x
**3*exp(2)+4*x**4)/exp(x**2/(exp(exp(x))**2-exp(2)*x+2*x**2)),x)

[Out]

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

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