3.37.40 \(\int \frac {e^{\frac {x^2}{9+e^{2 e^{-x+x^2 \log (x)} x}-6 x^2+x^4+e^{e^{-x+x^2 \log (x)} x} (6-2 x^2)}} (e^{x-x^2 \log (x)} (6 x+2 x^3)+e^{e^{-x+x^2 \log (x)} x} (2 e^{x-x^2 \log (x)} x-2 x^2+2 x^3-2 x^4-4 x^4 \log (x)))}{e^{x+3 e^{-x+x^2 \log (x)} x-x^2 \log (x)}+e^{x+2 e^{-x+x^2 \log (x)} x-x^2 \log (x)} (9-3 x^2)+e^{x+e^{-x+x^2 \log (x)} x-x^2 \log (x)} (27-18 x^2+3 x^4)+e^{x-x^2 \log (x)} (27-27 x^2+9 x^4-x^6)} \, dx\)

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

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Rubi [F]  time = 40.56, 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 {x^2}{9+e^{2 e^{-x+x^2 \log (x)} x}-6 x^2+x^4+e^{e^{-x+x^2 \log (x)} x} \left (6-2 x^2\right )}\right ) \left (e^{x-x^2 \log (x)} \left (6 x+2 x^3\right )+e^{e^{-x+x^2 \log (x)} x} \left (2 e^{x-x^2 \log (x)} x-2 x^2+2 x^3-2 x^4-4 x^4 \log (x)\right )\right )}{e^{x+3 e^{-x+x^2 \log (x)} x-x^2 \log (x)}+e^{x+2 e^{-x+x^2 \log (x)} x-x^2 \log (x)} \left (9-3 x^2\right )+e^{x+e^{-x+x^2 \log (x)} x-x^2 \log (x)} \left (27-18 x^2+3 x^4\right )+e^{x-x^2 \log (x)} \left (27-27 x^2+9 x^4-x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(x^2/(9 + E^(2*E^(-x + x^2*Log[x])*x) - 6*x^2 + x^4 + E^(E^(-x + x^2*Log[x])*x)*(6 - 2*x^2)))*(E^(x - x
^2*Log[x])*(6*x + 2*x^3) + E^(E^(-x + x^2*Log[x])*x)*(2*E^(x - x^2*Log[x])*x - 2*x^2 + 2*x^3 - 2*x^4 - 4*x^4*L
og[x])))/(E^(x + 3*E^(-x + x^2*Log[x])*x - x^2*Log[x]) + E^(x + 2*E^(-x + x^2*Log[x])*x - x^2*Log[x])*(9 - 3*x
^2) + E^(x + E^(-x + x^2*Log[x])*x - x^2*Log[x])*(27 - 18*x^2 + 3*x^4) + E^(x - x^2*Log[x])*(27 - 27*x^2 + 9*x
^4 - x^6)),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.54, size = 30, normalized size = 0.97 \begin {gather*} e^{\frac {x^2}{\left (3+e^{e^{-x} x^{1+x^2}}-x^2\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(x^2/(9 + E^(2*E^(-x + x^2*Log[x])*x) - 6*x^2 + x^4 + E^(E^(-x + x^2*Log[x])*x)*(6 - 2*x^2)))*(E^
(x - x^2*Log[x])*(6*x + 2*x^3) + E^(E^(-x + x^2*Log[x])*x)*(2*E^(x - x^2*Log[x])*x - 2*x^2 + 2*x^3 - 2*x^4 - 4
*x^4*Log[x])))/(E^(x + 3*E^(-x + x^2*Log[x])*x - x^2*Log[x]) + E^(x + 2*E^(-x + x^2*Log[x])*x - x^2*Log[x])*(9
- 3*x^2) + E^(x + E^(-x + x^2*Log[x])*x - x^2*Log[x])*(27 - 18*x^2 + 3*x^4) + E^(x - x^2*Log[x])*(27 - 27*x^2
+ 9*x^4 - x^6)),x]

[Out]

E^(x^2/(3 + E^(x^(1 + x^2)/E^x) - x^2)^2)

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fricas [A]  time = 0.97, size = 53, normalized size = 1.71 \begin {gather*} e^{\left (\frac {x^{2}}{x^{4} - 6 \, x^{2} - 2 \, {\left (x^{2} - 3\right )} e^{\left (x e^{\left (x^{2} \log \relax (x) - x\right )}\right )} + e^{\left (2 \, x e^{\left (x^{2} \log \relax (x) - x\right )}\right )} + 9}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*exp(-x^2*log(x)+x)-4*x^4*log(x)-2*x^4+2*x^3-2*x^2)*exp(x/exp(-x^2*log(x)+x))+(2*x^3+6*x)*exp(-
x^2*log(x)+x))*exp(x^2/(exp(x/exp(-x^2*log(x)+x))^2+(-2*x^2+6)*exp(x/exp(-x^2*log(x)+x))+x^4-6*x^2+9))/(exp(-x
^2*log(x)+x)*exp(x/exp(-x^2*log(x)+x))^3+(-3*x^2+9)*exp(-x^2*log(x)+x)*exp(x/exp(-x^2*log(x)+x))^2+(3*x^4-18*x
^2+27)*exp(-x^2*log(x)+x)*exp(x/exp(-x^2*log(x)+x))+(-x^6+9*x^4-27*x^2+27)*exp(-x^2*log(x)+x)),x, algorithm="f
ricas")

[Out]

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

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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(-x^2*log(x)+x)-4*x^4*log(x)-2*x^4+2*x^3-2*x^2)*exp(x/exp(-x^2*log(x)+x))+(2*x^3+6*x)*exp(-
x^2*log(x)+x))*exp(x^2/(exp(x/exp(-x^2*log(x)+x))^2+(-2*x^2+6)*exp(x/exp(-x^2*log(x)+x))+x^4-6*x^2+9))/(exp(-x
^2*log(x)+x)*exp(x/exp(-x^2*log(x)+x))^3+(-3*x^2+9)*exp(-x^2*log(x)+x)*exp(x/exp(-x^2*log(x)+x))^2+(3*x^4-18*x
^2+27)*exp(-x^2*log(x)+x)*exp(x/exp(-x^2*log(x)+x))+(-x^6+9*x^4-27*x^2+27)*exp(-x^2*log(x)+x)),x, algorithm="g
iac")

[Out]

Timed out

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maple [B]  time = 0.07, size = 74, normalized size = 2.39

method	result	size
risch	\({\mathrm e}^{\frac {x^{2}}{x^{4}-2 \,{\mathrm e}^{x \,x^{x^{2}} {\mathrm e}^{-x}} x^{2}-6 x^{2}+{\mathrm e}^{2 x \,x^{x^{2}} {\mathrm e}^{-x}}+6 \,{\mathrm e}^{x \,x^{x^{2}} {\mathrm e}^{-x}}+9}}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x*exp(-x^2*ln(x)+x)-4*x^4*ln(x)-2*x^4+2*x^3-2*x^2)*exp(x/exp(-x^2*ln(x)+x))+(2*x^3+6*x)*exp(-x^2*ln(x)
+x))*exp(x^2/(exp(x/exp(-x^2*ln(x)+x))^2+(-2*x^2+6)*exp(x/exp(-x^2*ln(x)+x))+x^4-6*x^2+9))/(exp(-x^2*ln(x)+x)*
exp(x/exp(-x^2*ln(x)+x))^3+(-3*x^2+9)*exp(-x^2*ln(x)+x)*exp(x/exp(-x^2*ln(x)+x))^2+(3*x^4-18*x^2+27)*exp(-x^2*
ln(x)+x)*exp(x/exp(-x^2*ln(x)+x))+(-x^6+9*x^4-27*x^2+27)*exp(-x^2*ln(x)+x)),x,method=_RETURNVERBOSE)

[Out]

exp(x^2/(x^4-2*exp(x/(x^(-x^2))*exp(-x))*x^2-6*x^2+exp(2*x/(x^(-x^2))*exp(-x))+6*exp(x/(x^(-x^2))*exp(-x))+9))

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maxima [B]  time = 2.33, size = 138, normalized size = 4.45 \begin {gather*} e^{\left (\frac {e^{\left (x e^{\left (x^{2} \log \relax (x) - x\right )}\right )}}{x^{4} - 6 \, x^{2} - 2 \, {\left (x^{2} - 3\right )} e^{\left (x e^{\left (x^{2} \log \relax (x) - x\right )}\right )} + e^{\left (2 \, x e^{\left (x^{2} \log \relax (x) - x\right )}\right )} + 9} + \frac {3}{x^{4} - 6 \, x^{2} - 2 \, {\left (x^{2} - 3\right )} e^{\left (x e^{\left (x^{2} \log \relax (x) - x\right )}\right )} + e^{\left (2 \, x e^{\left (x^{2} \log \relax (x) - x\right )}\right )} + 9} + \frac {1}{x^{2} - e^{\left (x e^{\left (x^{2} \log \relax (x) - x\right )}\right )} - 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*exp(-x^2*log(x)+x)-4*x^4*log(x)-2*x^4+2*x^3-2*x^2)*exp(x/exp(-x^2*log(x)+x))+(2*x^3+6*x)*exp(-
x^2*log(x)+x))*exp(x^2/(exp(x/exp(-x^2*log(x)+x))^2+(-2*x^2+6)*exp(x/exp(-x^2*log(x)+x))+x^4-6*x^2+9))/(exp(-x
^2*log(x)+x)*exp(x/exp(-x^2*log(x)+x))^3+(-3*x^2+9)*exp(-x^2*log(x)+x)*exp(x/exp(-x^2*log(x)+x))^2+(3*x^4-18*x
^2+27)*exp(-x^2*log(x)+x)*exp(x/exp(-x^2*log(x)+x))+(-x^6+9*x^4-27*x^2+27)*exp(-x^2*log(x)+x)),x, algorithm="m
axima")

[Out]

e^(e^(x*e^(x^2*log(x) - x))/(x^4 - 6*x^2 - 2*(x^2 - 3)*e^(x*e^(x^2*log(x) - x)) + e^(2*x*e^(x^2*log(x) - x)) +
9) + 3/(x^4 - 6*x^2 - 2*(x^2 - 3)*e^(x*e^(x^2*log(x) - x)) + e^(2*x*e^(x^2*log(x) - x)) + 9) + 1/(x^2 - e^(x*
e^(x^2*log(x) - x)) - 3))

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mupad [B]  time = 3.21, size = 64, normalized size = 2.06 \begin {gather*} {\mathrm {e}}^{\frac {x^2}{6\,{\mathrm {e}}^{x^{x^2+1}\,{\mathrm {e}}^{-x}}+{\mathrm {e}}^{2\,x^{x^2+1}\,{\mathrm {e}}^{-x}}-2\,x^2\,{\mathrm {e}}^{x^{x^2+1}\,{\mathrm {e}}^{-x}}-6\,x^2+x^4+9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x^2/(exp(2*x*exp(x^2*log(x) - x)) - 6*x^2 + x^4 - exp(x*exp(x^2*log(x) - x))*(2*x^2 - 6) + 9))*(exp(
x - x^2*log(x))*(6*x + 2*x^3) - exp(x*exp(x^2*log(x) - x))*(4*x^4*log(x) - 2*x*exp(x - x^2*log(x)) + 2*x^2 - 2
*x^3 + 2*x^4)))/(exp(x - x^2*log(x))*(27*x^2 - 9*x^4 + x^6 - 27) - exp(x - x^2*log(x))*exp(3*x*exp(x^2*log(x)
- x)) + exp(x - x^2*log(x))*exp(2*x*exp(x^2*log(x) - x))*(3*x^2 - 9) - exp(x - x^2*log(x))*exp(x*exp(x^2*log(x
) - x))*(3*x^4 - 18*x^2 + 27)),x)

[Out]

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

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sympy [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(-x**2*ln(x)+x)-4*x**4*ln(x)-2*x**4+2*x**3-2*x**2)*exp(x/exp(-x**2*ln(x)+x))+(2*x**3+6*x)*e
xp(-x**2*ln(x)+x))*exp(x**2/(exp(x/exp(-x**2*ln(x)+x))**2+(-2*x**2+6)*exp(x/exp(-x**2*ln(x)+x))+x**4-6*x**2+9)
)/(exp(-x**2*ln(x)+x)*exp(x/exp(-x**2*ln(x)+x))**3+(-3*x**2+9)*exp(-x**2*ln(x)+x)*exp(x/exp(-x**2*ln(x)+x))**2
+(3*x**4-18*x**2+27)*exp(-x**2*ln(x)+x)*exp(x/exp(-x**2*ln(x)+x))+(-x**6+9*x**4-27*x**2+27)*exp(-x**2*ln(x)+x)
),x)

[Out]

Timed out

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