3.16.62 \(\int e^{-e^x+2 e^{e^{-e^x} (e^{e^x} x^2-e^5 x^3)} x^2+e^{-e^x} (e^{e^x} x^2-e^5 x^3)} (-6 e^5 x^4+2 e^{5+x} x^5+e^{e^x} (4 x+4 x^3)) \, dx\)

Optimal. Leaf size=27 \[ e^{2 e^{x \left (x-e^{5-e^x} x^2\right )} x^2} \]

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Rubi [F]  time = 3.38, 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 \exp \left (-e^x+2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )\right ) \left (-6 e^5 x^4+2 e^{5+x} x^5+e^{e^x} \left (4 x+4 x^3\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-6 \exp \left (5-e^x+2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )\right ) x^4+2 \exp \left (5-e^x+x+2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )\right ) x^5+4 \exp \left (2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )\right ) x \left (1+x^2\right )\right ) \, dx\\ &=2 \int \exp \left (5-e^x+x+2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )\right ) x^5 \, dx+4 \int \exp \left (2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )\right ) x \left (1+x^2\right ) \, dx-6 \int \exp \left (5-e^x+2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )\right ) x^4 \, dx\\ &=2 \int \exp \left (5-e^x+x+2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )\right ) x^5 \, dx+4 \int \exp \left (x^2+2 e^{x^2-e^{5-e^x} x^3} x^2-e^{5-e^x} x^3\right ) x \left (1+x^2\right ) \, dx-6 \int \exp \left (5-e^x+2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )\right ) x^4 \, dx\\ &=2 \int \exp \left (5-e^x+x+2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )\right ) x^5 \, dx+4 \int \left (\exp \left (x^2+2 e^{x^2-e^{5-e^x} x^3} x^2-e^{5-e^x} x^3\right ) x+\exp \left (x^2+2 e^{x^2-e^{5-e^x} x^3} x^2-e^{5-e^x} x^3\right ) x^3\right ) \, dx-6 \int \exp \left (5-e^x+2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )\right ) x^4 \, dx\\ &=2 \int \exp \left (5-e^x+x+2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )\right ) x^5 \, dx+4 \int \exp \left (x^2+2 e^{x^2-e^{5-e^x} x^3} x^2-e^{5-e^x} x^3\right ) x \, dx+4 \int \exp \left (x^2+2 e^{x^2-e^{5-e^x} x^3} x^2-e^{5-e^x} x^3\right ) x^3 \, dx-6 \int \exp \left (5-e^x+2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )\right ) x^4 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 27, normalized size = 1.00 \begin {gather*} e^{2 e^{x^2-e^{5-e^x} x^3} x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(2*E^(x^2 - E^(5 - E^x)*x^3)*x^2)

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fricas [B]  time = 1.14, size = 91, normalized size = 3.37 \begin {gather*} e^{\left ({\left (x^{3} e^{5} - x^{2} e^{\left (e^{x}\right )}\right )} e^{\left (-e^{x}\right )} - {\left (x^{3} e^{10} - 2 \, x^{2} e^{\left (-{\left (x^{3} e^{5} - x^{2} e^{\left (e^{x}\right )}\right )} e^{\left (-e^{x}\right )} + e^{x} + 5\right )} - {\left (x^{2} e^{5} - e^{\left (x + 5\right )}\right )} e^{\left (e^{x}\right )}\right )} e^{\left (-e^{x} - 5\right )} + e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^((x^3*e^5 - x^2*e^(e^x))*e^(-e^x) - (x^3*e^10 - 2*x^2*e^(-(x^3*e^5 - x^2*e^(e^x))*e^(-e^x) + e^x + 5) - (x^2
*e^5 - e^(x + 5))*e^(e^x))*e^(-e^x - 5) + e^x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.08, size = 28, normalized size = 1.04

method	result	size
risch	\({\mathrm e}^{2 x^{2} {\mathrm e}^{-x^{2} \left (x \,{\mathrm e}^{5}-{\mathrm e}^{{\mathrm e}^{x}}\right ) {\mathrm e}^{-{\mathrm e}^{x}}}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(2*x^2*exp(-x^2*(x*exp(5)-exp(exp(x)))*exp(-exp(x))))

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maxima [A]  time = 1.00, size = 23, normalized size = 0.85 \begin {gather*} e^{\left (2 \, x^{2} e^{\left (-x^{3} e^{\left (-e^{x} + 5\right )} + x^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(2*x^2*e^(-x^3*e^(-e^x + 5) + x^2))

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mupad [B]  time = 1.25, size = 23, normalized size = 0.85 \begin {gather*} {\mathrm {e}}^{2\,x^2\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-x^3\,{\mathrm {e}}^5\,{\mathrm {e}}^{-{\mathrm {e}}^x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x^2*exp(exp(-exp(x))*(x^2*exp(exp(x)) - x^3*exp(5))))*exp(-exp(x))*exp(exp(-exp(x))*(x^2*exp(exp(x))
- x^3*exp(5)))*(exp(exp(x))*(4*x + 4*x^3) - 6*x^4*exp(5) + 2*x^5*exp(5)*exp(x)),x)

[Out]

exp(2*x^2*exp(x^2)*exp(-x^3*exp(5)*exp(-exp(x))))

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sympy [A]  time = 5.69, size = 27, normalized size = 1.00 \begin {gather*} e^{2 x^{2} e^{\left (- x^{3} e^{5} + x^{2} e^{e^{x}}\right ) e^{- e^{x}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(2*x**2*exp((-x**3*exp(5) + x**2*exp(exp(x)))*exp(-exp(x))))

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