3.93.28 \(\int \frac {1}{8} e^{-2 e^5+e^{\frac {1}{16} e^{-2 e^5} (16 e^{2+2 e^5}-4 x^4+4 x^5-x^6+(-4 x^3+2 x^4) \log (5)-x^2 \log ^2(5))}+\frac {1}{16} e^{-2 e^5} (16 e^{2+2 e^5}-4 x^4+4 x^5-x^6+(-4 x^3+2 x^4) \log (5)-x^2 \log ^2(5))} (-8 x^3+10 x^4-3 x^5+(-6 x^2+4 x^3) \log (5)-x \log ^2(5)) \, dx\)

Optimal. Leaf size=35 \[ e^{e^{e^2-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2}} \]

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Rubi [F]  time = 56.24, 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 {1}{8} \exp \left (-2 e^5+\exp \left (\frac {1}{16} e^{-2 e^5} \left (16 e^{2+2 e^5}-4 x^4+4 x^5-x^6+\left (-4 x^3+2 x^4\right ) \log (5)-x^2 \log ^2(5)\right )\right )+\frac {1}{16} e^{-2 e^5} \left (16 e^{2+2 e^5}-4 x^4+4 x^5-x^6+\left (-4 x^3+2 x^4\right ) \log (5)-x^2 \log ^2(5)\right )\right ) \left (-8 x^3+10 x^4-3 x^5+\left (-6 x^2+4 x^3\right ) \log (5)-x \log ^2(5)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-2*E^5 + E^((16*E^(2 + 2*E^5) - 4*x^4 + 4*x^5 - x^6 + (-4*x^3 + 2*x^4)*Log[5] - x^2*Log[5]^2)/(16*E^(2
*E^5))) + (16*E^(2 + 2*E^5) - 4*x^4 + 4*x^5 - x^6 + (-4*x^3 + 2*x^4)*Log[5] - x^2*Log[5]^2)/(16*E^(2*E^5)))*(-
8*x^3 + 10*x^4 - 3*x^5 + (-6*x^2 + 4*x^3)*Log[5] - x*Log[5]^2))/8,x]

[Out]

-1/8*(Log[5]^2*Defer[Int][E^(E^(E^2 - (x^2*(2*x - x^2 + Log[5])^2)/(16*E^(2*E^5))) + E^2*(1 - 2*E^3) - (x^2*(2
*x - x^2 + Log[5])^2)/(16*E^(2*E^5)))*x, x]) - (3*Log[5]*Defer[Int][E^(E^(E^2 - (x^2*(2*x - x^2 + Log[5])^2)/(
16*E^(2*E^5))) + E^2*(1 - 2*E^3) - (x^2*(2*x - x^2 + Log[5])^2)/(16*E^(2*E^5)))*x^2, x])/4 - ((2 - Log[5])*Def
er[Int][E^(E^(E^2 - (x^2*(2*x - x^2 + Log[5])^2)/(16*E^(2*E^5))) + E^2*(1 - 2*E^3) - (x^2*(2*x - x^2 + Log[5])
^2)/(16*E^(2*E^5)))*x^3, x])/2 + (5*Defer[Int][E^(E^(E^2 - (x^2*(2*x - x^2 + Log[5])^2)/(16*E^(2*E^5))) + E^2*
(1 - 2*E^3) - (x^2*(2*x - x^2 + Log[5])^2)/(16*E^(2*E^5)))*x^4, x])/4 - (3*Defer[Int][E^(E^(E^2 - (x^2*(2*x -
x^2 + Log[5])^2)/(16*E^(2*E^5))) + E^2*(1 - 2*E^3) - (x^2*(2*x - x^2 + Log[5])^2)/(16*E^(2*E^5)))*x^5, x])/8

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{8} \int \exp \left (-2 e^5+\exp \left (\frac {1}{16} e^{-2 e^5} \left (16 e^{2+2 e^5}-4 x^4+4 x^5-x^6+\left (-4 x^3+2 x^4\right ) \log (5)-x^2 \log ^2(5)\right )\right )+\frac {1}{16} e^{-2 e^5} \left (16 e^{2+2 e^5}-4 x^4+4 x^5-x^6+\left (-4 x^3+2 x^4\right ) \log (5)-x^2 \log ^2(5)\right )\right ) \left (-8 x^3+10 x^4-3 x^5+\left (-6 x^2+4 x^3\right ) \log (5)-x \log ^2(5)\right ) \, dx\\ &=\frac {1}{8} \int \exp \left (\exp \left (e^2-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right )+e^2 \left (1-2 e^3\right )-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right ) x \left (10 x^3-3 x^4-4 x^2 (2-\log (5))-6 x \log (5)-\log ^2(5)\right ) \, dx\\ &=\frac {1}{8} \int \left (10 \exp \left (\exp \left (e^2-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right )+e^2 \left (1-2 e^3\right )-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right ) x^4-3 \exp \left (\exp \left (e^2-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right )+e^2 \left (1-2 e^3\right )-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right ) x^5+4 \exp \left (\exp \left (e^2-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right )+e^2 \left (1-2 e^3\right )-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right ) x^3 (-2+\log (5))-6 \exp \left (\exp \left (e^2-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right )+e^2 \left (1-2 e^3\right )-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right ) x^2 \log (5)-\exp \left (\exp \left (e^2-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right )+e^2 \left (1-2 e^3\right )-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right ) x \log ^2(5)\right ) \, dx\\ &=-\left (\frac {3}{8} \int \exp \left (\exp \left (e^2-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right )+e^2 \left (1-2 e^3\right )-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right ) x^5 \, dx\right )+\frac {5}{4} \int \exp \left (\exp \left (e^2-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right )+e^2 \left (1-2 e^3\right )-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right ) x^4 \, dx+\frac {1}{2} (-2+\log (5)) \int \exp \left (\exp \left (e^2-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right )+e^2 \left (1-2 e^3\right )-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right ) x^3 \, dx-\frac {1}{4} (3 \log (5)) \int \exp \left (\exp \left (e^2-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right )+e^2 \left (1-2 e^3\right )-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right ) x^2 \, dx-\frac {1}{8} \log ^2(5) \int \exp \left (\exp \left (e^2-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right )+e^2 \left (1-2 e^3\right )-\frac {1}{16} e^{-2 e^5} x^2 \left (2 x-x^2+\log (5)\right )^2\right ) x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.40, size = 60, normalized size = 1.71 \begin {gather*} e^{5^{\frac {1}{8} e^{-2 e^5} (-2+x) x^3} e^{e^2-\frac {1}{16} e^{-2 e^5} x^2 \left (4 x^2-4 x^3+x^4+\log ^2(5)\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-2*E^5 + E^((16*E^(2 + 2*E^5) - 4*x^4 + 4*x^5 - x^6 + (-4*x^3 + 2*x^4)*Log[5] - x^2*Log[5]^2)/(1
6*E^(2*E^5))) + (16*E^(2 + 2*E^5) - 4*x^4 + 4*x^5 - x^6 + (-4*x^3 + 2*x^4)*Log[5] - x^2*Log[5]^2)/(16*E^(2*E^5
)))*(-8*x^3 + 10*x^4 - 3*x^5 + (-6*x^2 + 4*x^3)*Log[5] - x*Log[5]^2))/8,x]

[Out]

E^(5^(((-2 + x)*x^3)/(8*E^(2*E^5)))*E^(E^2 - (x^2*(4*x^2 - 4*x^3 + x^4 + Log[5]^2))/(16*E^(2*E^5))))

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fricas [B]  time = 0.90, size = 174, normalized size = 4.97 \begin {gather*} e^{\left (-\frac {1}{16} \, {\left (x^{6} - 4 \, x^{5} + 4 \, x^{4} + x^{2} \log \relax (5)^{2} + 16 \, {\left (2 \, e^{5} - e^{2}\right )} e^{\left (2 \, e^{5}\right )} - 2 \, {\left (x^{4} - 2 \, x^{3}\right )} \log \relax (5) - 16 \, e^{\left (-\frac {1}{16} \, {\left (x^{6} - 4 \, x^{5} + 4 \, x^{4} + x^{2} \log \relax (5)^{2} - 2 \, {\left (x^{4} - 2 \, x^{3}\right )} \log \relax (5) - 16 \, e^{\left (2 \, e^{5} + 2\right )}\right )} e^{\left (-2 \, e^{5}\right )} + 2 \, e^{5}\right )}\right )} e^{\left (-2 \, e^{5}\right )} + \frac {1}{16} \, {\left (x^{6} - 4 \, x^{5} + 4 \, x^{4} + x^{2} \log \relax (5)^{2} - 2 \, {\left (x^{4} - 2 \, x^{3}\right )} \log \relax (5) - 16 \, e^{\left (2 \, e^{5} + 2\right )}\right )} e^{\left (-2 \, e^{5}\right )} + 2 \, e^{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(-x*log(5)^2+(4*x^3-6*x^2)*log(5)-3*x^5+10*x^4-8*x^3)*exp(1/16*(16*exp(2)*exp(exp(5))^2-x^2*log(
5)^2+(2*x^4-4*x^3)*log(5)-x^6+4*x^5-4*x^4)/exp(exp(5))^2)*exp(exp(1/16*(16*exp(2)*exp(exp(5))^2-x^2*log(5)^2+(
2*x^4-4*x^3)*log(5)-x^6+4*x^5-4*x^4)/exp(exp(5))^2))/exp(exp(5))^2,x, algorithm="fricas")

[Out]

e^(-1/16*(x^6 - 4*x^5 + 4*x^4 + x^2*log(5)^2 + 16*(2*e^5 - e^2)*e^(2*e^5) - 2*(x^4 - 2*x^3)*log(5) - 16*e^(-1/
16*(x^6 - 4*x^5 + 4*x^4 + x^2*log(5)^2 - 2*(x^4 - 2*x^3)*log(5) - 16*e^(2*e^5 + 2))*e^(-2*e^5) + 2*e^5))*e^(-2
*e^5) + 1/16*(x^6 - 4*x^5 + 4*x^4 + x^2*log(5)^2 - 2*(x^4 - 2*x^3)*log(5) - 16*e^(2*e^5 + 2))*e^(-2*e^5) + 2*e
^5)

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giac [B]  time = 0.41, size = 73, normalized size = 2.09 \begin {gather*} e^{\left (e^{\left (-\frac {1}{16} \, x^{6} e^{\left (-2 \, e^{5}\right )} + \frac {1}{4} \, x^{5} e^{\left (-2 \, e^{5}\right )} + \frac {1}{8} \, x^{4} e^{\left (-2 \, e^{5}\right )} \log \relax (5) - \frac {1}{4} \, x^{4} e^{\left (-2 \, e^{5}\right )} - \frac {1}{4} \, x^{3} e^{\left (-2 \, e^{5}\right )} \log \relax (5) - \frac {1}{16} \, x^{2} e^{\left (-2 \, e^{5}\right )} \log \relax (5)^{2} + e^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(-x*log(5)^2+(4*x^3-6*x^2)*log(5)-3*x^5+10*x^4-8*x^3)*exp(1/16*(16*exp(2)*exp(exp(5))^2-x^2*log(
5)^2+(2*x^4-4*x^3)*log(5)-x^6+4*x^5-4*x^4)/exp(exp(5))^2)*exp(exp(1/16*(16*exp(2)*exp(exp(5))^2-x^2*log(5)^2+(
2*x^4-4*x^3)*log(5)-x^6+4*x^5-4*x^4)/exp(exp(5))^2))/exp(exp(5))^2,x, algorithm="giac")

[Out]

e^(e^(-1/16*x^6*e^(-2*e^5) + 1/4*x^5*e^(-2*e^5) + 1/8*x^4*e^(-2*e^5)*log(5) - 1/4*x^4*e^(-2*e^5) - 1/4*x^3*e^(
-2*e^5)*log(5) - 1/16*x^2*e^(-2*e^5)*log(5)^2 + e^2))

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maple [A]  time = 0.23, size = 55, normalized size = 1.57

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{-\frac {\left (x^{6}-2 x^{4} \ln \relax (5)-4 x^{5}+x^{2} \ln \relax (5)^{2}+4 x^{3} \ln \relax (5)+4 x^{4}-16 \,{\mathrm e}^{2 \,{\mathrm e}^{5}+2}\right ) {\mathrm e}^{-2 \,{\mathrm e}^{5}}}{16}}}\)	\(55\)
norman	\({\mathrm e}^{{\mathrm e}^{\frac {\left (16 \,{\mathrm e}^{2} {\mathrm e}^{2 \,{\mathrm e}^{5}}-x^{2} \ln \relax (5)^{2}+\left (2 x^{4}-4 x^{3}\right ) \ln \relax (5)-x^{6}+4 x^{5}-4 x^{4}\right ) {\mathrm e}^{-2 \,{\mathrm e}^{5}}}{16}}}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/8*(-x*ln(5)^2+(4*x^3-6*x^2)*ln(5)-3*x^5+10*x^4-8*x^3)*exp(1/16*(16*exp(2)*exp(exp(5))^2-x^2*ln(5)^2+(2*x
^4-4*x^3)*ln(5)-x^6+4*x^5-4*x^4)/exp(exp(5))^2)*exp(exp(1/16*(16*exp(2)*exp(exp(5))^2-x^2*ln(5)^2+(2*x^4-4*x^3
)*ln(5)-x^6+4*x^5-4*x^4)/exp(exp(5))^2))/exp(exp(5))^2,x,method=_RETURNVERBOSE)

[Out]

exp(exp(-1/16*(x^6-2*x^4*ln(5)-4*x^5+x^2*ln(5)^2+4*x^3*ln(5)+4*x^4-16*exp(2*exp(5)+2))*exp(-2*exp(5))))

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maxima [B]  time = 2.17, size = 73, normalized size = 2.09 \begin {gather*} e^{\left (e^{\left (-\frac {1}{16} \, x^{6} e^{\left (-2 \, e^{5}\right )} + \frac {1}{4} \, x^{5} e^{\left (-2 \, e^{5}\right )} + \frac {1}{8} \, x^{4} e^{\left (-2 \, e^{5}\right )} \log \relax (5) - \frac {1}{4} \, x^{4} e^{\left (-2 \, e^{5}\right )} - \frac {1}{4} \, x^{3} e^{\left (-2 \, e^{5}\right )} \log \relax (5) - \frac {1}{16} \, x^{2} e^{\left (-2 \, e^{5}\right )} \log \relax (5)^{2} + e^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(-x*log(5)^2+(4*x^3-6*x^2)*log(5)-3*x^5+10*x^4-8*x^3)*exp(1/16*(16*exp(2)*exp(exp(5))^2-x^2*log(
5)^2+(2*x^4-4*x^3)*log(5)-x^6+4*x^5-4*x^4)/exp(exp(5))^2)*exp(exp(1/16*(16*exp(2)*exp(exp(5))^2-x^2*log(5)^2+(
2*x^4-4*x^3)*log(5)-x^6+4*x^5-4*x^4)/exp(exp(5))^2))/exp(exp(5))^2,x, algorithm="maxima")

[Out]

e^(e^(-1/16*x^6*e^(-2*e^5) + 1/4*x^5*e^(-2*e^5) + 1/8*x^4*e^(-2*e^5)*log(5) - 1/4*x^4*e^(-2*e^5) - 1/4*x^3*e^(
-2*e^5)*log(5) - 1/16*x^2*e^(-2*e^5)*log(5)^2 + e^2))

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mupad [B]  time = 10.16, size = 75, normalized size = 2.14 \begin {gather*} {\mathrm {e}}^{\frac {{\mathrm {e}}^{-\frac {x^2\,{\mathrm {e}}^{-2\,{\mathrm {e}}^5}\,{\ln \relax (5)}^2}{16}}\,{\mathrm {e}}^{-\frac {x^4\,{\mathrm {e}}^{-2\,{\mathrm {e}}^5}}{4}}\,{\mathrm {e}}^{\frac {x^5\,{\mathrm {e}}^{-2\,{\mathrm {e}}^5}}{4}}\,{\mathrm {e}}^{-\frac {x^6\,{\mathrm {e}}^{-2\,{\mathrm {e}}^5}}{16}}\,{\mathrm {e}}^{{\mathrm {e}}^2}}{5^{\frac {{\mathrm {e}}^{-2\,{\mathrm {e}}^5}\,\left (2\,x^3-x^4\right )}{8}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-2*exp(5))*exp(-exp(-2*exp(5))*((x^2*log(5)^2)/16 + (log(5)*(4*x^3 - 2*x^4))/16 + x^4/4 - x^5/4 + x^
6/16 - exp(2*exp(5))*exp(2)))*exp(exp(-exp(-2*exp(5))*((x^2*log(5)^2)/16 + (log(5)*(4*x^3 - 2*x^4))/16 + x^4/4
- x^5/4 + x^6/16 - exp(2*exp(5))*exp(2))))*(log(5)*(6*x^2 - 4*x^3) + x*log(5)^2 + 8*x^3 - 10*x^4 + 3*x^5))/8,
x)

[Out]

exp((exp(-(x^2*exp(-2*exp(5))*log(5)^2)/16)*exp(-(x^4*exp(-2*exp(5)))/4)*exp((x^5*exp(-2*exp(5)))/4)*exp(-(x^6
*exp(-2*exp(5)))/16)*exp(exp(2)))/5^((exp(-2*exp(5))*(2*x^3 - x^4))/8))

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sympy [B]  time = 0.75, size = 60, normalized size = 1.71 \begin {gather*} e^{e^{\frac {- \frac {x^{6}}{16} + \frac {x^{5}}{4} - \frac {x^{4}}{4} - \frac {x^{2} \log {\relax (5 )}^{2}}{16} + \frac {\left (2 x^{4} - 4 x^{3}\right ) \log {\relax (5 )}}{16} + e^{2} e^{2 e^{5}}}{e^{2 e^{5}}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(-x*ln(5)**2+(4*x**3-6*x**2)*ln(5)-3*x**5+10*x**4-8*x**3)*exp(1/16*(16*exp(2)*exp(exp(5))**2-x**
2*ln(5)**2+(2*x**4-4*x**3)*ln(5)-x**6+4*x**5-4*x**4)/exp(exp(5))**2)*exp(exp(1/16*(16*exp(2)*exp(exp(5))**2-x*
*2*ln(5)**2+(2*x**4-4*x**3)*ln(5)-x**6+4*x**5-4*x**4)/exp(exp(5))**2))/exp(exp(5))**2,x)

[Out]

exp(exp((-x**6/16 + x**5/4 - x**4/4 - x**2*log(5)**2/16 + (2*x**4 - 4*x**3)*log(5)/16 + exp(2)*exp(2*exp(5)))*
exp(-2*exp(5))))

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