3.25.42 \(\int \frac {1}{3} e^{e^{\frac {1}{3} (-3+e^{5 e^{e^x}-5 x}+6 x)}+\frac {1}{3} (-3+e^{5 e^{e^x}-5 x}+6 x)} (6+e^{5 e^{e^x}-5 x} (-5+5 e^{e^x+x})) \, dx\)

Optimal. Leaf size=26 \[ e^{e^{-1+\frac {1}{3} e^{5 \left (e^{e^x}-x\right )}+2 x}} \]

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Rubi [F]  time = 1.46, 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}{3} \exp \left (e^{\frac {1}{3} \left (-3+e^{5 e^{e^x}-5 x}+6 x\right )}+\frac {1}{3} \left (-3+e^{5 e^{e^x}-5 x}+6 x\right )\right ) \left (6+e^{5 e^{e^x}-5 x} \left (-5+5 e^{e^x+x}\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.15, size = 26, normalized size = 1.00 \begin {gather*} e^{e^{-1+\frac {1}{3} e^{5 e^{e^x}-5 x}+2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^E^(-1 + E^(5*E^E^x - 5*x)/3 + 2*x)

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fricas [A]  time = 1.21, size = 28, normalized size = 1.08 \begin {gather*} e^{\left (e^{\left (2 \, x + \frac {1}{3} \, e^{\left (-5 \, {\left (x e^{x} - e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )}\right )} - 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((5*exp(x)*exp(exp(x))-5)*exp(5*exp(exp(x))-5*x)+6)*exp(1/3*exp(5*exp(exp(x))-5*x)+2*x-1)*exp(ex
p(1/3*exp(5*exp(exp(x))-5*x)+2*x-1)),x, algorithm="fricas")

[Out]

e^(e^(2*x + 1/3*e^(-5*(x*e^x - e^(x + e^x))*e^(-x)) - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{3} \, {\left (5 \, {\left (e^{\left (x + e^{x}\right )} - 1\right )} e^{\left (-5 \, x + 5 \, e^{\left (e^{x}\right )}\right )} + 6\right )} e^{\left (2 \, x + e^{\left (2 \, x + \frac {1}{3} \, e^{\left (-5 \, x + 5 \, e^{\left (e^{x}\right )}\right )} - 1\right )} + \frac {1}{3} \, e^{\left (-5 \, x + 5 \, e^{\left (e^{x}\right )}\right )} - 1\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((5*exp(x)*exp(exp(x))-5)*exp(5*exp(exp(x))-5*x)+6)*exp(1/3*exp(5*exp(exp(x))-5*x)+2*x-1)*exp(ex
p(1/3*exp(5*exp(exp(x))-5*x)+2*x-1)),x, algorithm="giac")

[Out]

integrate(1/3*(5*(e^(x + e^x) - 1)*e^(-5*x + 5*e^(e^x)) + 6)*e^(2*x + e^(2*x + 1/3*e^(-5*x + 5*e^(e^x)) - 1) +
1/3*e^(-5*x + 5*e^(e^x)) - 1), x)

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maple [A]  time = 0.17, size = 20, normalized size = 0.77

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{\frac {{\mathrm e}^{5 \,{\mathrm e}^{{\mathrm e}^{x}}-5 x}}{3}+2 x -1}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((5*exp(x)*exp(exp(x))-5)*exp(5*exp(exp(x))-5*x)+6)*exp(1/3*exp(5*exp(exp(x))-5*x)+2*x-1)*exp(exp(1/3*
exp(5*exp(exp(x))-5*x)+2*x-1)),x,method=_RETURNVERBOSE)

[Out]

exp(exp(1/3*exp(5*exp(exp(x))-5*x)+2*x-1))

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maxima [A]  time = 1.19, size = 19, normalized size = 0.73 \begin {gather*} e^{\left (e^{\left (2 \, x + \frac {1}{3} \, e^{\left (-5 \, x + 5 \, e^{\left (e^{x}\right )}\right )} - 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((5*exp(x)*exp(exp(x))-5)*exp(5*exp(exp(x))-5*x)+6)*exp(1/3*exp(5*exp(exp(x))-5*x)+2*x-1)*exp(ex
p(1/3*exp(5*exp(exp(x))-5*x)+2*x-1)),x, algorithm="maxima")

[Out]

e^(e^(2*x + 1/3*e^(-5*x + 5*e^(e^x)) - 1))

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mupad [B]  time = 1.57, size = 21, normalized size = 0.81 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^{\frac {{\mathrm {e}}^{5\,{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^{-5\,x}}{3}}\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x + exp(5*exp(exp(x)) - 5*x)/3 - 1)*exp(exp(2*x + exp(5*exp(exp(x)) - 5*x)/3 - 1))*(exp(5*exp(exp(x
)) - 5*x)*(5*exp(exp(x))*exp(x) - 5) + 6))/3,x)

[Out]

exp(exp((exp(5*exp(exp(x)))*exp(-5*x))/3)*exp(2*x)*exp(-1))

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sympy [A]  time = 14.91, size = 20, normalized size = 0.77 \begin {gather*} e^{e^{2 x + \frac {e^{- 5 x + 5 e^{e^{x}}}}{3} - 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((5*exp(x)*exp(exp(x))-5)*exp(5*exp(exp(x))-5*x)+6)*exp(1/3*exp(5*exp(exp(x))-5*x)+2*x-1)*exp(ex
p(1/3*exp(5*exp(exp(x))-5*x)+2*x-1)),x)

[Out]

exp(exp(2*x + exp(-5*x + 5*exp(exp(x)))/3 - 1))

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