3.86.15 \(\int \frac {1}{3} e^{-e^{e^x}} (-e^{e^{e^x}}-e^{1+x}+e^{e^x} (-6 e^x+e^{1+2 x})) \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{3} \left (e^{-e^{e^x}} \left (6-e^{1+x}\right )-x\right ) \]

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Rubi [F]  time = 0.56, 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 {1}{3} e^{-e^{e^x}} \left (-e^{e^{e^x}}-e^{1+x}+e^{e^x} \left (-6 e^x+e^{1+2 x}\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-E^E^E^x - E^(1 + x) + E^E^x*(-6*E^x + E^(1 + 2*x)))/(3*E^E^E^x),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.16, size = 26, normalized size = 0.96 \begin {gather*} \frac {1}{3} \left (-e^{-e^{e^x}} \left (-6+e^{1+x}\right )-x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-E^E^E^x - E^(1 + x) + E^E^x*(-6*E^x + E^(1 + 2*x)))/(3*E^E^E^x),x]

[Out]

(-((-6 + E^(1 + x))/E^E^E^x) - x)/3

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fricas [A]  time = 0.93, size = 20, normalized size = 0.74 \begin {gather*} -\frac {1}{3} \, {\left (x e^{\left (e^{\left (e^{x}\right )}\right )} + e^{\left (x + 1\right )} - 6\right )} e^{\left (-e^{\left (e^{x}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-exp(exp(exp(x)))+(exp(x)*exp(x+1)-6*exp(x))*exp(exp(x))-exp(x+1))/exp(exp(exp(x))),x, algorith
m="fricas")

[Out]

-1/3*(x*e^(e^(e^x)) + e^(x + 1) - 6)*e^(-e^(e^x))

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giac [A]  time = 0.14, size = 35, normalized size = 1.30 \begin {gather*} -\frac {1}{3} \, {\left (x e^{\left (e^{x}\right )} + e^{\left (x + e^{x} - e^{\left (e^{x}\right )} + 1\right )} - 6 \, e^{\left (e^{x} - e^{\left (e^{x}\right )}\right )}\right )} e^{\left (-e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-exp(exp(exp(x)))+(exp(x)*exp(x+1)-6*exp(x))*exp(exp(x))-exp(x+1))/exp(exp(exp(x))),x, algorith
m="giac")

[Out]

-1/3*(x*e^(e^x) + e^(x + e^x - e^(e^x) + 1) - 6*e^(e^x - e^(e^x)))*e^(-e^x)

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maple [A]  time = 0.12, size = 21, normalized size = 0.78

method	result	size
risch	\(-\frac {x}{3}+\frac {\left (6-{\mathrm e}^{x +1}\right ) {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x}}}}{3}\)	\(21\)
norman	\(\left (2-\frac {{\mathrm e} \,{\mathrm e}^{x}}{3}-\frac {{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}} x}{3}\right ) {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x}}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(-exp(exp(exp(x)))+(exp(x)*exp(x+1)-6*exp(x))*exp(exp(x))-exp(x+1))/exp(exp(exp(x))),x,method=_RETURNV
ERBOSE)

[Out]

-1/3*x+1/3*(6-exp(x+1))*exp(-exp(exp(x)))

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maxima [A]  time = 0.44, size = 18, normalized size = 0.67 \begin {gather*} -\frac {1}{3} \, {\left (e^{\left (x + 1\right )} - 6\right )} e^{\left (-e^{\left (e^{x}\right )}\right )} - \frac {1}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-exp(exp(exp(x)))+(exp(x)*exp(x+1)-6*exp(x))*exp(exp(x))-exp(x+1))/exp(exp(exp(x))),x, algorith
m="maxima")

[Out]

-1/3*(e^(x + 1) - 6)*e^(-e^(e^x)) - 1/3*x

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mupad [B]  time = 5.30, size = 24, normalized size = 0.89 \begin {gather*} 2\,{\mathrm {e}}^{-{\mathrm {e}}^{{\mathrm {e}}^x}}-\frac {x}{3}-\frac {{\mathrm {e}}^{-{\mathrm {e}}^{{\mathrm {e}}^x}}\,\mathrm {e}\,{\mathrm {e}}^x}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.32, size = 19, normalized size = 0.70 \begin {gather*} - \frac {x}{3} + \frac {\left (- e e^{x} + 6\right ) e^{- e^{e^{x}}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-exp(exp(exp(x)))+(exp(x)*exp(x+1)-6*exp(x))*exp(exp(x))-exp(x+1))/exp(exp(exp(x))),x)

[Out]

-x/3 + (-E*exp(x) + 6)*exp(-exp(exp(x)))/3

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