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

Optimal. Leaf size=25 \[ \frac {4}{3}+e^{e^{e^{-3 x/2} x \left (-1+\frac {\log (3)}{x}\right )}} \]

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Rubi [F]  time = 3.17, 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 (e^{e^{\frac {1}{2} (-3 x+2 \log (-x+\log (3)))}}+e^{\frac {1}{2} (-3 x+2 \log (-x+\log (3)))}+\frac {1}{2} (-3 x+2 \log (-x+\log (3)))\right ) (-2+3 x-3 \log (3))}{-2 x+2 \log (3)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-1/2*((2 + Log[27])*Defer[Int][3^E^((-3*x)/2)*E^(3^E^((-3*x)/2)/E^(x/E^((3*x)/2)) - (3*x)/2 - x/E^((3*x)/2)),
x]) + Defer[Int][3^(1 + E^((-3*x)/2))*E^(3^E^((-3*x)/2)/E^(x/E^((3*x)/2)) - (3*x)/2 - x/E^((3*x)/2))*x, x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1}{2} 3^{e^{-3 x/2}} \exp \left (3^{e^{-3 x/2}} e^{-e^{-3 x/2} x}-\frac {3 x}{2}-e^{-3 x/2} x\right ) (-2+3 x-3 \log (3)) \, dx\\ &=\frac {1}{2} \int 3^{e^{-3 x/2}} \exp \left (3^{e^{-3 x/2}} e^{-e^{-3 x/2} x}-\frac {3 x}{2}-e^{-3 x/2} x\right ) (-2+3 x-3 \log (3)) \, dx\\ &=\frac {1}{2} \int \left (3^{1+e^{-3 x/2}} \exp \left (3^{e^{-3 x/2}} e^{-e^{-3 x/2} x}-\frac {3 x}{2}-e^{-3 x/2} x\right ) x-2\ 3^{e^{-3 x/2}} \exp \left (3^{e^{-3 x/2}} e^{-e^{-3 x/2} x}-\frac {3 x}{2}-e^{-3 x/2} x\right ) \left (1+\frac {3 \log (3)}{2}\right )\right ) \, dx\\ &=\frac {1}{2} \int 3^{1+e^{-3 x/2}} \exp \left (3^{e^{-3 x/2}} e^{-e^{-3 x/2} x}-\frac {3 x}{2}-e^{-3 x/2} x\right ) x \, dx+\frac {1}{2} (-2-\log (27)) \int 3^{e^{-3 x/2}} \exp \left (3^{e^{-3 x/2}} e^{-e^{-3 x/2} x}-\frac {3 x}{2}-e^{-3 x/2} x\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.27, size = 24, normalized size = 0.96 \begin {gather*} e^{3^{e^{-3 x/2}} e^{-e^{-3 x/2} x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(3^E^((-3*x)/2)/E^(x/E^((3*x)/2)))

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fricas [A]  time = 0.68, size = 14, normalized size = 0.56 \begin {gather*} e^{\left (e^{\left (e^{\left (-\frac {3}{2} \, x + \log \left (-x + \log \relax (3)\right )\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(3)+3*x-2)*exp(log(log(3)-x)-3/2*x)*exp(exp(log(log(3)-x)-3/2*x))*exp(exp(exp(log(log(3)-x)-3
/2*x)))/(2*log(3)-2*x),x, algorithm="fricas")

[Out]

e^(e^(e^(-3/2*x + log(-x + log(3)))))

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giac [A]  time = 0.20, size = 17, normalized size = 0.68 \begin {gather*} e^{\left (e^{\left (-x e^{\left (-\frac {3}{2} \, x\right )} + e^{\left (-\frac {3}{2} \, x\right )} \log \relax (3)\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(3)+3*x-2)*exp(log(log(3)-x)-3/2*x)*exp(exp(log(log(3)-x)-3/2*x))*exp(exp(exp(log(log(3)-x)-3
/2*x)))/(2*log(3)-2*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.14, size = 14, normalized size = 0.56

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{\left (\ln \relax (3)-x \right ) {\mathrm e}^{-\frac {3 x}{2}}}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*ln(3)+3*x-2)*exp(ln(ln(3)-x)-3/2*x)*exp(exp(ln(ln(3)-x)-3/2*x))*exp(exp(exp(ln(ln(3)-x)-3/2*x)))/(2*ln
(3)-2*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.76, size = 17, normalized size = 0.68 \begin {gather*} e^{\left (e^{\left (-x e^{\left (-\frac {3}{2} \, x\right )} + e^{\left (-\frac {3}{2} \, x\right )} \log \relax (3)\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(3)+3*x-2)*exp(log(log(3)-x)-3/2*x)*exp(exp(log(log(3)-x)-3/2*x))*exp(exp(exp(log(log(3)-x)-3
/2*x)))/(2*log(3)-2*x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.45, size = 16, normalized size = 0.64 \begin {gather*} {\mathrm {e}}^{3^{{\mathrm {e}}^{-\frac {3\,x}{2}}}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{-\frac {3\,x}{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(log(log(3) - x) - (3*x)/2))*exp(exp(exp(log(log(3) - x) - (3*x)/2)))*exp(log(log(3) - x) - (3*x)/
2)*(3*log(3) - 3*x + 2))/(2*x - 2*log(3)),x)

[Out]

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

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sympy [A]  time = 0.61, size = 14, normalized size = 0.56 \begin {gather*} e^{e^{\left (- x + \log {\relax (3 )}\right ) e^{- \frac {3 x}{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*ln(3)+3*x-2)*exp(ln(ln(3)-x)-3/2*x)*exp(exp(ln(ln(3)-x)-3/2*x))*exp(exp(exp(ln(ln(3)-x)-3/2*x)))
/(2*ln(3)-2*x),x)

[Out]

exp(exp((-x + log(3))*exp(-3*x/2)))

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