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

Optimal. Leaf size=15 \[ e^{x^{e^{1-\frac {e^x}{2}}}} \]

________________________________________________________________________________________

Rubi [F]  time = 1.15, 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}{2} e^{\frac {1}{2} \left (2-e^x\right )+x^{e^{\frac {1}{2} \left (2-e^x\right )}}} x^{-1+e^{\frac {1}{2} \left (2-e^x\right )}} \left (2-e^x x \log (x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.28, size = 15, normalized size = 1.00 \begin {gather*} e^{x^{e^{1-\frac {e^x}{2}}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^x^E^(1 - E^x/2)

________________________________________________________________________________________

fricas [A]  time = 0.50, size = 10, normalized size = 0.67 \begin {gather*} e^{\left (x^{e^{\left (-\frac {1}{2} \, e^{x} + 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-x*exp(x)*log(x)+2)*exp(log(x)/exp(1/2*exp(x)-1))*exp(exp(log(x)/exp(1/2*exp(x)-1)))/x/exp(1/2*
exp(x)-1),x, algorithm="fricas")

[Out]

e^(x^e^(-1/2*e^x + 1))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-x*exp(x)*log(x)+2)*exp(log(x)/exp(1/2*exp(x)-1))*exp(exp(log(x)/exp(1/2*exp(x)-1)))/x/exp(1/2*
exp(x)-1),x, algorithm="giac")

[Out]

integrate(-1/2*(x*e^x*log(x) - 2)*x^e^(-1/2*e^x + 1)*e^(x^e^(-1/2*e^x + 1) - 1/2*e^x + 1)/x, x)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 11, normalized size = 0.73

method	result	size
risch	\({\mathrm e}^{x^{{\mathrm e}^{1-\frac {{\mathrm e}^{x}}{2}}}}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(-x*exp(x)*ln(x)+2)*exp(ln(x)/exp(1/2*exp(x)-1))*exp(exp(ln(x)/exp(1/2*exp(x)-1)))/x/exp(1/2*exp(x)-1)
,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.51, size = 10, normalized size = 0.67 \begin {gather*} e^{\left (x^{e^{\left (-\frac {1}{2} \, e^{x} + 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-x*exp(x)*log(x)+2)*exp(log(x)/exp(1/2*exp(x)-1))*exp(exp(log(x)/exp(1/2*exp(x)-1)))/x/exp(1/2*
exp(x)-1),x, algorithm="maxima")

[Out]

e^(x^e^(-1/2*e^x + 1))

________________________________________________________________________________________

mupad [B]  time = 4.46, size = 11, normalized size = 0.73 \begin {gather*} {\mathrm {e}}^{x^{\frac {\mathrm {e}}{\sqrt {{\mathrm {e}}^{{\mathrm {e}}^x}}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(1 - exp(x)/2)*exp(exp(exp(1 - exp(x)/2)*log(x)))*exp(exp(1 - exp(x)/2)*log(x))*(x*exp(x)*log(x) - 2)
)/(2*x),x)

[Out]

exp(x^(exp(1)/exp(exp(x))^(1/2)))

________________________________________________________________________________________

sympy [A]  time = 41.79, size = 14, normalized size = 0.93 \begin {gather*} e^{e^{e^{1 - \frac {e^{x}}{2}} \log {\relax (x )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-x*exp(x)*ln(x)+2)*exp(ln(x)/exp(1/2*exp(x)-1))*exp(exp(ln(x)/exp(1/2*exp(x)-1)))/x/exp(1/2*exp
(x)-1),x)

[Out]

exp(exp(exp(1 - exp(x)/2)*log(x)))

________________________________________________________________________________________