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

Optimal. Leaf size=22 \[ e^{-e^x x-\log ^2(-x+\log (x))} x \]

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Rubi [F]  time = 3.74, 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 {e^{-e^x x-\log ^2(-x+\log (x))} x \left (-x+e^x \left (x^2+x^3\right )+\left (1+e^x \left (-x-x^2\right )\right ) \log (x)+(-2+2 x) \log (-x+\log (x))\right )}{-x^2+x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.66, size = 22, normalized size = 1.00 \begin {gather*} e^{-e^x x-\log ^2(-x+\log (x))} x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(-(E^x*x) - Log[-x + Log[x]]^2)*x

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fricas [A]  time = 0.59, size = 20, normalized size = 0.91 \begin {gather*} e^{\left (-x e^{x} - \log \left (-x + \log \relax (x)\right )^{2} + \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 2.10, size = 20, normalized size = 0.91 \begin {gather*} e^{\left (-x e^{x} - \log \left (-x + \log \relax (x)\right )^{2} + \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 21, normalized size = 0.95

method	result	size
risch	\(x \,{\mathrm e}^{-\ln \left (\ln \relax (x )-x \right )^{2}-{\mathrm e}^{x} x}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x-2)*ln(ln(x)-x)+((-x^2-x)*exp(x)+1)*ln(x)+(x^3+x^2)*exp(x)-x)*exp(-ln(ln(x)-x)^2+ln(x)-exp(x)*x)/(x*l
n(x)-x^2),x,method=_RETURNVERBOSE)

[Out]

x*exp(-ln(ln(x)-x)^2-exp(x)*x)

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maxima [A]  time = 0.56, size = 20, normalized size = 0.91 \begin {gather*} x e^{\left (-x e^{x} - \log \left (-x + \log \relax (x)\right )^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(-x*e^x - log(-x + log(x))^2)

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mupad [B]  time = 4.18, size = 20, normalized size = 0.91 \begin {gather*} x\,{\mathrm {e}}^{-x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-{\ln \left (\ln \relax (x)-x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(log(x) - x*exp(x) - log(log(x) - x)^2)*(x - log(log(x) - x)*(2*x - 2) - exp(x)*(x^2 + x^3) + log(x)*
(exp(x)*(x + x^2) - 1)))/(x*log(x) - x^2),x)

[Out]

x*exp(-x*exp(x))*exp(-log(log(x) - x)^2)

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sympy [A]  time = 7.13, size = 17, normalized size = 0.77 \begin {gather*} x e^{- x e^{x} - \log {\left (- x + \log {\relax (x )} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-2)*ln(ln(x)-x)+((-x**2-x)*exp(x)+1)*ln(x)+(x**3+x**2)*exp(x)-x)*exp(-ln(ln(x)-x)**2+ln(x)-exp(
x)*x)/(x*ln(x)-x**2),x)

[Out]

x*exp(-x*exp(x) - log(-x + log(x))**2)

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