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

Optimal. Leaf size=28 \[ -e^3+x+\log (x) \log \left (-e^{-x+x \log (x)}+x \log (x)\right ) \]

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Rubi [F]  time = 3.11, 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 {\left (-x-x^2\right ) \log (x)-x \log ^2(x)+e^{-x+x \log (x)} \left (x+x \log ^2(x)\right )+\left (e^{-x+x \log (x)}-x \log (x)\right ) \log \left (-e^{-x+x \log (x)}+x \log (x)\right )}{e^{-x+x \log (x)} x-x^2 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 21, normalized size = 0.75 \begin {gather*} x+\log (x) \log \left (-e^{-x} x^x+x \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x + Log[x]*Log[-(x^x/E^x) + x*Log[x]]

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fricas [A]  time = 0.78, size = 22, normalized size = 0.79 \begin {gather*} \log \left (x \log \relax (x) - e^{\left (x \log \relax (x) - x\right )}\right ) \log \relax (x) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A]  time = 0.04, size = 21, normalized size = 0.75

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.53, size = 23, normalized size = 0.82 \begin {gather*} -x \log \relax (x) + \log \left (x e^{x} \log \relax (x) - x^{x}\right ) \log \relax (x) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.93, size = 20, normalized size = 0.71 \begin {gather*} x+\ln \left (x\,\ln \relax (x)-x^x\,{\mathrm {e}}^{-x}\right )\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.73, size = 19, normalized size = 0.68 \begin {gather*} x + \log {\relax (x )} \log {\left (x \log {\relax (x )} - e^{x \log {\relax (x )} - x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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