∫ −e+elog2(32)−2 log(32) log(3)+log2(3)−x−x2+(e−elog2(32)−2 log(32) log(3)+log2(3)+2x) log(x) ___________________________________________________________________________________________________________________

Optimal. Leaf size=28 \[ \frac {x \left (e-e^{\left (-\log \left (\frac {3}{2}\right )+\log (3)\right )^2}+x\right )}{-x+\log (x)} \]

________________________________________________________________________________________

Rubi [F] time = 0.80, 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^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}-x-x^2+\left (e-e^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}+2 x\right ) \log (x)}{x^2-2 x \log (x)+\log ^2(x)} \, dx \end {gather*}

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

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

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

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

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

________________________________________________________________________________________

Mathematica [A] time = 0.28, size = 21, normalized size = 0.75 \begin {gather*} \frac {x \left (e-e^{\log ^2(2)}+x\right )}{-x+\log (x)} \end {gather*}

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

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

________________________________________________________________________________________

fricas [A] time = 0.49, size = 37, normalized size = 1.32 \begin {gather*} -\frac {x^{2} + x e - x e^{\left (\log \relax (3)^{2} + 2 \, \log \relax (3) \log \left (\frac {2}{3}\right ) + \log \left (\frac {2}{3}\right )^{2}\right )}}{x - \log \relax (x)} \end {gather*}

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

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

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

________________________________________________________________________________________

giac [A] time = 0.22, size = 26, normalized size = 0.93 \begin {gather*} -\frac {x^{2} + x e - x e^{\left (\log \relax (2)^{2}\right )}}{x - \log \relax (x)} \end {gather*}

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

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

________________________________________________________________________________________


method	result	size

risch	\(-\frac {\left (x +{\mathrm e}-{\mathrm e}^{\ln \relax (2)^{2}}\right ) x}{x -\ln \relax (x )}\)	\(23\)
norman	\(\frac {\left (-{\mathrm e}+{\mathrm e}^{\ln \relax (2)^{2}}\right ) \ln \relax (x )-x^{2}}{x -\ln \relax (x )}\)	\(29\)

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

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

________________________________________________________________________________________

maxima [A] time = 0.54, size = 26, normalized size = 0.93 \begin {gather*} -\frac {x^{2} + x {\left (e - e^{\left (\log \relax (2)^{2}\right )}\right )}}{x - \log \relax (x)} \end {gather*}

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

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

________________________________________________________________________________________

mupad [B] time = 8.08, size = 22, normalized size = 0.79 \begin {gather*} -\frac {x\,\left (x-{\mathrm {e}}^{{\ln \relax (2)}^2}+\mathrm {e}\right )}{x-\ln \relax (x)} \end {gather*}

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

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

________________________________________________________________________________________

sympy [A] time = 0.14, size = 20, normalized size = 0.71 \begin {gather*} \frac {x^{2} - x e^{\log {\relax (2 )}^{2}} + e x}{- x + \log {\relax (x )}} \end {gather*}

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

________________________________________________________________________________________

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