\(\int \frac {-e+e^{\log ^2(\frac {3}{2})-2 \log (\frac {3}{2}) \log (3)+\log ^2(3)}-x-x^2+(e-e^{\log ^2(\frac {3}{2})-2 \log (\frac {3}{2}) \log (3)+\log ^2(3)}+2 x) \log (x)}{x^2-2 x \log (x)+\log ^2(x)} \, dx\) [9144]

Optimal result

Integrand size = 80, antiderivative size = 28 \[ \int \frac {-e+e^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}-x-x^2+\left (e-e^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}+2 x\right ) \log (x)}{x^2-2 x \log (x)+\log ^2(x)} \, dx=\frac {x \left (e-e^{\left (-\log \left (\frac {3}{2}\right )+\log (3)\right )^2}+x\right )}{-x+\log (x)} \]

[Out]

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

Rubi [F]

\[ \int \frac {-e+e^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}-x-x^2+\left (e-e^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}+2 x\right ) \log (x)}{x^2-2 x \log (x)+\log ^2(x)} \, dx=\int \frac {-e+e^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}-x-x^2+\left (e-e^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}+2 x\right ) \log (x)}{x^2-2 x \log (x)+\log ^2(x)} \, dx \]

[In]

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]

[Out]

-((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
g[x]), x]

Rubi steps \begin{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {-e+e^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}-x-x^2+\left (e-e^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}+2 x\right ) \log (x)}{x^2-2 x \log (x)+\log ^2(x)} \, dx=\frac {x \left (e-e^{\log ^2(2)}+x\right )}{-x+\log (x)} \]

[In]

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]

[Out]

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

Maple [A] (verified)

Time = 1.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82

method	result	size
risch	\(-\frac {\left (x +{\mathrm e}-{\mathrm e}^{\ln \left (2\right )^{2}}\right ) x}{x -\ln \left (x \right )}\)	\(23\)
default	\(\frac {x^{2}+\left ({\mathrm e}-{\mathrm e}^{\ln \left (2\right )^{2}}\right ) x}{\ln \left (x \right )-x}\)	\(26\)
norman	\(\frac {\left (-{\mathrm e}+{\mathrm e}^{\ln \left (2\right )^{2}}\right ) \ln \left (x \right )-x^{2}}{x -\ln \left (x \right )}\)	\(29\)
parallelrisch	\(-\frac {x \,{\mathrm e}-{\mathrm e}^{\ln \left (3\right )^{2}+2 \ln \left (\frac {2}{3}\right ) \ln \left (3\right )+\ln \left (\frac {2}{3}\right )^{2}} x +x^{2}}{x -\ln \left (x \right )}\)	\(38\)

[In]

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)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {-e+e^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}-x-x^2+\left (e-e^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}+2 x\right ) \log (x)}{x^2-2 x \log (x)+\log ^2(x)} \, dx=-\frac {x^{2} + x e - x e^{\left (\log \left (3\right )^{2} + 2 \, \log \left (3\right ) \log \left (\frac {2}{3}\right ) + \log \left (\frac {2}{3}\right )^{2}\right )}}{x - \log \left (x\right )} \]

[In]

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")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {-e+e^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}-x-x^2+\left (e-e^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}+2 x\right ) \log (x)}{x^2-2 x \log (x)+\log ^2(x)} \, dx=\frac {x^{2} - x e^{\log {\left (2 \right )}^{2}} + e x}{- x + \log {\left (x \right )}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {-e+e^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}-x-x^2+\left (e-e^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}+2 x\right ) \log (x)}{x^2-2 x \log (x)+\log ^2(x)} \, dx=-\frac {x^{2} + x {\left (e - e^{\left (\log \left (2\right )^{2}\right )}\right )}}{x - \log \left (x\right )} \]

[In]

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")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {-e+e^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}-x-x^2+\left (e-e^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}+2 x\right ) \log (x)}{x^2-2 x \log (x)+\log ^2(x)} \, dx=-\frac {x^{2} + x e - x e^{\left (\log \left (2\right )^{2}\right )}}{x - \log \left (x\right )} \]

[In]

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")

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.86 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {-e+e^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}-x-x^2+\left (e-e^{\log ^2\left (\frac {3}{2}\right )-2 \log \left (\frac {3}{2}\right ) \log (3)+\log ^2(3)}+2 x\right ) \log (x)}{x^2-2 x \log (x)+\log ^2(x)} \, dx=-\frac {x\,\left (x-{\mathrm {e}}^{{\ln \left (2\right )}^2}+\mathrm {e}\right )}{x-\ln \left (x\right )} \]

[In]

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)

[Out]

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