\(\int \frac {1-e^{e^{20}}-x^2-\log (x)}{e^{2 e^{20}}+4 x^2-4 x^3+x^4+e^{e^{20}} (4 x-2 x^2)+(2 e^{e^{20}}+4 x-2 x^2) \log (x)+\log ^2(x)} \, dx\) [5314]

Optimal result

Integrand size = 80, antiderivative size = 22 \[ \int \frac {1-e^{e^{20}}-x^2-\log (x)}{e^{2 e^{20}}+4 x^2-4 x^3+x^4+e^{e^{20}} \left (4 x-2 x^2\right )+\left (2 e^{e^{20}}+4 x-2 x^2\right ) \log (x)+\log ^2(x)} \, dx=\frac {x}{-e^{e^{20}}-2 x+x^2-\log (x)} \]

[Out]

x/(x^2-ln(x)-exp(exp(20))-2*x)

Rubi [F]

\[ \int \frac {1-e^{e^{20}}-x^2-\log (x)}{e^{2 e^{20}}+4 x^2-4 x^3+x^4+e^{e^{20}} \left (4 x-2 x^2\right )+\left (2 e^{e^{20}}+4 x-2 x^2\right ) \log (x)+\log ^2(x)} \, dx=\int \frac {1-e^{e^{20}}-x^2-\log (x)}{e^{2 e^{20}}+4 x^2-4 x^3+x^4+e^{e^{20}} \left (4 x-2 x^2\right )+\left (2 e^{e^{20}}+4 x-2 x^2\right ) \log (x)+\log ^2(x)} \, dx \]

[In]

Int[(1 - E^E^20 - x^2 - Log[x])/(E^(2*E^20) + 4*x^2 - 4*x^3 + x^4 + E^E^20*(4*x - 2*x^2) + (2*E^E^20 + 4*x - 2
*x^2)*Log[x] + Log[x]^2),x]

[Out]

2*Defer[Int][x/(-E^E^20 - 2*x + x^2 - Log[x])^2, x] - 2*Defer[Int][x^2/(-E^E^20 - 2*x + x^2 - Log[x])^2, x] +
Defer[Int][(E^E^20 + 2*x - x^2 + Log[x])^(-2), x] - Defer[Int][(E^E^20 + 2*x - x^2 + Log[x])^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1-e^{e^{20}}-x^2-\log (x)}{\left (e^{e^{20}}-(-2+x) x+\log (x)\right )^2} \, dx \\ & = \int \left (-\frac {-1-2 x+2 x^2}{\left (-e^{e^{20}}-2 x+x^2-\log (x)\right )^2}-\frac {1}{e^{e^{20}}+2 x-x^2+\log (x)}\right ) \, dx \\ & = -\int \frac {-1-2 x+2 x^2}{\left (-e^{e^{20}}-2 x+x^2-\log (x)\right )^2} \, dx-\int \frac {1}{e^{e^{20}}+2 x-x^2+\log (x)} \, dx \\ & = -\int \frac {1}{e^{e^{20}}+2 x-x^2+\log (x)} \, dx-\int \left (-\frac {2 x}{\left (-e^{e^{20}}-2 x+x^2-\log (x)\right )^2}+\frac {2 x^2}{\left (-e^{e^{20}}-2 x+x^2-\log (x)\right )^2}-\frac {1}{\left (e^{e^{20}}+2 x-x^2+\log (x)\right )^2}\right ) \, dx \\ & = 2 \int \frac {x}{\left (-e^{e^{20}}-2 x+x^2-\log (x)\right )^2} \, dx-2 \int \frac {x^2}{\left (-e^{e^{20}}-2 x+x^2-\log (x)\right )^2} \, dx+\int \frac {1}{\left (e^{e^{20}}+2 x-x^2+\log (x)\right )^2} \, dx-\int \frac {1}{e^{e^{20}}+2 x-x^2+\log (x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {1-e^{e^{20}}-x^2-\log (x)}{e^{2 e^{20}}+4 x^2-4 x^3+x^4+e^{e^{20}} \left (4 x-2 x^2\right )+\left (2 e^{e^{20}}+4 x-2 x^2\right ) \log (x)+\log ^2(x)} \, dx=-\frac {x}{e^{e^{20}}+2 x-x^2+\log (x)} \]

[In]

Integrate[(1 - E^E^20 - x^2 - Log[x])/(E^(2*E^20) + 4*x^2 - 4*x^3 + x^4 + E^E^20*(4*x - 2*x^2) + (2*E^E^20 + 4
*x - 2*x^2)*Log[x] + Log[x]^2),x]

[Out]

-(x/(E^E^20 + 2*x - x^2 + Log[x]))

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

method	result	size
default	\(-\frac {x}{-x^{2}+{\mathrm e}^{{\mathrm e}^{20}}+\ln \left (x \right )+2 x}\)	\(20\)
norman	\(-\frac {x}{-x^{2}+{\mathrm e}^{{\mathrm e}^{20}}+\ln \left (x \right )+2 x}\)	\(20\)
risch	\(-\frac {x}{-x^{2}+{\mathrm e}^{{\mathrm e}^{20}}+\ln \left (x \right )+2 x}\)	\(20\)
parallelrisch	\(-\frac {x}{-x^{2}+{\mathrm e}^{{\mathrm e}^{20}}+\ln \left (x \right )+2 x}\)	\(20\)

[In]

int((-ln(x)-exp(exp(20))-x^2+1)/(ln(x)^2+(2*exp(exp(20))-2*x^2+4*x)*ln(x)+exp(exp(20))^2+(-2*x^2+4*x)*exp(exp(
20))+x^4-4*x^3+4*x^2),x,method=_RETURNVERBOSE)

[Out]

-x/(-x^2+exp(exp(20))+ln(x)+2*x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1-e^{e^{20}}-x^2-\log (x)}{e^{2 e^{20}}+4 x^2-4 x^3+x^4+e^{e^{20}} \left (4 x-2 x^2\right )+\left (2 e^{e^{20}}+4 x-2 x^2\right ) \log (x)+\log ^2(x)} \, dx=\frac {x}{x^{2} - 2 \, x - e^{\left (e^{20}\right )} - \log \left (x\right )} \]

[In]

integrate((-log(x)-exp(exp(20))-x^2+1)/(log(x)^2+(2*exp(exp(20))-2*x^2+4*x)*log(x)+exp(exp(20))^2+(-2*x^2+4*x)
*exp(exp(20))+x^4-4*x^3+4*x^2),x, algorithm="fricas")

[Out]

x/(x^2 - 2*x - e^(e^20) - log(x))

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {1-e^{e^{20}}-x^2-\log (x)}{e^{2 e^{20}}+4 x^2-4 x^3+x^4+e^{e^{20}} \left (4 x-2 x^2\right )+\left (2 e^{e^{20}}+4 x-2 x^2\right ) \log (x)+\log ^2(x)} \, dx=- \frac {x}{- x^{2} + 2 x + \log {\left (x \right )} + e^{e^{20}}} \]

[In]

integrate((-ln(x)-exp(exp(20))-x**2+1)/(ln(x)**2+(2*exp(exp(20))-2*x**2+4*x)*ln(x)+exp(exp(20))**2+(-2*x**2+4*
x)*exp(exp(20))+x**4-4*x**3+4*x**2),x)

[Out]

-x/(-x**2 + 2*x + log(x) + exp(exp(20)))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1-e^{e^{20}}-x^2-\log (x)}{e^{2 e^{20}}+4 x^2-4 x^3+x^4+e^{e^{20}} \left (4 x-2 x^2\right )+\left (2 e^{e^{20}}+4 x-2 x^2\right ) \log (x)+\log ^2(x)} \, dx=\frac {x}{x^{2} - 2 \, x - e^{\left (e^{20}\right )} - \log \left (x\right )} \]

[In]

integrate((-log(x)-exp(exp(20))-x^2+1)/(log(x)^2+(2*exp(exp(20))-2*x^2+4*x)*log(x)+exp(exp(20))^2+(-2*x^2+4*x)
*exp(exp(20))+x^4-4*x^3+4*x^2),x, algorithm="maxima")

[Out]

x/(x^2 - 2*x - e^(e^20) - log(x))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1-e^{e^{20}}-x^2-\log (x)}{e^{2 e^{20}}+4 x^2-4 x^3+x^4+e^{e^{20}} \left (4 x-2 x^2\right )+\left (2 e^{e^{20}}+4 x-2 x^2\right ) \log (x)+\log ^2(x)} \, dx=\frac {x}{x^{2} - 2 \, x - e^{\left (e^{20}\right )} - \log \left (x\right )} \]

[In]

integrate((-log(x)-exp(exp(20))-x^2+1)/(log(x)^2+(2*exp(exp(20))-2*x^2+4*x)*log(x)+exp(exp(20))^2+(-2*x^2+4*x)
*exp(exp(20))+x^4-4*x^3+4*x^2),x, algorithm="giac")

[Out]

x/(x^2 - 2*x - e^(e^20) - log(x))

Mupad [B] (verification not implemented)

Time = 11.71 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {1-e^{e^{20}}-x^2-\log (x)}{e^{2 e^{20}}+4 x^2-4 x^3+x^4+e^{e^{20}} \left (4 x-2 x^2\right )+\left (2 e^{e^{20}}+4 x-2 x^2\right ) \log (x)+\log ^2(x)} \, dx=-\frac {x}{2\,x+{\mathrm {e}}^{{\mathrm {e}}^{20}}+\ln \left (x\right )-x^2} \]

[In]

int(-(exp(exp(20)) + log(x) + x^2 - 1)/(exp(2*exp(20)) + log(x)^2 + exp(exp(20))*(4*x - 2*x^2) + 4*x^2 - 4*x^3
+ x^4 + log(x)*(4*x + 2*exp(exp(20)) - 2*x^2)),x)

[Out]

-x/(2*x + exp(exp(20)) + log(x) - x^2)