\(\int \frac {-4 x-6 x^2-2 x^3+e^x (2 x+2 x^2)+e^{\frac {x^2+\log ^2(-2 x+e^x x-x^2)}{x}} (-2 x-3 x^2-x^3+e^x (x+x^2)+(-4-4 x+e^x (2+2 x)) \log (-2 x+e^x x-x^2)+(2-e^x+x) \log ^2(-2 x+e^x x-x^2))}{-2 x+e^x x-x^2} \, dx\) [821]

Optimal result

Integrand size = 151, antiderivative size = 28 \[ \int \frac {-4 x-6 x^2-2 x^3+e^x \left (2 x+2 x^2\right )+e^{\frac {x^2+\log ^2\left (-2 x+e^x x-x^2\right )}{x}} \left (-2 x-3 x^2-x^3+e^x \left (x+x^2\right )+\left (-4-4 x+e^x (2+2 x)\right ) \log \left (-2 x+e^x x-x^2\right )+\left (2-e^x+x\right ) \log ^2\left (-2 x+e^x x-x^2\right )\right )}{-2 x+e^x x-x^2} \, dx=-1+x \left (2+e^{x+\frac {\log ^2\left (\left (-2+e^x-x\right ) x\right )}{x}}+x\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {-4 x-6 x^2-2 x^3+e^x \left (2 x+2 x^2\right )+e^{\frac {x^2+\log ^2\left (-2 x+e^x x-x^2\right )}{x}} \left (-2 x-3 x^2-x^3+e^x \left (x+x^2\right )+\left (-4-4 x+e^x (2+2 x)\right ) \log \left (-2 x+e^x x-x^2\right )+\left (2-e^x+x\right ) \log ^2\left (-2 x+e^x x-x^2\right )\right )}{-2 x+e^x x-x^2} \, dx=x \left (2+e^{x+\frac {\log ^2\left (-x \left (2-e^x+x\right )\right )}{x}}+x\right ) \] Input:

Integrate[(-4*x - 6*x^2 - 2*x^3 + E^x*(2*x + 2*x^2) + E^((x^2 + Log[-2*x + 
 E^x*x - x^2]^2)/x)*(-2*x - 3*x^2 - x^3 + E^x*(x + x^2) + (-4 - 4*x + E^x* 
(2 + 2*x))*Log[-2*x + E^x*x - x^2] + (2 - E^x + x)*Log[-2*x + E^x*x - x^2] 
^2))/(-2*x + E^x*x - x^2),x]
 

Output:

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

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {-2 x^3-6 x^2+e^x \left (2 x^2+2 x\right )+e^{\frac {x^2+\log ^2\left (-x^2+e^x x-2 x\right )}{x}} \left (-x^3-3 x^2+e^x \left (x^2+x\right )+\left (x-e^x+2\right ) \log ^2\left (-x^2+e^x x-2 x\right )+\left (-4 x+e^x (2 x+2)-4\right ) \log \left (-x^2+e^x x-2 x\right )-2 x\right )-4 x}{-x^2+e^x x-2 x} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{x+\frac {\log ^2\left (-x \left (x-e^x+2\right )\right )}{x}} \left (-x^3+e^x x^2-3 x^2+e^x x-2 x+x \log ^2\left (-x \left (x-e^x+2\right )\right )-e^x \log ^2\left (-x \left (x-e^x+2\right )\right )+2 \log ^2\left (-x \left (x-e^x+2\right )\right )+2 e^x x \log \left (-x \left (x-e^x+2\right )\right )-4 x \log \left (-x \left (x-e^x+2\right )\right )+2 e^x \log \left (-x \left (x-e^x+2\right )\right )-4 \log \left (-x \left (x-e^x+2\right )\right )\right )}{\left (-x+e^x-2\right ) x}+2 (x+1)\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \left (2 (x+1)+\frac {e^{x+\frac {\log ^2\left (-x \left (x-e^x+2\right )\right )}{x}} \left (-x (x+1) \left (x-e^x+2\right )+\left (x-e^x+2\right ) \log ^2\left (-x \left (x-e^x+2\right )\right )+2 \left (e^x-2\right ) (x+1) \log \left (-x \left (x-e^x+2\right )\right )\right )}{\left (-x+e^x-2\right ) x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \int e^{\frac {\log ^2\left (-x \left (x-e^x+2\right )\right )}{x}+x} \log \left (-x^2+e^x x-2 x\right )dx+2 \int \frac {e^{\frac {\log ^2\left (-x \left (x-e^x+2\right )\right )}{x}+x} \log \left (-x^2+e^x x-2 x\right )}{-x+e^x-2}dx+2 \int \frac {e^{\frac {\log ^2\left (-x \left (x-e^x+2\right )\right )}{x}+x} \log \left (-x^2+e^x x-2 x\right )}{x}dx+2 \int \frac {e^{\frac {\log ^2\left (-x \left (x-e^x+2\right )\right )}{x}+x} x \log \left (-x^2+e^x x-2 x\right )}{-x+e^x-2}dx-\int \frac {e^{\frac {\log ^2\left (-x \left (x-e^x+2\right )\right )}{x}+x} \log ^2\left (-x^2+e^x x-2 x\right )}{x}dx+\int e^{\frac {\log ^2\left (-x \left (x-e^x+2\right )\right )}{x}+x}dx+\int e^{\frac {\log ^2\left (-x \left (x-e^x+2\right )\right )}{x}+x} xdx+(x+1)^2\)

Input:

Int[(-4*x - 6*x^2 - 2*x^3 + E^x*(2*x + 2*x^2) + E^((x^2 + Log[-2*x + E^x*x 
 - x^2]^2)/x)*(-2*x - 3*x^2 - x^3 + E^x*(x + x^2) + (-4 - 4*x + E^x*(2 + 2 
*x))*Log[-2*x + E^x*x - x^2] + (2 - E^x + x)*Log[-2*x + E^x*x - x^2]^2))/( 
-2*x + E^x*x - x^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.82 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14

method	result	size
parallelrisch	\(-4+x^{2}+{\mathrm e}^{\frac {{\ln \left (x \left ({\mathrm e}^{x}-2-x \right )\right )}^{2}+x^{2}}{x}} x +2 x\)	\(32\)
risch	\(\text {Expression too large to display}\)	\(812\)

Input:

int((((x-exp(x)+2)*ln(exp(x)*x-x^2-2*x)^2+((2+2*x)*exp(x)-4*x-4)*ln(exp(x) 
*x-x^2-2*x)+(x^2+x)*exp(x)-x^3-3*x^2-2*x)*exp((ln(exp(x)*x-x^2-2*x)^2+x^2) 
/x)+(2*x^2+2*x)*exp(x)-2*x^3-6*x^2-4*x)/(exp(x)*x-x^2-2*x),x,method=_RETUR 
NVERBOSE)
 

Output:

-4+x^2+exp((ln(x*(exp(x)-2-x))^2+x^2)/x)*x+2*x
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {-4 x-6 x^2-2 x^3+e^x \left (2 x+2 x^2\right )+e^{\frac {x^2+\log ^2\left (-2 x+e^x x-x^2\right )}{x}} \left (-2 x-3 x^2-x^3+e^x \left (x+x^2\right )+\left (-4-4 x+e^x (2+2 x)\right ) \log \left (-2 x+e^x x-x^2\right )+\left (2-e^x+x\right ) \log ^2\left (-2 x+e^x x-x^2\right )\right )}{-2 x+e^x x-x^2} \, dx=x^{2} + x e^{\left (\frac {x^{2} + \log \left (-x^{2} + x e^{x} - 2 \, x\right )^{2}}{x}\right )} + 2 \, x \] Input:

integrate((((x-exp(x)+2)*log(exp(x)*x-x^2-2*x)^2+((2+2*x)*exp(x)-4*x-4)*lo 
g(exp(x)*x-x^2-2*x)+(x^2+x)*exp(x)-x^3-3*x^2-2*x)*exp((log(exp(x)*x-x^2-2* 
x)^2+x^2)/x)+(2*x^2+2*x)*exp(x)-2*x^3-6*x^2-4*x)/(exp(x)*x-x^2-2*x),x, alg 
orithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {-4 x-6 x^2-2 x^3+e^x \left (2 x+2 x^2\right )+e^{\frac {x^2+\log ^2\left (-2 x+e^x x-x^2\right )}{x}} \left (-2 x-3 x^2-x^3+e^x \left (x+x^2\right )+\left (-4-4 x+e^x (2+2 x)\right ) \log \left (-2 x+e^x x-x^2\right )+\left (2-e^x+x\right ) \log ^2\left (-2 x+e^x x-x^2\right )\right )}{-2 x+e^x x-x^2} \, dx=\text {Timed out} \] Input:

integrate((((x-exp(x)+2)*ln(exp(x)*x-x**2-2*x)**2+((2+2*x)*exp(x)-4*x-4)*l 
n(exp(x)*x-x**2-2*x)+(x**2+x)*exp(x)-x**3-3*x**2-2*x)*exp((ln(exp(x)*x-x** 
2-2*x)**2+x**2)/x)+(2*x**2+2*x)*exp(x)-2*x**3-6*x**2-4*x)/(exp(x)*x-x**2-2 
*x),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {-4 x-6 x^2-2 x^3+e^x \left (2 x+2 x^2\right )+e^{\frac {x^2+\log ^2\left (-2 x+e^x x-x^2\right )}{x}} \left (-2 x-3 x^2-x^3+e^x \left (x+x^2\right )+\left (-4-4 x+e^x (2+2 x)\right ) \log \left (-2 x+e^x x-x^2\right )+\left (2-e^x+x\right ) \log ^2\left (-2 x+e^x x-x^2\right )\right )}{-2 x+e^x x-x^2} \, dx=x^{2} + x e^{\left (x + \frac {\log \left (x\right )^{2}}{x} + \frac {2 \, \log \left (x\right ) \log \left (-x + e^{x} - 2\right )}{x} + \frac {\log \left (-x + e^{x} - 2\right )^{2}}{x}\right )} + 2 \, x \] Input:

integrate((((x-exp(x)+2)*log(exp(x)*x-x^2-2*x)^2+((2+2*x)*exp(x)-4*x-4)*lo 
g(exp(x)*x-x^2-2*x)+(x^2+x)*exp(x)-x^3-3*x^2-2*x)*exp((log(exp(x)*x-x^2-2* 
x)^2+x^2)/x)+(2*x^2+2*x)*exp(x)-2*x^3-6*x^2-4*x)/(exp(x)*x-x^2-2*x),x, alg 
orithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {-4 x-6 x^2-2 x^3+e^x \left (2 x+2 x^2\right )+e^{\frac {x^2+\log ^2\left (-2 x+e^x x-x^2\right )}{x}} \left (-2 x-3 x^2-x^3+e^x \left (x+x^2\right )+\left (-4-4 x+e^x (2+2 x)\right ) \log \left (-2 x+e^x x-x^2\right )+\left (2-e^x+x\right ) \log ^2\left (-2 x+e^x x-x^2\right )\right )}{-2 x+e^x x-x^2} \, dx=\int { \frac {2 \, x^{3} + 6 \, x^{2} - 2 \, {\left (x^{2} + x\right )} e^{x} + {\left (x^{3} - {\left (x - e^{x} + 2\right )} \log \left (-x^{2} + x e^{x} - 2 \, x\right )^{2} + 3 \, x^{2} - {\left (x^{2} + x\right )} e^{x} - 2 \, {\left ({\left (x + 1\right )} e^{x} - 2 \, x - 2\right )} \log \left (-x^{2} + x e^{x} - 2 \, x\right ) + 2 \, x\right )} e^{\left (\frac {x^{2} + \log \left (-x^{2} + x e^{x} - 2 \, x\right )^{2}}{x}\right )} + 4 \, x}{x^{2} - x e^{x} + 2 \, x} \,d x } \] Input:

integrate((((x-exp(x)+2)*log(exp(x)*x-x^2-2*x)^2+((2+2*x)*exp(x)-4*x-4)*lo 
g(exp(x)*x-x^2-2*x)+(x^2+x)*exp(x)-x^3-3*x^2-2*x)*exp((log(exp(x)*x-x^2-2* 
x)^2+x^2)/x)+(2*x^2+2*x)*exp(x)-2*x^3-6*x^2-4*x)/(exp(x)*x-x^2-2*x),x, alg 
orithm="giac")
 

Output:

integrate((2*x^3 + 6*x^2 - 2*(x^2 + x)*e^x + (x^3 - (x - e^x + 2)*log(-x^2 
 + x*e^x - 2*x)^2 + 3*x^2 - (x^2 + x)*e^x - 2*((x + 1)*e^x - 2*x - 2)*log( 
-x^2 + x*e^x - 2*x) + 2*x)*e^((x^2 + log(-x^2 + x*e^x - 2*x)^2)/x) + 4*x)/ 
(x^2 - x*e^x + 2*x), x)
 

Mupad [B] (verification not implemented)

Time = 2.93 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {-4 x-6 x^2-2 x^3+e^x \left (2 x+2 x^2\right )+e^{\frac {x^2+\log ^2\left (-2 x+e^x x-x^2\right )}{x}} \left (-2 x-3 x^2-x^3+e^x \left (x+x^2\right )+\left (-4-4 x+e^x (2+2 x)\right ) \log \left (-2 x+e^x x-x^2\right )+\left (2-e^x+x\right ) \log ^2\left (-2 x+e^x x-x^2\right )\right )}{-2 x+e^x x-x^2} \, dx=2\,x+x^2+x\,{\mathrm {e}}^{\frac {{\ln \left (x\,{\mathrm {e}}^x-2\,x-x^2\right )}^2}{x}}\,{\mathrm {e}}^x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {-4 x-6 x^2-2 x^3+e^x \left (2 x+2 x^2\right )+e^{\frac {x^2+\log ^2\left (-2 x+e^x x-x^2\right )}{x}} \left (-2 x-3 x^2-x^3+e^x \left (x+x^2\right )+\left (-4-4 x+e^x (2+2 x)\right ) \log \left (-2 x+e^x x-x^2\right )+\left (2-e^x+x\right ) \log ^2\left (-2 x+e^x x-x^2\right )\right )}{-2 x+e^x x-x^2} \, dx=x \left (e^{\frac {\mathrm {log}\left (e^{x} x -x^{2}-2 x \right )^{2}+x^{2}}{x}}+x +2\right ) \] Input:

int((((x-exp(x)+2)*log(exp(x)*x-x^2-2*x)^2+((2+2*x)*exp(x)-4*x-4)*log(exp( 
x)*x-x^2-2*x)+(x^2+x)*exp(x)-x^3-3*x^2-2*x)*exp((log(exp(x)*x-x^2-2*x)^2+x 
^2)/x)+(2*x^2+2*x)*exp(x)-2*x^3-6*x^2-4*x)/(exp(x)*x-x^2-2*x),x)
 

Output:

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