\(\int \frac {1242+3728 x+3730 x^2+1244 x^3+e^{\frac {2 x-x^2}{2+2 x}} (2 x^3-2 x^4-x^5)+(-2 x-4 x^2-2 x^3+e^{\frac {2 x-x^2}{2+2 x}} (4 x^2-4 x^3-2 x^4)) \log (x)+e^{\frac {2 x-x^2}{2+2 x}} (2 x-2 x^2-x^3) \log ^2(x)}{2 x^3+4 x^4+2 x^5+(4 x^2+8 x^3+4 x^4) \log (x)+(2 x+4 x^2+2 x^3) \log ^2(x)} \, dx\) [1819]

Optimal result

Integrand size = 199, antiderivative size = 28 \[ \int \frac {1242+3728 x+3730 x^2+1244 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x^3-2 x^4-x^5\right )+\left (-2 x-4 x^2-2 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (4 x^2-4 x^3-2 x^4\right )\right ) \log (x)+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x-2 x^2-x^3\right ) \log ^2(x)}{2 x^3+4 x^4+2 x^5+\left (4 x^2+8 x^3+4 x^4\right ) \log (x)+\left (2 x+4 x^2+2 x^3\right ) \log ^2(x)} \, dx=e^{x-\frac {3 x}{2+\frac {2}{x}}}-\frac {621+x}{x+\log (x)} \] Output:

exp(x-3/(2/x+2)*x)-(621+x)/(x+ln(x))
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {1242+3728 x+3730 x^2+1244 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x^3-2 x^4-x^5\right )+\left (-2 x-4 x^2-2 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (4 x^2-4 x^3-2 x^4\right )\right ) \log (x)+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x-2 x^2-x^3\right ) \log ^2(x)}{2 x^3+4 x^4+2 x^5+\left (4 x^2+8 x^3+4 x^4\right ) \log (x)+\left (2 x+4 x^2+2 x^3\right ) \log ^2(x)} \, dx=\frac {1}{2} \left (2 e^{-\frac {(-2+x) x}{2 (1+x)}}-\frac {2 (621+x)}{x+\log (x)}\right ) \] Input:

Integrate[(1242 + 3728*x + 3730*x^2 + 1244*x^3 + E^((2*x - x^2)/(2 + 2*x)) 
*(2*x^3 - 2*x^4 - x^5) + (-2*x - 4*x^2 - 2*x^3 + E^((2*x - x^2)/(2 + 2*x)) 
*(4*x^2 - 4*x^3 - 2*x^4))*Log[x] + E^((2*x - x^2)/(2 + 2*x))*(2*x - 2*x^2 
- x^3)*Log[x]^2)/(2*x^3 + 4*x^4 + 2*x^5 + (4*x^2 + 8*x^3 + 4*x^4)*Log[x] + 
 (2*x + 4*x^2 + 2*x^3)*Log[x]^2),x]
 

Output:

(2/E^(((-2 + x)*x)/(2*(1 + x))) - (2*(621 + x))/(x + Log[x]))/2
 

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.

\(\displaystyle \int \frac {1244 x^3+3730 x^2+e^{\frac {2 x-x^2}{2 x+2}} \left (-x^3-2 x^2+2 x\right ) \log ^2(x)+\left (-2 x^3-4 x^2+e^{\frac {2 x-x^2}{2 x+2}} \left (-2 x^4-4 x^3+4 x^2\right )-2 x\right ) \log (x)+e^{\frac {2 x-x^2}{2 x+2}} \left (-x^5-2 x^4+2 x^3\right )+3728 x+1242}{2 x^5+4 x^4+2 x^3+\left (2 x^3+4 x^2+2 x\right ) \log ^2(x)+\left (4 x^4+8 x^3+4 x^2\right ) \log (x)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {1244 x^3+3730 x^2+e^{\frac {2 x-x^2}{2 x+2}} \left (-x^3-2 x^2+2 x\right ) \log ^2(x)+\left (-2 x^3-4 x^2+e^{\frac {2 x-x^2}{2 x+2}} \left (-2 x^4-4 x^3+4 x^2\right )-2 x\right ) \log (x)+e^{\frac {2 x-x^2}{2 x+2}} \left (-x^5-2 x^4+2 x^3\right )+3728 x+1242}{2 x (x+1)^2 (x+\log (x))^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \int \frac {1244 x^3+3730 x^2+3728 x+e^{\frac {2 x-x^2}{2 (x+1)}} \left (-x^3-2 x^2+2 x\right ) \log ^2(x)+e^{\frac {2 x-x^2}{2 (x+1)}} \left (-x^5-2 x^4+2 x^3\right )-2 \left (x^3+2 x^2+x-e^{\frac {2 x-x^2}{2 (x+1)}} \left (-x^4-2 x^3+2 x^2\right )\right ) \log (x)+1242}{x (x+1)^2 (x+\log (x))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (-\frac {2 \log (x) x^2}{(x+1)^2 (x+\log (x))^2}+\frac {1244 x^2}{(x+1)^2 (x+\log (x))^2}-\frac {4 \log (x) x}{(x+1)^2 (x+\log (x))^2}+\frac {3730 x}{(x+1)^2 (x+\log (x))^2}-\frac {e^{-\frac {(x-2) x}{2 (x+1)}} \left (x^2+2 x-2\right )}{(x+1)^2}-\frac {2 \log (x)}{(x+1)^2 (x+\log (x))^2}+\frac {3728}{(x+1)^2 (x+\log (x))^2}+\frac {1242}{(x+1)^2 (x+\log (x))^2 x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (1244 \int \frac {1}{(x+\log (x))^2}dx+1242 \int \frac {1}{x (x+\log (x))^2}dx+2 \int \frac {x}{(x+\log (x))^2}dx-2 \int \frac {1}{x+\log (x)}dx+2 e^{\frac {(2-x) x}{2 (x+1)}}\right )\)

Input:

Int[(1242 + 3728*x + 3730*x^2 + 1244*x^3 + E^((2*x - x^2)/(2 + 2*x))*(2*x^ 
3 - 2*x^4 - x^5) + (-2*x - 4*x^2 - 2*x^3 + E^((2*x - x^2)/(2 + 2*x))*(4*x^ 
2 - 4*x^3 - 2*x^4))*Log[x] + E^((2*x - x^2)/(2 + 2*x))*(2*x - 2*x^2 - x^3) 
*Log[x]^2)/(2*x^3 + 4*x^4 + 2*x^5 + (4*x^2 + 8*x^3 + 4*x^4)*Log[x] + (2*x 
+ 4*x^2 + 2*x^3)*Log[x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 169.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89

method	result	size
risch	\({\mathrm e}^{-\frac {\left (-2+x \right ) x}{2 \left (1+x \right )}}-\frac {621+x}{x +\ln \left (x \right )}\)	\(25\)
parallelrisch	\(-\frac {-8 x \,{\mathrm e}^{\frac {-x^{2}+2 x}{2+2 x}}+4968-8 \ln \left (x \right ) {\mathrm e}^{\frac {-x^{2}+2 x}{2+2 x}}+8 x}{8 \left (x +\ln \left (x \right )\right )}\)	\(55\)

Input:

int(((-x^3-2*x^2+2*x)*exp((-x^2+2*x)/(2+2*x))*ln(x)^2+((-2*x^4-4*x^3+4*x^2 
)*exp((-x^2+2*x)/(2+2*x))-2*x^3-4*x^2-2*x)*ln(x)+(-x^5-2*x^4+2*x^3)*exp((- 
x^2+2*x)/(2+2*x))+1244*x^3+3730*x^2+3728*x+1242)/((2*x^3+4*x^2+2*x)*ln(x)^ 
2+(4*x^4+8*x^3+4*x^2)*ln(x)+2*x^5+4*x^4+2*x^3),x,method=_RETURNVERBOSE)
 

Output:

exp(-1/2*(-2+x)*x/(1+x))-(621+x)/(x+ln(x))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {1242+3728 x+3730 x^2+1244 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x^3-2 x^4-x^5\right )+\left (-2 x-4 x^2-2 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (4 x^2-4 x^3-2 x^4\right )\right ) \log (x)+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x-2 x^2-x^3\right ) \log ^2(x)}{2 x^3+4 x^4+2 x^5+\left (4 x^2+8 x^3+4 x^4\right ) \log (x)+\left (2 x+4 x^2+2 x^3\right ) \log ^2(x)} \, dx=\frac {x e^{\left (-\frac {x^{2} - 2 \, x}{2 \, {\left (x + 1\right )}}\right )} + e^{\left (-\frac {x^{2} - 2 \, x}{2 \, {\left (x + 1\right )}}\right )} \log \left (x\right ) - x - 621}{x + \log \left (x\right )} \] Input:

integrate(((-x^3-2*x^2+2*x)*exp((-x^2+2*x)/(2+2*x))*log(x)^2+((-2*x^4-4*x^ 
3+4*x^2)*exp((-x^2+2*x)/(2+2*x))-2*x^3-4*x^2-2*x)*log(x)+(-x^5-2*x^4+2*x^3 
)*exp((-x^2+2*x)/(2+2*x))+1244*x^3+3730*x^2+3728*x+1242)/((2*x^3+4*x^2+2*x 
)*log(x)^2+(4*x^4+8*x^3+4*x^2)*log(x)+2*x^5+4*x^4+2*x^3),x, algorithm="fri 
cas")
 

Output:

(x*e^(-1/2*(x^2 - 2*x)/(x + 1)) + e^(-1/2*(x^2 - 2*x)/(x + 1))*log(x) - x 
- 621)/(x + log(x))
 

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {1242+3728 x+3730 x^2+1244 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x^3-2 x^4-x^5\right )+\left (-2 x-4 x^2-2 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (4 x^2-4 x^3-2 x^4\right )\right ) \log (x)+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x-2 x^2-x^3\right ) \log ^2(x)}{2 x^3+4 x^4+2 x^5+\left (4 x^2+8 x^3+4 x^4\right ) \log (x)+\left (2 x+4 x^2+2 x^3\right ) \log ^2(x)} \, dx=\frac {- x - 621}{x + \log {\left (x \right )}} + e^{\frac {- x^{2} + 2 x}{2 x + 2}} \] Input:

integrate(((-x**3-2*x**2+2*x)*exp((-x**2+2*x)/(2+2*x))*ln(x)**2+((-2*x**4- 
4*x**3+4*x**2)*exp((-x**2+2*x)/(2+2*x))-2*x**3-4*x**2-2*x)*ln(x)+(-x**5-2* 
x**4+2*x**3)*exp((-x**2+2*x)/(2+2*x))+1244*x**3+3730*x**2+3728*x+1242)/((2 
*x**3+4*x**2+2*x)*ln(x)**2+(4*x**4+8*x**3+4*x**2)*ln(x)+2*x**5+4*x**4+2*x* 
*3),x)
 

Output:

(-x - 621)/(x + log(x)) + exp((-x**2 + 2*x)/(2*x + 2))
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {1242+3728 x+3730 x^2+1244 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x^3-2 x^4-x^5\right )+\left (-2 x-4 x^2-2 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (4 x^2-4 x^3-2 x^4\right )\right ) \log (x)+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x-2 x^2-x^3\right ) \log ^2(x)}{2 x^3+4 x^4+2 x^5+\left (4 x^2+8 x^3+4 x^4\right ) \log (x)+\left (2 x+4 x^2+2 x^3\right ) \log ^2(x)} \, dx=-\frac {{\left ({\left (x + 621\right )} e^{\left (\frac {1}{2} \, x\right )} - {\left (x e^{\frac {3}{2}} + e^{\frac {3}{2}} \log \left (x\right )\right )} e^{\left (-\frac {3}{2 \, {\left (x + 1\right )}}\right )}\right )} e^{\left (-\frac {1}{2} \, x\right )}}{x + \log \left (x\right )} \] Input:

integrate(((-x^3-2*x^2+2*x)*exp((-x^2+2*x)/(2+2*x))*log(x)^2+((-2*x^4-4*x^ 
3+4*x^2)*exp((-x^2+2*x)/(2+2*x))-2*x^3-4*x^2-2*x)*log(x)+(-x^5-2*x^4+2*x^3 
)*exp((-x^2+2*x)/(2+2*x))+1244*x^3+3730*x^2+3728*x+1242)/((2*x^3+4*x^2+2*x 
)*log(x)^2+(4*x^4+8*x^3+4*x^2)*log(x)+2*x^5+4*x^4+2*x^3),x, algorithm="max 
ima")
 

Output:

-((x + 621)*e^(1/2*x) - (x*e^(3/2) + e^(3/2)*log(x))*e^(-3/2/(x + 1)))*e^( 
-1/2*x)/(x + log(x))
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {1242+3728 x+3730 x^2+1244 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x^3-2 x^4-x^5\right )+\left (-2 x-4 x^2-2 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (4 x^2-4 x^3-2 x^4\right )\right ) \log (x)+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x-2 x^2-x^3\right ) \log ^2(x)}{2 x^3+4 x^4+2 x^5+\left (4 x^2+8 x^3+4 x^4\right ) \log (x)+\left (2 x+4 x^2+2 x^3\right ) \log ^2(x)} \, dx=\frac {x e^{\left (-\frac {x^{2} - 2 \, x}{2 \, {\left (x + 1\right )}}\right )} + e^{\left (-\frac {x^{2} - 2 \, x}{2 \, {\left (x + 1\right )}}\right )} \log \left (x\right ) - x - 621}{x + \log \left (x\right )} \] Input:

integrate(((-x^3-2*x^2+2*x)*exp((-x^2+2*x)/(2+2*x))*log(x)^2+((-2*x^4-4*x^ 
3+4*x^2)*exp((-x^2+2*x)/(2+2*x))-2*x^3-4*x^2-2*x)*log(x)+(-x^5-2*x^4+2*x^3 
)*exp((-x^2+2*x)/(2+2*x))+1244*x^3+3730*x^2+3728*x+1242)/((2*x^3+4*x^2+2*x 
)*log(x)^2+(4*x^4+8*x^3+4*x^2)*log(x)+2*x^5+4*x^4+2*x^3),x, algorithm="gia 
c")
 

Output:

(x*e^(-1/2*(x^2 - 2*x)/(x + 1)) + e^(-1/2*(x^2 - 2*x)/(x + 1))*log(x) - x 
- 621)/(x + log(x))
 

Mupad [B] (verification not implemented)

Time = 3.32 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {1242+3728 x+3730 x^2+1244 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x^3-2 x^4-x^5\right )+\left (-2 x-4 x^2-2 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (4 x^2-4 x^3-2 x^4\right )\right ) \log (x)+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x-2 x^2-x^3\right ) \log ^2(x)}{2 x^3+4 x^4+2 x^5+\left (4 x^2+8 x^3+4 x^4\right ) \log (x)+\left (2 x+4 x^2+2 x^3\right ) \log ^2(x)} \, dx={\mathrm {e}}^{\frac {2\,x}{2\,x+2}-\frac {x^2}{2\,x+2}}+\frac {1}{x+1}-\frac {\frac {622\,x+621}{x+1}-\frac {x\,\ln \left (x\right )}{x+1}}{x+\ln \left (x\right )} \] Input:

int((3728*x - log(x)*(2*x + exp((2*x - x^2)/(2*x + 2))*(4*x^3 - 4*x^2 + 2* 
x^4) + 4*x^2 + 2*x^3) - exp((2*x - x^2)/(2*x + 2))*(2*x^4 - 2*x^3 + x^5) + 
 3730*x^2 + 1244*x^3 - exp((2*x - x^2)/(2*x + 2))*log(x)^2*(2*x^2 - 2*x + 
x^3) + 1242)/(log(x)^2*(2*x + 4*x^2 + 2*x^3) + log(x)*(4*x^2 + 8*x^3 + 4*x 
^4) + 2*x^3 + 4*x^4 + 2*x^5),x)
 

Output:

exp((2*x)/(2*x + 2) - x^2/(2*x + 2)) + 1/(x + 1) - ((622*x + 621)/(x + 1) 
- (x*log(x))/(x + 1))/(x + log(x))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.39 \[ \int \frac {1242+3728 x+3730 x^2+1244 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x^3-2 x^4-x^5\right )+\left (-2 x-4 x^2-2 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (4 x^2-4 x^3-2 x^4\right )\right ) \log (x)+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x-2 x^2-x^3\right ) \log ^2(x)}{2 x^3+4 x^4+2 x^5+\left (4 x^2+8 x^3+4 x^4\right ) \log (x)+\left (2 x+4 x^2+2 x^3\right ) \log ^2(x)} \, dx=\frac {e^{\frac {x^{2}+2}{2 x +2}} \mathrm {log}\left (x \right )-621 e^{\frac {x^{2}+2}{2 x +2}}+\mathrm {log}\left (x \right ) e +e x}{e^{\frac {x^{2}+2}{2 x +2}} \left (\mathrm {log}\left (x \right )+x \right )} \] Input:

int(((-x^3-2*x^2+2*x)*exp((-x^2+2*x)/(2+2*x))*log(x)^2+((-2*x^4-4*x^3+4*x^ 
2)*exp((-x^2+2*x)/(2+2*x))-2*x^3-4*x^2-2*x)*log(x)+(-x^5-2*x^4+2*x^3)*exp( 
(-x^2+2*x)/(2+2*x))+1244*x^3+3730*x^2+3728*x+1242)/((2*x^3+4*x^2+2*x)*log( 
x)^2+(4*x^4+8*x^3+4*x^2)*log(x)+2*x^5+4*x^4+2*x^3),x)
 

Output:

(e**((x**2 + 2)/(2*x + 2))*log(x) - 621*e**((x**2 + 2)/(2*x + 2)) + log(x) 
*e + e*x)/(e**((x**2 + 2)/(2*x + 2))*(log(x) + x))