\(\int \frac {8 x+15 x^3+2 x^5+e^{5 x} (-2 x^2-2 x^3-4 x^4-4 x^5)+e^{4 x} (-6 x^3-6 x^4-12 x^5-12 x^6)+e^x (8 x-3 x^2+16 x^3+10 x^4+8 x^6)+e^{3 x} (-1+4 x+2 x^2+8 x^3-6 x^4-6 x^5-12 x^6-12 x^7)+e^{2 x} (-3 x+4 x^2+10 x^3+8 x^4+6 x^5-2 x^6-4 x^7-4 x^8)}{x^4+2 x^6+e^{3 x} (x+2 x^3)+e^{2 x} (3 x^2+6 x^4)+e^x (3 x^3+6 x^5)} \, dx\) [14]

Optimal result

Integrand size = 239, antiderivative size = 36 \[ \int \frac {8 x+15 x^3+2 x^5+e^{5 x} \left (-2 x^2-2 x^3-4 x^4-4 x^5\right )+e^{4 x} \left (-6 x^3-6 x^4-12 x^5-12 x^6\right )+e^x \left (8 x-3 x^2+16 x^3+10 x^4+8 x^6\right )+e^{3 x} \left (-1+4 x+2 x^2+8 x^3-6 x^4-6 x^5-12 x^6-12 x^7\right )+e^{2 x} \left (-3 x+4 x^2+10 x^3+8 x^4+6 x^5-2 x^6-4 x^7-4 x^8\right )}{x^4+2 x^6+e^{3 x} \left (x+2 x^3\right )+e^{2 x} \left (3 x^2+6 x^4\right )+e^x \left (3 x^3+6 x^5\right )} \, dx=-\left (e^x x-\frac {2}{e^x+x}\right )^2+\log \left (-x+\frac {1}{5} \left (-\frac {2}{x}+x\right )\right ) \] Output:

ln(-4/5*x-2/5/x)-(exp(x)*x-2/(exp(x)+x))^2
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 9.69 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.11 \[ \int \frac {8 x+15 x^3+2 x^5+e^{5 x} \left (-2 x^2-2 x^3-4 x^4-4 x^5\right )+e^{4 x} \left (-6 x^3-6 x^4-12 x^5-12 x^6\right )+e^x \left (8 x-3 x^2+16 x^3+10 x^4+8 x^6\right )+e^{3 x} \left (-1+4 x+2 x^2+8 x^3-6 x^4-6 x^5-12 x^6-12 x^7\right )+e^{2 x} \left (-3 x+4 x^2+10 x^3+8 x^4+6 x^5-2 x^6-4 x^7-4 x^8\right )}{x^4+2 x^6+e^{3 x} \left (x+2 x^3\right )+e^{2 x} \left (3 x^2+6 x^4\right )+e^x \left (3 x^3+6 x^5\right )} \, dx=-\frac {\left (-2+e^{2 x} x+e^x x^2\right )^2}{\left (e^x+x\right )^2}-\log (x)+\log \left (1+2 x^2\right ) \] Input:

Integrate[(8*x + 15*x^3 + 2*x^5 + E^(5*x)*(-2*x^2 - 2*x^3 - 4*x^4 - 4*x^5) 
 + E^(4*x)*(-6*x^3 - 6*x^4 - 12*x^5 - 12*x^6) + E^x*(8*x - 3*x^2 + 16*x^3 
+ 10*x^4 + 8*x^6) + E^(3*x)*(-1 + 4*x + 2*x^2 + 8*x^3 - 6*x^4 - 6*x^5 - 12 
*x^6 - 12*x^7) + E^(2*x)*(-3*x + 4*x^2 + 10*x^3 + 8*x^4 + 6*x^5 - 2*x^6 - 
4*x^7 - 4*x^8))/(x^4 + 2*x^6 + E^(3*x)*(x + 2*x^3) + E^(2*x)*(3*x^2 + 6*x^ 
4) + E^x*(3*x^3 + 6*x^5)),x]
 

Output:

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

\(\displaystyle \int \frac {2 x^5+15 x^3+e^{4 x} \left (-12 x^6-12 x^5-6 x^4-6 x^3\right )+e^x \left (8 x^6+10 x^4+16 x^3-3 x^2+8 x\right )+e^{5 x} \left (-4 x^5-4 x^4-2 x^3-2 x^2\right )+e^{3 x} \left (-12 x^7-12 x^6-6 x^5-6 x^4+8 x^3+2 x^2+4 x-1\right )+e^{2 x} \left (-4 x^8-4 x^7-2 x^6+6 x^5+8 x^4+10 x^3+4 x^2-3 x\right )+8 x}{2 x^6+x^4+e^{3 x} \left (2 x^3+x\right )+e^x \left (6 x^5+3 x^3\right )+e^{2 x} \left (6 x^4+3 x^2\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {2 x^5+15 x^3+e^{4 x} \left (-12 x^6-12 x^5-6 x^4-6 x^3\right )+e^x \left (8 x^6+10 x^4+16 x^3-3 x^2+8 x\right )+e^{5 x} \left (-4 x^5-4 x^4-2 x^3-2 x^2\right )+e^{3 x} \left (-12 x^7-12 x^6-6 x^5-6 x^4+8 x^3+2 x^2+4 x-1\right )+e^{2 x} \left (-4 x^8-4 x^7-2 x^6+6 x^5+8 x^4+10 x^3+4 x^2-3 x\right )+8 x}{x \left (x+e^x\right )^3 \left (2 x^2+1\right )}dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (-\frac {4 \left (x^3-x^2-2\right )}{\left (x+e^x\right )^2}+\frac {8 x^3+2 x^2+4 x-1}{x \left (2 x^2+1\right )}-\frac {8 (x-1)}{\left (x+e^x\right )^3}-2 e^{2 x} x (x+1)+\frac {4 (x-2) x}{x+e^x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -4 \int \frac {x^3}{\left (x+e^x\right )^2}dx+4 \int \frac {x^2}{\left (x+e^x\right )^2}dx+4 \int \frac {x^2}{x+e^x}dx-8 \int \frac {x}{x+e^x}dx-e^{2 x} x^2+\log \left (2 x^2+1\right )+4 x-\frac {4}{\left (x+e^x\right )^2}-\log (x)\)

Input:

Int[(8*x + 15*x^3 + 2*x^5 + E^(5*x)*(-2*x^2 - 2*x^3 - 4*x^4 - 4*x^5) + E^( 
4*x)*(-6*x^3 - 6*x^4 - 12*x^5 - 12*x^6) + E^x*(8*x - 3*x^2 + 16*x^3 + 10*x 
^4 + 8*x^6) + E^(3*x)*(-1 + 4*x + 2*x^2 + 8*x^3 - 6*x^4 - 6*x^5 - 12*x^6 - 
 12*x^7) + E^(2*x)*(-3*x + 4*x^2 + 10*x^3 + 8*x^4 + 6*x^5 - 2*x^6 - 4*x^7 
- 4*x^8))/(x^4 + 2*x^6 + E^(3*x)*(x + 2*x^3) + E^(2*x)*(3*x^2 + 6*x^4) + E 
^x*(3*x^3 + 6*x^5)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.97 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.25

method	result	size
risch	\(4 x -\ln \left (x \right )+\ln \left (2 x^{2}+1\right )-{\mathrm e}^{2 x} x^{2}-\frac {4 \left (x^{3}+{\mathrm e}^{x} x^{2}+1\right )}{\left ({\mathrm e}^{x}+x \right )^{2}}\)	\(45\)
parallelrisch	\(-\frac {{\mathrm e}^{2 x} x^{4}+2 x^{3} {\mathrm e}^{3 x}+x^{2} {\mathrm e}^{4 x}+x^{2} \ln \left (x \right )+2 x \,{\mathrm e}^{x} \ln \left (x \right )+{\mathrm e}^{2 x} \ln \left (x \right )-\ln \left (\frac {1}{2}+x^{2}\right ) x^{2}-2 \ln \left (\frac {1}{2}+x^{2}\right ) x \,{\mathrm e}^{x}-\ln \left (\frac {1}{2}+x^{2}\right ) {\mathrm e}^{2 x}-4 \,{\mathrm e}^{x} x^{2}+4-4 x \,{\mathrm e}^{2 x}}{{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x +x^{2}}\)	\(113\)

Input:

int(((-4*x^5-4*x^4-2*x^3-2*x^2)*exp(x)^5+(-12*x^6-12*x^5-6*x^4-6*x^3)*exp( 
x)^4+(-12*x^7-12*x^6-6*x^5-6*x^4+8*x^3+2*x^2+4*x-1)*exp(x)^3+(-4*x^8-4*x^7 
-2*x^6+6*x^5+8*x^4+10*x^3+4*x^2-3*x)*exp(x)^2+(8*x^6+10*x^4+16*x^3-3*x^2+8 
*x)*exp(x)+2*x^5+15*x^3+8*x)/((2*x^3+x)*exp(x)^3+(6*x^4+3*x^2)*exp(x)^2+(6 
*x^5+3*x^3)*exp(x)+2*x^6+x^4),x,method=_RETURNVERBOSE)
 

Output:

4*x-ln(x)+ln(2*x^2+1)-exp(x)^2*x^2-4*(x^3+exp(x)*x^2+1)/(exp(x)+x)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (28) = 56\).

Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.61 \[ \int \frac {8 x+15 x^3+2 x^5+e^{5 x} \left (-2 x^2-2 x^3-4 x^4-4 x^5\right )+e^{4 x} \left (-6 x^3-6 x^4-12 x^5-12 x^6\right )+e^x \left (8 x-3 x^2+16 x^3+10 x^4+8 x^6\right )+e^{3 x} \left (-1+4 x+2 x^2+8 x^3-6 x^4-6 x^5-12 x^6-12 x^7\right )+e^{2 x} \left (-3 x+4 x^2+10 x^3+8 x^4+6 x^5-2 x^6-4 x^7-4 x^8\right )}{x^4+2 x^6+e^{3 x} \left (x+2 x^3\right )+e^{2 x} \left (3 x^2+6 x^4\right )+e^x \left (3 x^3+6 x^5\right )} \, dx=-\frac {2 \, x^{3} e^{\left (3 \, x\right )} + x^{2} e^{\left (4 \, x\right )} - 4 \, x^{2} e^{x} + {\left (x^{4} - 4 \, x\right )} e^{\left (2 \, x\right )} - {\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )} \log \left (2 \, x^{2} + 1\right ) + {\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )} \log \left (x\right ) + 4}{x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}} \] Input:

integrate(((-4*x^5-4*x^4-2*x^3-2*x^2)*exp(x)^5+(-12*x^6-12*x^5-6*x^4-6*x^3 
)*exp(x)^4+(-12*x^7-12*x^6-6*x^5-6*x^4+8*x^3+2*x^2+4*x-1)*exp(x)^3+(-4*x^8 
-4*x^7-2*x^6+6*x^5+8*x^4+10*x^3+4*x^2-3*x)*exp(x)^2+(8*x^6+10*x^4+16*x^3-3 
*x^2+8*x)*exp(x)+2*x^5+15*x^3+8*x)/((2*x^3+x)*exp(x)^3+(6*x^4+3*x^2)*exp(x 
)^2+(6*x^5+3*x^3)*exp(x)+2*x^6+x^4),x, algorithm="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (26) = 52\).

Time = 0.15 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.50 \[ \int \frac {8 x+15 x^3+2 x^5+e^{5 x} \left (-2 x^2-2 x^3-4 x^4-4 x^5\right )+e^{4 x} \left (-6 x^3-6 x^4-12 x^5-12 x^6\right )+e^x \left (8 x-3 x^2+16 x^3+10 x^4+8 x^6\right )+e^{3 x} \left (-1+4 x+2 x^2+8 x^3-6 x^4-6 x^5-12 x^6-12 x^7\right )+e^{2 x} \left (-3 x+4 x^2+10 x^3+8 x^4+6 x^5-2 x^6-4 x^7-4 x^8\right )}{x^4+2 x^6+e^{3 x} \left (x+2 x^3\right )+e^{2 x} \left (3 x^2+6 x^4\right )+e^x \left (3 x^3+6 x^5\right )} \, dx=- x^{2} e^{2 x} + 4 x - \log {\left (x \right )} + \log {\left (2 x^{2} + 1 \right )} + \frac {- 4 x^{3} - 4 x^{2} e^{x} - 4}{x^{2} + 2 x e^{x} + e^{2 x}} \] Input:

integrate(((-4*x**5-4*x**4-2*x**3-2*x**2)*exp(x)**5+(-12*x**6-12*x**5-6*x* 
*4-6*x**3)*exp(x)**4+(-12*x**7-12*x**6-6*x**5-6*x**4+8*x**3+2*x**2+4*x-1)* 
exp(x)**3+(-4*x**8-4*x**7-2*x**6+6*x**5+8*x**4+10*x**3+4*x**2-3*x)*exp(x)* 
*2+(8*x**6+10*x**4+16*x**3-3*x**2+8*x)*exp(x)+2*x**5+15*x**3+8*x)/((2*x**3 
+x)*exp(x)**3+(6*x**4+3*x**2)*exp(x)**2+(6*x**5+3*x**3)*exp(x)+2*x**6+x**4 
),x)
 

Output:

-x**2*exp(2*x) + 4*x - log(x) + log(2*x**2 + 1) + (-4*x**3 - 4*x**2*exp(x) 
 - 4)/(x**2 + 2*x*exp(x) + exp(2*x))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (28) = 56\).

Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.89 \[ \int \frac {8 x+15 x^3+2 x^5+e^{5 x} \left (-2 x^2-2 x^3-4 x^4-4 x^5\right )+e^{4 x} \left (-6 x^3-6 x^4-12 x^5-12 x^6\right )+e^x \left (8 x-3 x^2+16 x^3+10 x^4+8 x^6\right )+e^{3 x} \left (-1+4 x+2 x^2+8 x^3-6 x^4-6 x^5-12 x^6-12 x^7\right )+e^{2 x} \left (-3 x+4 x^2+10 x^3+8 x^4+6 x^5-2 x^6-4 x^7-4 x^8\right )}{x^4+2 x^6+e^{3 x} \left (x+2 x^3\right )+e^{2 x} \left (3 x^2+6 x^4\right )+e^x \left (3 x^3+6 x^5\right )} \, dx=-\frac {2 \, x^{3} e^{\left (3 \, x\right )} + x^{2} e^{\left (4 \, x\right )} - 4 \, x^{2} e^{x} + {\left (x^{4} - 4 \, x\right )} e^{\left (2 \, x\right )} + 4}{x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}} + \log \left (2 \, x^{2} + 1\right ) - \log \left (x\right ) \] Input:

integrate(((-4*x^5-4*x^4-2*x^3-2*x^2)*exp(x)^5+(-12*x^6-12*x^5-6*x^4-6*x^3 
)*exp(x)^4+(-12*x^7-12*x^6-6*x^5-6*x^4+8*x^3+2*x^2+4*x-1)*exp(x)^3+(-4*x^8 
-4*x^7-2*x^6+6*x^5+8*x^4+10*x^3+4*x^2-3*x)*exp(x)^2+(8*x^6+10*x^4+16*x^3-3 
*x^2+8*x)*exp(x)+2*x^5+15*x^3+8*x)/((2*x^3+x)*exp(x)^3+(6*x^4+3*x^2)*exp(x 
)^2+(6*x^5+3*x^3)*exp(x)+2*x^6+x^4),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (28) = 56\).

Time = 0.14 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.28 \[ \int \frac {8 x+15 x^3+2 x^5+e^{5 x} \left (-2 x^2-2 x^3-4 x^4-4 x^5\right )+e^{4 x} \left (-6 x^3-6 x^4-12 x^5-12 x^6\right )+e^x \left (8 x-3 x^2+16 x^3+10 x^4+8 x^6\right )+e^{3 x} \left (-1+4 x+2 x^2+8 x^3-6 x^4-6 x^5-12 x^6-12 x^7\right )+e^{2 x} \left (-3 x+4 x^2+10 x^3+8 x^4+6 x^5-2 x^6-4 x^7-4 x^8\right )}{x^4+2 x^6+e^{3 x} \left (x+2 x^3\right )+e^{2 x} \left (3 x^2+6 x^4\right )+e^x \left (3 x^3+6 x^5\right )} \, dx=-\frac {x^{4} e^{\left (2 \, x\right )} + 2 \, x^{3} e^{\left (3 \, x\right )} + x^{2} e^{\left (4 \, x\right )} - 4 \, x^{2} e^{x} - x^{2} \log \left (2 \, x^{2} + 1\right ) - 2 \, x e^{x} \log \left (2 \, x^{2} + 1\right ) + x^{2} \log \left (x\right ) + 2 \, x e^{x} \log \left (x\right ) - 4 \, x e^{\left (2 \, x\right )} - e^{\left (2 \, x\right )} \log \left (2 \, x^{2} + 1\right ) + e^{\left (2 \, x\right )} \log \left (x\right ) + 4}{x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}} \] Input:

integrate(((-4*x^5-4*x^4-2*x^3-2*x^2)*exp(x)^5+(-12*x^6-12*x^5-6*x^4-6*x^3 
)*exp(x)^4+(-12*x^7-12*x^6-6*x^5-6*x^4+8*x^3+2*x^2+4*x-1)*exp(x)^3+(-4*x^8 
-4*x^7-2*x^6+6*x^5+8*x^4+10*x^3+4*x^2-3*x)*exp(x)^2+(8*x^6+10*x^4+16*x^3-3 
*x^2+8*x)*exp(x)+2*x^5+15*x^3+8*x)/((2*x^3+x)*exp(x)^3+(6*x^4+3*x^2)*exp(x 
)^2+(6*x^5+3*x^3)*exp(x)+2*x^6+x^4),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.72 \[ \int \frac {8 x+15 x^3+2 x^5+e^{5 x} \left (-2 x^2-2 x^3-4 x^4-4 x^5\right )+e^{4 x} \left (-6 x^3-6 x^4-12 x^5-12 x^6\right )+e^x \left (8 x-3 x^2+16 x^3+10 x^4+8 x^6\right )+e^{3 x} \left (-1+4 x+2 x^2+8 x^3-6 x^4-6 x^5-12 x^6-12 x^7\right )+e^{2 x} \left (-3 x+4 x^2+10 x^3+8 x^4+6 x^5-2 x^6-4 x^7-4 x^8\right )}{x^4+2 x^6+e^{3 x} \left (x+2 x^3\right )+e^{2 x} \left (3 x^2+6 x^4\right )+e^x \left (3 x^3+6 x^5\right )} \, dx=4\,x+\ln \left (x^2+\frac {1}{2}\right )-\ln \left (x\right )-x^2\,{\mathrm {e}}^{2\,x}-\frac {4}{{\mathrm {e}}^{2\,x}+2\,x\,{\mathrm {e}}^x+x^2}+\frac {4\,\left (x^2-x^3\right )}{\left (x+{\mathrm {e}}^x\right )\,\left (x-1\right )} \] Input:

int((8*x - exp(3*x)*(6*x^4 - 2*x^2 - 8*x^3 - 4*x + 6*x^5 + 12*x^6 + 12*x^7 
 + 1) + exp(x)*(8*x - 3*x^2 + 16*x^3 + 10*x^4 + 8*x^6) - exp(2*x)*(3*x - 4 
*x^2 - 10*x^3 - 8*x^4 - 6*x^5 + 2*x^6 + 4*x^7 + 4*x^8) - exp(5*x)*(2*x^2 + 
 2*x^3 + 4*x^4 + 4*x^5) - exp(4*x)*(6*x^3 + 6*x^4 + 12*x^5 + 12*x^6) + 15* 
x^3 + 2*x^5)/(exp(x)*(3*x^3 + 6*x^5) + exp(2*x)*(3*x^2 + 6*x^4) + exp(3*x) 
*(x + 2*x^3) + x^4 + 2*x^6),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 130, normalized size of antiderivative = 3.61 \[ \int \frac {8 x+15 x^3+2 x^5+e^{5 x} \left (-2 x^2-2 x^3-4 x^4-4 x^5\right )+e^{4 x} \left (-6 x^3-6 x^4-12 x^5-12 x^6\right )+e^x \left (8 x-3 x^2+16 x^3+10 x^4+8 x^6\right )+e^{3 x} \left (-1+4 x+2 x^2+8 x^3-6 x^4-6 x^5-12 x^6-12 x^7\right )+e^{2 x} \left (-3 x+4 x^2+10 x^3+8 x^4+6 x^5-2 x^6-4 x^7-4 x^8\right )}{x^4+2 x^6+e^{3 x} \left (x+2 x^3\right )+e^{2 x} \left (3 x^2+6 x^4\right )+e^x \left (3 x^3+6 x^5\right )} \, dx=\frac {-e^{4 x} x^{2}-2 e^{3 x} x^{3}+e^{2 x} \mathrm {log}\left (2 x^{2}+1\right )-e^{2 x} \mathrm {log}\left (x \right )-e^{2 x} x^{4}+4 e^{2 x} x +2 e^{x} \mathrm {log}\left (2 x^{2}+1\right ) x -2 e^{x} \mathrm {log}\left (x \right ) x +4 e^{x} x^{2}+\mathrm {log}\left (2 x^{2}+1\right ) x^{2}-\mathrm {log}\left (x \right ) x^{2}-4}{e^{2 x}+2 e^{x} x +x^{2}} \] Input:

int(((-4*x^5-4*x^4-2*x^3-2*x^2)*exp(x)^5+(-12*x^6-12*x^5-6*x^4-6*x^3)*exp( 
x)^4+(-12*x^7-12*x^6-6*x^5-6*x^4+8*x^3+2*x^2+4*x-1)*exp(x)^3+(-4*x^8-4*x^7 
-2*x^6+6*x^5+8*x^4+10*x^3+4*x^2-3*x)*exp(x)^2+(8*x^6+10*x^4+16*x^3-3*x^2+8 
*x)*exp(x)+2*x^5+15*x^3+8*x)/((2*x^3+x)*exp(x)^3+(6*x^4+3*x^2)*exp(x)^2+(6 
*x^5+3*x^3)*exp(x)+2*x^6+x^4),x)
 

Output:

( - e**(4*x)*x**2 - 2*e**(3*x)*x**3 + e**(2*x)*log(2*x**2 + 1) - e**(2*x)* 
log(x) - e**(2*x)*x**4 + 4*e**(2*x)*x + 2*e**x*log(2*x**2 + 1)*x - 2*e**x* 
log(x)*x + 4*e**x*x**2 + log(2*x**2 + 1)*x**2 - log(x)*x**2 - 4)/(e**(2*x) 
 + 2*e**x*x + x**2)