\(\int \frac {-x^2+e^x (3 x+3 x^2-3 x^4)+(3 x^2+2 x^3-x^4+e^x (-6 x-3 x^2-3 x^4)) \log (x)+(x+x^2-2 x^3+e^x (-3+3 x+6 x^2-6 x^3)) \log ^2(x)+(-x^2+e^x (3+3 x-3 x^2)) \log ^3(x)}{x^4+2 x^5+x^6+(2 x^3+4 x^4+2 x^5) \log (x)+(x^2+2 x^3+x^4) \log ^2(x)} \, dx\) [2627]

Optimal result

Integrand size = 171, antiderivative size = 26 \[ \int \frac {-x^2+e^x \left (3 x+3 x^2-3 x^4\right )+\left (3 x^2+2 x^3-x^4+e^x \left (-6 x-3 x^2-3 x^4\right )\right ) \log (x)+\left (x+x^2-2 x^3+e^x \left (-3+3 x+6 x^2-6 x^3\right )\right ) \log ^2(x)+\left (-x^2+e^x \left (3+3 x-3 x^2\right )\right ) \log ^3(x)}{x^4+2 x^5+x^6+\left (2 x^3+4 x^4+2 x^5\right ) \log (x)+\left (x^2+2 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {\left (-3 e^x+x\right ) \left (\log (x)+\frac {x}{x+\log (x)}\right )}{x+x^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-x^2+e^x \left (3 x+3 x^2-3 x^4\right )+\left (3 x^2+2 x^3-x^4+e^x \left (-6 x-3 x^2-3 x^4\right )\right ) \log (x)+\left (x+x^2-2 x^3+e^x \left (-3+3 x+6 x^2-6 x^3\right )\right ) \log ^2(x)+\left (-x^2+e^x \left (3+3 x-3 x^2\right )\right ) \log ^3(x)}{x^4+2 x^5+x^6+\left (2 x^3+4 x^4+2 x^5\right ) \log (x)+\left (x^2+2 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {\left (-3 e^x+x\right ) \left (x+x \log (x)+\log ^2(x)\right )}{x (1+x) (x+\log (x))} \] Input:

Integrate[(-x^2 + E^x*(3*x + 3*x^2 - 3*x^4) + (3*x^2 + 2*x^3 - x^4 + E^x*( 
-6*x - 3*x^2 - 3*x^4))*Log[x] + (x + x^2 - 2*x^3 + E^x*(-3 + 3*x + 6*x^2 - 
 6*x^3))*Log[x]^2 + (-x^2 + E^x*(3 + 3*x - 3*x^2))*Log[x]^3)/(x^4 + 2*x^5 
+ x^6 + (2*x^3 + 4*x^4 + 2*x^5)*Log[x] + (x^2 + 2*x^3 + x^4)*Log[x]^2),x]
 

Output:

((-3*E^x + x)*(x + x*Log[x] + Log[x]^2))/(x*(1 + x)*(x + Log[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.

\(\displaystyle \int \frac {-x^2+\left (e^x \left (-3 x^2+3 x+3\right )-x^2\right ) \log ^3(x)+e^x \left (-3 x^4+3 x^2+3 x\right )+\left (-2 x^3+x^2+e^x \left (-6 x^3+6 x^2+3 x-3\right )+x\right ) \log ^2(x)+\left (-x^4+2 x^3+3 x^2+e^x \left (-3 x^4-3 x^2-6 x\right )\right ) \log (x)}{x^6+2 x^5+x^4+\left (2 x^5+4 x^4+2 x^3\right ) \log (x)+\left (x^4+2 x^3+x^2\right ) \log ^2(x)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-x^2+\left (e^x \left (-3 x^2+3 x+3\right )-x^2\right ) \log ^3(x)+e^x \left (-3 x^4+3 x^2+3 x\right )+\left (-2 x^3+x^2+e^x \left (-6 x^3+6 x^2+3 x-3\right )+x\right ) \log ^2(x)+\left (-x^4+2 x^3+3 x^2+e^x \left (-3 x^4-3 x^2-6 x\right )\right ) \log (x)}{x^2 (x+1)^2 (x+\log (x))^2}dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-x \left (3 e^x \left (x^3-x-1\right )+x\right )-\left (\left (x^2+3 e^x \left (x^2-x-1\right )\right ) \log ^3(x)\right )-(x-1) \left (e^x \left (6 x^2-3\right )+x (2 x+1)\right ) \log ^2(x)-x (x+1) \left (3 e^x \left (x^2-x+2\right )+(x-3) x\right ) \log (x)}{x^2 (x+1)^2 (x+\log (x))^2}dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7299

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

Input:

Int[(-x^2 + E^x*(3*x + 3*x^2 - 3*x^4) + (3*x^2 + 2*x^3 - x^4 + E^x*(-6*x - 
 3*x^2 - 3*x^4))*Log[x] + (x + x^2 - 2*x^3 + E^x*(-3 + 3*x + 6*x^2 - 6*x^3 
))*Log[x]^2 + (-x^2 + E^x*(3 + 3*x - 3*x^2))*Log[x]^3)/(x^4 + 2*x^5 + x^6 
+ (2*x^3 + 4*x^4 + 2*x^5)*Log[x] + (x^2 + 2*x^3 + x^4)*Log[x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42

\[\frac {\left (x -3 \,{\mathrm e}^{x}\right ) \ln \left (x \right )}{\left (1+x \right ) x}+\frac {x -3 \,{\mathrm e}^{x}}{\left (1+x \right ) \left (x +\ln \left (x \right )\right )}\]

Input:

int((((-3*x^2+3*x+3)*exp(x)-x^2)*ln(x)^3+((-6*x^3+6*x^2+3*x-3)*exp(x)-2*x^ 
3+x^2+x)*ln(x)^2+((-3*x^4-3*x^2-6*x)*exp(x)-x^4+2*x^3+3*x^2)*ln(x)+(-3*x^4 
+3*x^2+3*x)*exp(x)-x^2)/((x^4+2*x^3+x^2)*ln(x)^2+(2*x^5+4*x^4+2*x^3)*ln(x) 
+x^6+2*x^5+x^4),x)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.92 \[ \int \frac {-x^2+e^x \left (3 x+3 x^2-3 x^4\right )+\left (3 x^2+2 x^3-x^4+e^x \left (-6 x-3 x^2-3 x^4\right )\right ) \log (x)+\left (x+x^2-2 x^3+e^x \left (-3+3 x+6 x^2-6 x^3\right )\right ) \log ^2(x)+\left (-x^2+e^x \left (3+3 x-3 x^2\right )\right ) \log ^3(x)}{x^4+2 x^5+x^6+\left (2 x^3+4 x^4+2 x^5\right ) \log (x)+\left (x^2+2 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {{\left (x - 3 \, e^{x}\right )} \log \left (x\right )^{2} + x^{2} - 3 \, x e^{x} + {\left (x^{2} - 3 \, x e^{x}\right )} \log \left (x\right )}{x^{3} + x^{2} + {\left (x^{2} + x\right )} \log \left (x\right )} \] Input:

integrate((((-3*x^2+3*x+3)*exp(x)-x^2)*log(x)^3+((-6*x^3+6*x^2+3*x-3)*exp( 
x)-2*x^3+x^2+x)*log(x)^2+((-3*x^4-3*x^2-6*x)*exp(x)-x^4+2*x^3+3*x^2)*log(x 
)+(-3*x^4+3*x^2+3*x)*exp(x)-x^2)/((x^4+2*x^3+x^2)*log(x)^2+(2*x^5+4*x^4+2* 
x^3)*log(x)+x^6+2*x^5+x^4),x, algorithm="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (20) = 40\).

Time = 0.20 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \frac {-x^2+e^x \left (3 x+3 x^2-3 x^4\right )+\left (3 x^2+2 x^3-x^4+e^x \left (-6 x-3 x^2-3 x^4\right )\right ) \log (x)+\left (x+x^2-2 x^3+e^x \left (-3+3 x+6 x^2-6 x^3\right )\right ) \log ^2(x)+\left (-x^2+e^x \left (3+3 x-3 x^2\right )\right ) \log ^3(x)}{x^4+2 x^5+x^6+\left (2 x^3+4 x^4+2 x^5\right ) \log (x)+\left (x^2+2 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {x}{x^{2} + x + \left (x + 1\right ) \log {\left (x \right )}} + \frac {\left (- 3 x \log {\left (x \right )} - 3 x - 3 \log {\left (x \right )}^{2}\right ) e^{x}}{x^{3} + x^{2} \log {\left (x \right )} + x^{2} + x \log {\left (x \right )}} + \frac {\log {\left (x \right )}}{x + 1} \] Input:

integrate((((-3*x**2+3*x+3)*exp(x)-x**2)*ln(x)**3+((-6*x**3+6*x**2+3*x-3)* 
exp(x)-2*x**3+x**2+x)*ln(x)**2+((-3*x**4-3*x**2-6*x)*exp(x)-x**4+2*x**3+3* 
x**2)*ln(x)+(-3*x**4+3*x**2+3*x)*exp(x)-x**2)/((x**4+2*x**3+x**2)*ln(x)**2 
+(2*x**5+4*x**4+2*x**3)*ln(x)+x**6+2*x**5+x**4),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \frac {-x^2+e^x \left (3 x+3 x^2-3 x^4\right )+\left (3 x^2+2 x^3-x^4+e^x \left (-6 x-3 x^2-3 x^4\right )\right ) \log (x)+\left (x+x^2-2 x^3+e^x \left (-3+3 x+6 x^2-6 x^3\right )\right ) \log ^2(x)+\left (-x^2+e^x \left (3+3 x-3 x^2\right )\right ) \log ^3(x)}{x^4+2 x^5+x^6+\left (2 x^3+4 x^4+2 x^5\right ) \log (x)+\left (x^2+2 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} + x^{2} - 3 \, {\left (x \log \left (x\right ) + \log \left (x\right )^{2} + x\right )} e^{x}}{x^{3} + x^{2} + {\left (x^{2} + x\right )} \log \left (x\right )} \] Input:

integrate((((-3*x^2+3*x+3)*exp(x)-x^2)*log(x)^3+((-6*x^3+6*x^2+3*x-3)*exp( 
x)-2*x^3+x^2+x)*log(x)^2+((-3*x^4-3*x^2-6*x)*exp(x)-x^4+2*x^3+3*x^2)*log(x 
)+(-3*x^4+3*x^2+3*x)*exp(x)-x^2)/((x^4+2*x^3+x^2)*log(x)^2+(2*x^5+4*x^4+2* 
x^3)*log(x)+x^6+2*x^5+x^4),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (25) = 50\).

Time = 0.16 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \[ \int \frac {-x^2+e^x \left (3 x+3 x^2-3 x^4\right )+\left (3 x^2+2 x^3-x^4+e^x \left (-6 x-3 x^2-3 x^4\right )\right ) \log (x)+\left (x+x^2-2 x^3+e^x \left (-3+3 x+6 x^2-6 x^3\right )\right ) \log ^2(x)+\left (-x^2+e^x \left (3+3 x-3 x^2\right )\right ) \log ^3(x)}{x^4+2 x^5+x^6+\left (2 x^3+4 x^4+2 x^5\right ) \log (x)+\left (x^2+2 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {x^{2} \log \left (x\right ) - 3 \, x e^{x} \log \left (x\right ) + x \log \left (x\right )^{2} - 3 \, e^{x} \log \left (x\right )^{2} + x^{2} - 3 \, x e^{x}}{x^{3} + x^{2} \log \left (x\right ) + x^{2} + x \log \left (x\right )} \] Input:

integrate((((-3*x^2+3*x+3)*exp(x)-x^2)*log(x)^3+((-6*x^3+6*x^2+3*x-3)*exp( 
x)-2*x^3+x^2+x)*log(x)^2+((-3*x^4-3*x^2-6*x)*exp(x)-x^4+2*x^3+3*x^2)*log(x 
)+(-3*x^4+3*x^2+3*x)*exp(x)-x^2)/((x^4+2*x^3+x^2)*log(x)^2+(2*x^5+4*x^4+2* 
x^3)*log(x)+x^6+2*x^5+x^4),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {-x^2+e^x \left (3 x+3 x^2-3 x^4\right )+\left (3 x^2+2 x^3-x^4+e^x \left (-6 x-3 x^2-3 x^4\right )\right ) \log (x)+\left (x+x^2-2 x^3+e^x \left (-3+3 x+6 x^2-6 x^3\right )\right ) \log ^2(x)+\left (-x^2+e^x \left (3+3 x-3 x^2\right )\right ) \log ^3(x)}{x^4+2 x^5+x^6+\left (2 x^3+4 x^4+2 x^5\right ) \log (x)+\left (x^2+2 x^3+x^4\right ) \log ^2(x)} \, dx=\int \frac {\ln \left (x\right )\,\left (3\,x^2+2\,x^3-x^4-{\mathrm {e}}^x\,\left (3\,x^4+3\,x^2+6\,x\right )\right )+{\ln \left (x\right )}^2\,\left (x+x^2-2\,x^3+{\mathrm {e}}^x\,\left (-6\,x^3+6\,x^2+3\,x-3\right )\right )-x^2+{\mathrm {e}}^x\,\left (-3\,x^4+3\,x^2+3\,x\right )+{\ln \left (x\right )}^3\,\left ({\mathrm {e}}^x\,\left (-3\,x^2+3\,x+3\right )-x^2\right )}{\ln \left (x\right )\,\left (2\,x^5+4\,x^4+2\,x^3\right )+x^4+2\,x^5+x^6+{\ln \left (x\right )}^2\,\left (x^4+2\,x^3+x^2\right )} \,d x \] Input:

int((log(x)*(3*x^2 + 2*x^3 - x^4 - exp(x)*(6*x + 3*x^2 + 3*x^4)) + log(x)^ 
2*(x + x^2 - 2*x^3 + exp(x)*(3*x + 6*x^2 - 6*x^3 - 3)) - x^2 + exp(x)*(3*x 
 + 3*x^2 - 3*x^4) + log(x)^3*(exp(x)*(3*x - 3*x^2 + 3) - x^2))/(log(x)*(2* 
x^3 + 4*x^4 + 2*x^5) + x^4 + 2*x^5 + x^6 + log(x)^2*(x^2 + 2*x^3 + x^4)),x 
)
 

Output:

int((log(x)*(3*x^2 + 2*x^3 - x^4 - exp(x)*(6*x + 3*x^2 + 3*x^4)) + log(x)^ 
2*(x + x^2 - 2*x^3 + exp(x)*(3*x + 6*x^2 - 6*x^3 - 3)) - x^2 + exp(x)*(3*x 
 + 3*x^2 - 3*x^4) + log(x)^3*(exp(x)*(3*x - 3*x^2 + 3) - x^2))/(log(x)*(2* 
x^3 + 4*x^4 + 2*x^5) + x^4 + 2*x^5 + x^6 + log(x)^2*(x^2 + 2*x^3 + x^4)), 
x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \[ \int \frac {-x^2+e^x \left (3 x+3 x^2-3 x^4\right )+\left (3 x^2+2 x^3-x^4+e^x \left (-6 x-3 x^2-3 x^4\right )\right ) \log (x)+\left (x+x^2-2 x^3+e^x \left (-3+3 x+6 x^2-6 x^3\right )\right ) \log ^2(x)+\left (-x^2+e^x \left (3+3 x-3 x^2\right )\right ) \log ^3(x)}{x^4+2 x^5+x^6+\left (2 x^3+4 x^4+2 x^5\right ) \log (x)+\left (x^2+2 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {-3 e^{x} \mathrm {log}\left (x \right )^{2}-3 e^{x} \mathrm {log}\left (x \right ) x -3 e^{x} x +\mathrm {log}\left (x \right )^{2} x +\mathrm {log}\left (x \right ) x^{2}+x^{2}}{x \left (\mathrm {log}\left (x \right ) x +\mathrm {log}\left (x \right )+x^{2}+x \right )} \] Input:

int((((-3*x^2+3*x+3)*exp(x)-x^2)*log(x)^3+((-6*x^3+6*x^2+3*x-3)*exp(x)-2*x 
^3+x^2+x)*log(x)^2+((-3*x^4-3*x^2-6*x)*exp(x)-x^4+2*x^3+3*x^2)*log(x)+(-3* 
x^4+3*x^2+3*x)*exp(x)-x^2)/((x^4+2*x^3+x^2)*log(x)^2+(2*x^5+4*x^4+2*x^3)*l 
og(x)+x^6+2*x^5+x^4),x)
 

Output:

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