\(\int \frac {e^4 (x^3+x^4)+e^{4+x^2} (2 x^2+2 x^3-2 x^4-2 x^5)+(e^{4+x^2} (1+x)+e^4 (x+x^2)) \log (1+x)+\log (3 x) (e^{4+x^2} x+e^4 x^2+(e^4 (-x-x^2)+e^{4+x^2} (-2 x^2-2 x^3)) \log (1+x))}{x^3+x^4+e^{2 x^2} (x+x^2)+e^{x^2} (2 x^2+2 x^3)} \, dx\) [275]

Optimal result

Integrand size = 167, antiderivative size = 26 \[ \int \frac {e^4 \left (x^3+x^4\right )+e^{4+x^2} \left (2 x^2+2 x^3-2 x^4-2 x^5\right )+\left (e^{4+x^2} (1+x)+e^4 \left (x+x^2\right )\right ) \log (1+x)+\log (3 x) \left (e^{4+x^2} x+e^4 x^2+\left (e^4 \left (-x-x^2\right )+e^{4+x^2} \left (-2 x^2-2 x^3\right )\right ) \log (1+x)\right )}{x^3+x^4+e^{2 x^2} \left (x+x^2\right )+e^{x^2} \left (2 x^2+2 x^3\right )} \, dx=\frac {e^4 \left (x^2+\log (3 x) \log (1+x)\right )}{e^{x^2}+x} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^4 \left (x^3+x^4\right )+e^{4+x^2} \left (2 x^2+2 x^3-2 x^4-2 x^5\right )+\left (e^{4+x^2} (1+x)+e^4 \left (x+x^2\right )\right ) \log (1+x)+\log (3 x) \left (e^{4+x^2} x+e^4 x^2+\left (e^4 \left (-x-x^2\right )+e^{4+x^2} \left (-2 x^2-2 x^3\right )\right ) \log (1+x)\right )}{x^3+x^4+e^{2 x^2} \left (x+x^2\right )+e^{x^2} \left (2 x^2+2 x^3\right )} \, dx=\frac {e^4 \left (x^2+\log (3 x) \log (1+x)\right )}{e^{x^2}+x} \] Input:

Integrate[(E^4*(x^3 + x^4) + E^(4 + x^2)*(2*x^2 + 2*x^3 - 2*x^4 - 2*x^5) + 
 (E^(4 + x^2)*(1 + x) + E^4*(x + x^2))*Log[1 + x] + Log[3*x]*(E^(4 + x^2)* 
x + E^4*x^2 + (E^4*(-x - x^2) + E^(4 + x^2)*(-2*x^2 - 2*x^3))*Log[1 + x])) 
/(x^3 + x^4 + E^(2*x^2)*(x + x^2) + E^x^2*(2*x^2 + 2*x^3)),x]
 

Output:

(E^4*(x^2 + Log[3*x]*Log[1 + x]))/(E^x^2 + 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.

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7299

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

Input:

Int[(E^4*(x^3 + x^4) + E^(4 + x^2)*(2*x^2 + 2*x^3 - 2*x^4 - 2*x^5) + (E^(4 
 + x^2)*(1 + x) + E^4*(x + x^2))*Log[1 + x] + Log[3*x]*(E^(4 + x^2)*x + E^ 
4*x^2 + (E^4*(-x - x^2) + E^(4 + x^2)*(-2*x^2 - 2*x^3))*Log[1 + x]))/(x^3 
+ x^4 + E^(2*x^2)*(x + x^2) + E^x^2*(2*x^2 + 2*x^3)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35

\[\frac {{\mathrm e}^{4} \ln \left (3 x \right ) \ln \left (1+x \right )}{{\mathrm e}^{x^{2}}+x}+\frac {x^{2} {\mathrm e}^{4}}{{\mathrm e}^{x^{2}}+x}\]

Input:

int(((((-2*x^3-2*x^2)*exp(4)*exp(x^2)+(-x^2-x)*exp(4))*ln(1+x)+x*exp(4)*ex 
p(x^2)+x^2*exp(4))*ln(3*x)+((1+x)*exp(4)*exp(x^2)+(x^2+x)*exp(4))*ln(1+x)+ 
(-2*x^5-2*x^4+2*x^3+2*x^2)*exp(4)*exp(x^2)+(x^4+x^3)*exp(4))/((x^2+x)*exp( 
x^2)^2+(2*x^3+2*x^2)*exp(x^2)+x^4+x^3),x)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {e^4 \left (x^3+x^4\right )+e^{4+x^2} \left (2 x^2+2 x^3-2 x^4-2 x^5\right )+\left (e^{4+x^2} (1+x)+e^4 \left (x+x^2\right )\right ) \log (1+x)+\log (3 x) \left (e^{4+x^2} x+e^4 x^2+\left (e^4 \left (-x-x^2\right )+e^{4+x^2} \left (-2 x^2-2 x^3\right )\right ) \log (1+x)\right )}{x^3+x^4+e^{2 x^2} \left (x+x^2\right )+e^{x^2} \left (2 x^2+2 x^3\right )} \, dx=\frac {x^{2} e^{8} + e^{8} \log \left (3 \, x\right ) \log \left (x + 1\right )}{x e^{4} + e^{\left (x^{2} + 4\right )}} \] Input:

integrate(((((-2*x^3-2*x^2)*exp(4)*exp(x^2)+(-x^2-x)*exp(4))*log(1+x)+x*ex 
p(4)*exp(x^2)+x^2*exp(4))*log(3*x)+((1+x)*exp(4)*exp(x^2)+(x^2+x)*exp(4))* 
log(1+x)+(-2*x^5-2*x^4+2*x^3+2*x^2)*exp(4)*exp(x^2)+(x^4+x^3)*exp(4))/((x^ 
2+x)*exp(x^2)^2+(2*x^3+2*x^2)*exp(x^2)+x^4+x^3),x, algorithm="fricas")
 

Output:

(x^2*e^8 + e^8*log(3*x)*log(x + 1))/(x*e^4 + e^(x^2 + 4))
 

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^4 \left (x^3+x^4\right )+e^{4+x^2} \left (2 x^2+2 x^3-2 x^4-2 x^5\right )+\left (e^{4+x^2} (1+x)+e^4 \left (x+x^2\right )\right ) \log (1+x)+\log (3 x) \left (e^{4+x^2} x+e^4 x^2+\left (e^4 \left (-x-x^2\right )+e^{4+x^2} \left (-2 x^2-2 x^3\right )\right ) \log (1+x)\right )}{x^3+x^4+e^{2 x^2} \left (x+x^2\right )+e^{x^2} \left (2 x^2+2 x^3\right )} \, dx=\frac {x^{2} e^{4} + e^{4} \log {\left (3 x \right )} \log {\left (x + 1 \right )}}{x + e^{x^{2}}} \] Input:

integrate(((((-2*x**3-2*x**2)*exp(4)*exp(x**2)+(-x**2-x)*exp(4))*ln(1+x)+x 
*exp(4)*exp(x**2)+x**2*exp(4))*ln(3*x)+((1+x)*exp(4)*exp(x**2)+(x**2+x)*ex 
p(4))*ln(1+x)+(-2*x**5-2*x**4+2*x**3+2*x**2)*exp(4)*exp(x**2)+(x**4+x**3)* 
exp(4))/((x**2+x)*exp(x**2)**2+(2*x**3+2*x**2)*exp(x**2)+x**4+x**3),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {e^4 \left (x^3+x^4\right )+e^{4+x^2} \left (2 x^2+2 x^3-2 x^4-2 x^5\right )+\left (e^{4+x^2} (1+x)+e^4 \left (x+x^2\right )\right ) \log (1+x)+\log (3 x) \left (e^{4+x^2} x+e^4 x^2+\left (e^4 \left (-x-x^2\right )+e^{4+x^2} \left (-2 x^2-2 x^3\right )\right ) \log (1+x)\right )}{x^3+x^4+e^{2 x^2} \left (x+x^2\right )+e^{x^2} \left (2 x^2+2 x^3\right )} \, dx=\frac {x^{2} e^{4} + {\left (e^{4} \log \left (3\right ) + e^{4} \log \left (x\right )\right )} \log \left (x + 1\right )}{x + e^{\left (x^{2}\right )}} \] Input:

integrate(((((-2*x^3-2*x^2)*exp(4)*exp(x^2)+(-x^2-x)*exp(4))*log(1+x)+x*ex 
p(4)*exp(x^2)+x^2*exp(4))*log(3*x)+((1+x)*exp(4)*exp(x^2)+(x^2+x)*exp(4))* 
log(1+x)+(-2*x^5-2*x^4+2*x^3+2*x^2)*exp(4)*exp(x^2)+(x^4+x^3)*exp(4))/((x^ 
2+x)*exp(x^2)^2+(2*x^3+2*x^2)*exp(x^2)+x^4+x^3),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {e^4 \left (x^3+x^4\right )+e^{4+x^2} \left (2 x^2+2 x^3-2 x^4-2 x^5\right )+\left (e^{4+x^2} (1+x)+e^4 \left (x+x^2\right )\right ) \log (1+x)+\log (3 x) \left (e^{4+x^2} x+e^4 x^2+\left (e^4 \left (-x-x^2\right )+e^{4+x^2} \left (-2 x^2-2 x^3\right )\right ) \log (1+x)\right )}{x^3+x^4+e^{2 x^2} \left (x+x^2\right )+e^{x^2} \left (2 x^2+2 x^3\right )} \, dx=\frac {x^{2} e^{4} + e^{4} \log \left (3\right ) \log \left (x + 1\right ) + e^{4} \log \left (x + 1\right ) \log \left (x\right )}{x + e^{\left (x^{2}\right )}} \] Input:

integrate(((((-2*x^3-2*x^2)*exp(4)*exp(x^2)+(-x^2-x)*exp(4))*log(1+x)+x*ex 
p(4)*exp(x^2)+x^2*exp(4))*log(3*x)+((1+x)*exp(4)*exp(x^2)+(x^2+x)*exp(4))* 
log(1+x)+(-2*x^5-2*x^4+2*x^3+2*x^2)*exp(4)*exp(x^2)+(x^4+x^3)*exp(4))/((x^ 
2+x)*exp(x^2)^2+(2*x^3+2*x^2)*exp(x^2)+x^4+x^3),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 2.76 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {e^4 \left (x^3+x^4\right )+e^{4+x^2} \left (2 x^2+2 x^3-2 x^4-2 x^5\right )+\left (e^{4+x^2} (1+x)+e^4 \left (x+x^2\right )\right ) \log (1+x)+\log (3 x) \left (e^{4+x^2} x+e^4 x^2+\left (e^4 \left (-x-x^2\right )+e^{4+x^2} \left (-2 x^2-2 x^3\right )\right ) \log (1+x)\right )}{x^3+x^4+e^{2 x^2} \left (x+x^2\right )+e^{x^2} \left (2 x^2+2 x^3\right )} \, dx=\frac {{\mathrm {e}}^4\,\left (x^2+\ln \left (3\,x\right )\,\ln \left (x+1\right )\right )}{x+{\mathrm {e}}^{x^2}} \] Input:

int((log(x + 1)*(exp(4)*(x + x^2) + exp(x^2)*exp(4)*(x + 1)) + log(3*x)*(x 
^2*exp(4) - log(x + 1)*(exp(4)*(x + x^2) + exp(x^2)*exp(4)*(2*x^2 + 2*x^3) 
) + x*exp(x^2)*exp(4)) + exp(4)*(x^3 + x^4) + exp(x^2)*exp(4)*(2*x^2 + 2*x 
^3 - 2*x^4 - 2*x^5))/(exp(x^2)*(2*x^2 + 2*x^3) + exp(2*x^2)*(x + x^2) + x^ 
3 + x^4),x)
                                                                                    
                                                                                    
 

Output:

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

Reduce [F]

\[ \int \frac {e^4 \left (x^3+x^4\right )+e^{4+x^2} \left (2 x^2+2 x^3-2 x^4-2 x^5\right )+\left (e^{4+x^2} (1+x)+e^4 \left (x+x^2\right )\right ) \log (1+x)+\log (3 x) \left (e^{4+x^2} x+e^4 x^2+\left (e^4 \left (-x-x^2\right )+e^{4+x^2} \left (-2 x^2-2 x^3\right )\right ) \log (1+x)\right )}{x^3+x^4+e^{2 x^2} \left (x+x^2\right )+e^{x^2} \left (2 x^2+2 x^3\right )} \, dx=\int \frac {\left (\left (\left (-2 x^{3}-2 x^{2}\right ) {\mathrm e}^{4} {\mathrm e}^{x^{2}}+\left (-x^{2}-x \right ) {\mathrm e}^{4}\right ) \mathrm {log}\left (x +1\right )+x \,{\mathrm e}^{4} {\mathrm e}^{x^{2}}+x^{2} {\mathrm e}^{4}\right ) \mathrm {log}\left (3 x \right )+\left (\left (x +1\right ) {\mathrm e}^{4} {\mathrm e}^{x^{2}}+\left (x^{2}+x \right ) {\mathrm e}^{4}\right ) \mathrm {log}\left (x +1\right )+\left (-2 x^{5}-2 x^{4}+2 x^{3}+2 x^{2}\right ) {\mathrm e}^{4} {\mathrm e}^{x^{2}}+\left (x^{4}+x^{3}\right ) {\mathrm e}^{4}}{\left (x^{2}+x \right ) \left ({\mathrm e}^{x^{2}}\right )^{2}+\left (2 x^{3}+2 x^{2}\right ) {\mathrm e}^{x^{2}}+x^{4}+x^{3}}d x \] Input:

int(((((-2*x^3-2*x^2)*exp(4)*exp(x^2)+(-x^2-x)*exp(4))*log(1+x)+x*exp(4)*e 
xp(x^2)+x^2*exp(4))*log(3*x)+((1+x)*exp(4)*exp(x^2)+(x^2+x)*exp(4))*log(1+ 
x)+(-2*x^5-2*x^4+2*x^3+2*x^2)*exp(4)*exp(x^2)+(x^4+x^3)*exp(4))/((x^2+x)*e 
xp(x^2)^2+(2*x^3+2*x^2)*exp(x^2)+x^4+x^3),x)
 

Output:

int(((((-2*x^3-2*x^2)*exp(4)*exp(x^2)+(-x^2-x)*exp(4))*log(1+x)+x*exp(4)*e 
xp(x^2)+x^2*exp(4))*log(3*x)+((1+x)*exp(4)*exp(x^2)+(x^2+x)*exp(4))*log(1+ 
x)+(-2*x^5-2*x^4+2*x^3+2*x^2)*exp(4)*exp(x^2)+(x^4+x^3)*exp(4))/((x^2+x)*e 
xp(x^2)^2+(2*x^3+2*x^2)*exp(x^2)+x^4+x^3),x)