\(\int \frac {2 e^x x+2 x^3+e^{(e^x+x^2)^{\frac {x+\log (x)}{x}}} (e^x+x^2)^{\frac {x+\log (x)}{x}} (e^x x^2+2 x^3+(e^x x+2 x^2) \log (x)+(e^x+x^2+(-e^x-x^2) \log (x)) \log (e^x+x^2))}{-e^x x^2-x^4+e^{(e^x+x^2)^{\frac {x+\log (x)}{x}}} (e^x x^2+x^4)+(e^x x^2+x^4) \log (x^2)} \, dx\) [1326]

Optimal result

Integrand size = 167, antiderivative size = 25 \[ \int \frac {2 e^x x+2 x^3+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x+x^2\right )^{\frac {x+\log (x)}{x}} \left (e^x x^2+2 x^3+\left (e^x x+2 x^2\right ) \log (x)+\left (e^x+x^2+\left (-e^x-x^2\right ) \log (x)\right ) \log \left (e^x+x^2\right )\right )}{-e^x x^2-x^4+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x x^2+x^4\right )+\left (e^x x^2+x^4\right ) \log \left (x^2\right )} \, dx=\log \left (-1+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}}+\log \left (x^2\right )\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {2 e^x x+2 x^3+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x+x^2\right )^{\frac {x+\log (x)}{x}} \left (e^x x^2+2 x^3+\left (e^x x+2 x^2\right ) \log (x)+\left (e^x+x^2+\left (-e^x-x^2\right ) \log (x)\right ) \log \left (e^x+x^2\right )\right )}{-e^x x^2-x^4+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x x^2+x^4\right )+\left (e^x x^2+x^4\right ) \log \left (x^2\right )} \, dx=\log \left (-1+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}}+\log \left (x^2\right )\right ) \] Input:

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

Output:

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

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.18 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.96

\[\ln \left ({\mathrm e}^{\left (x^{2}+{\mathrm e}^{x}\right )^{\frac {x +\ln \left (x \right )}{x}}}-\frac {i \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 i \ln \left (x \right )-2 i\right )}{2}\right )\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {2 e^x x+2 x^3+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x+x^2\right )^{\frac {x+\log (x)}{x}} \left (e^x x^2+2 x^3+\left (e^x x+2 x^2\right ) \log (x)+\left (e^x+x^2+\left (-e^x-x^2\right ) \log (x)\right ) \log \left (e^x+x^2\right )\right )}{-e^x x^2-x^4+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x x^2+x^4\right )+\left (e^x x^2+x^4\right ) \log \left (x^2\right )} \, dx=\log \left (e^{\left ({\left (x^{2} + e^{x}\right )}^{\frac {x + \log \left (x\right )}{x}}\right )} + 2 \, \log \left (x\right ) - 1\right ) \] Input:

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

Output:

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

Sympy [F(-1)]

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

integrate(((((-exp(x)-x**2)*ln(x)+x**2+exp(x))*ln(x**2+exp(x))+(exp(x)*x+2 
*x**2)*ln(x)+exp(x)*x**2+2*x**3)*exp((x+ln(x))*ln(x**2+exp(x))/x)*exp(exp( 
(x+ln(x))*ln(x**2+exp(x))/x))+2*exp(x)*x+2*x**3)/((exp(x)*x**2+x**4)*exp(e 
xp((x+ln(x))*ln(x**2+exp(x))/x))+(exp(x)*x**2+x**4)*ln(x**2)-exp(x)*x**2-x 
**4),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (23) = 46\).

Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.32 \[ \int \frac {2 e^x x+2 x^3+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x+x^2\right )^{\frac {x+\log (x)}{x}} \left (e^x x^2+2 x^3+\left (e^x x+2 x^2\right ) \log (x)+\left (e^x+x^2+\left (-e^x-x^2\right ) \log (x)\right ) \log \left (e^x+x^2\right )\right )}{-e^x x^2-x^4+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x x^2+x^4\right )+\left (e^x x^2+x^4\right ) \log \left (x^2\right )} \, dx=x^{2} e^{\left (\frac {\log \left (x^{2} + e^{x}\right ) \log \left (x\right )}{x}\right )} + \log \left ({\left (e^{\left (x^{2} e^{\left (\frac {\log \left (x^{2} + e^{x}\right ) \log \left (x\right )}{x}\right )} + e^{\left (x + \frac {\log \left (x^{2} + e^{x}\right ) \log \left (x\right )}{x}\right )}\right )} + 2 \, \log \left (x\right ) - 1\right )} e^{\left (-x^{2} e^{\left (\frac {\log \left (x^{2} + e^{x}\right ) \log \left (x\right )}{x}\right )}\right )}\right ) \] Input:

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

Output:

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

Giac [F]

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

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

Output:

integrate(-(2*x^3 + (2*x^3 + x^2*e^x + (x^2 - (x^2 + e^x)*log(x) + e^x)*lo 
g(x^2 + e^x) + (2*x^2 + x*e^x)*log(x))*(x^2 + e^x)^((x + log(x))/x)*e^((x^ 
2 + e^x)^((x + log(x))/x)) + 2*x*e^x)/(x^4 + x^2*e^x - (x^4 + x^2*e^x)*e^( 
(x^2 + e^x)^((x + log(x))/x)) - (x^4 + x^2*e^x)*log(x^2)), x)
 

Mupad [B] (verification not implemented)

Time = 2.38 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {2 e^x x+2 x^3+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x+x^2\right )^{\frac {x+\log (x)}{x}} \left (e^x x^2+2 x^3+\left (e^x x+2 x^2\right ) \log (x)+\left (e^x+x^2+\left (-e^x-x^2\right ) \log (x)\right ) \log \left (e^x+x^2\right )\right )}{-e^x x^2-x^4+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x x^2+x^4\right )+\left (e^x x^2+x^4\right ) \log \left (x^2\right )} \, dx=\ln \left (\ln \left (x^2\right )+{\mathrm {e}}^{x^{\frac {\ln \left ({\mathrm {e}}^x+x^2\right )}{x}}\,x^2+x^{\frac {\ln \left ({\mathrm {e}}^x+x^2\right )}{x}}\,{\mathrm {e}}^x}-1\right ) \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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