\(\int \frac {e^{\frac {\log (x) \log (x^2)}{5 x+x^3+e^{x^2} \log (x^2)}} ((10 x+2 x^3) \log (x)+(5 x+x^3+(-5 x-3 x^3) \log (x)) \log (x^2)+(e^{x^2}-2 e^{x^2} x^2 \log (x)) \log ^2(x^2))}{25 x^3+10 x^5+x^7+e^{x^2} (10 x^2+2 x^4) \log (x^2)+e^{2 x^2} x \log ^2(x^2)} \, dx\) [2817]

Optimal result

Integrand size = 143, antiderivative size = 23 \[ \int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \left (\left (10 x+2 x^3\right ) \log (x)+\left (5 x+x^3+\left (-5 x-3 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (e^{x^2}-2 e^{x^2} x^2 \log (x)\right ) \log ^2\left (x^2\right )\right )}{25 x^3+10 x^5+x^7+e^{x^2} \left (10 x^2+2 x^4\right ) \log \left (x^2\right )+e^{2 x^2} x \log ^2\left (x^2\right )} \, dx=x^{\frac {1}{e^{x^2}+\frac {x \left (5+x^2\right )}{\log \left (x^2\right )}}} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(123\) vs. \(2(23)=46\).

Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 5.35 \[ \int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \left (\left (10 x+2 x^3\right ) \log (x)+\left (5 x+x^3+\left (-5 x-3 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (e^{x^2}-2 e^{x^2} x^2 \log (x)\right ) \log ^2\left (x^2\right )\right )}{25 x^3+10 x^5+x^7+e^{x^2} \left (10 x^2+2 x^4\right ) \log \left (x^2\right )+e^{2 x^2} x \log ^2\left (x^2\right )} \, dx=e^{-\frac {1}{2} e^{-2 x^2} x \left (5+x^2\right )} x^{\frac {e^{-2 x^2} \left (x^2 \left (5+x^2\right )^2+2 e^{x^2} \left (5 x+x^3+e^{x^2} \log (x)\right ) \log \left (x^2\right )+e^{2 x^2} \log ^2\left (x^2\right )\right )}{2 \log (x) \left (5 x+x^3+e^{x^2} \log \left (x^2\right )\right )}} \left (x^2\right )^{-\frac {e^{-x^2}}{2}} \] Input:

Integrate[(E^((Log[x]*Log[x^2])/(5*x + x^3 + E^x^2*Log[x^2]))*((10*x + 2*x 
^3)*Log[x] + (5*x + x^3 + (-5*x - 3*x^3)*Log[x])*Log[x^2] + (E^x^2 - 2*E^x 
^2*x^2*Log[x])*Log[x^2]^2))/(25*x^3 + 10*x^5 + x^7 + E^x^2*(10*x^2 + 2*x^4 
)*Log[x^2] + E^(2*x^2)*x*Log[x^2]^2),x]
 

Output:

x^((x^2*(5 + x^2)^2 + 2*E^x^2*(5*x + x^3 + E^x^2*Log[x])*Log[x^2] + E^(2*x 
^2)*Log[x^2]^2)/(2*E^(2*x^2)*Log[x]*(5*x + x^3 + E^x^2*Log[x^2])))/(E^((x* 
(5 + x^2))/(2*E^(2*x^2)))*(x^2)^(1/(2*E^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.

\(\displaystyle \int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{x^3+e^{x^2} \log \left (x^2\right )+5 x}} \left (\left (2 x^3+10 x\right ) \log (x)+\left (e^{x^2}-2 e^{x^2} x^2 \log (x)\right ) \log ^2\left (x^2\right )+\left (x^3+\left (-3 x^3-5 x\right ) \log (x)+5 x\right ) \log \left (x^2\right )\right )}{x^7+10 x^5+25 x^3+e^{2 x^2} x \log ^2\left (x^2\right )+e^{x^2} \left (2 x^4+10 x^2\right ) \log \left (x^2\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{x^3+e^{x^2} \log \left (x^2\right )+5 x}} \left (\left (2 x^3+10 x\right ) \log (x)+\left (e^{x^2}-2 e^{x^2} x^2 \log (x)\right ) \log ^2\left (x^2\right )+\left (x^3+\left (-3 x^3-5 x\right ) \log (x)+5 x\right ) \log \left (x^2\right )\right )}{x \left (x^3+e^{x^2} \log \left (x^2\right )+5 x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{\frac {\log (x) \log \left (x^2\right )}{x^3+e^{x^2} \log \left (x^2\right )+5 x}} \log (x) \left (2 x^2+7 x^2 \log \left (x^2\right )-5 \log \left (x^2\right )+2 x^4 \log \left (x^2\right )+10\right )}{\left (x^3+e^{x^2} \log \left (x^2\right )+5 x\right )^2}-\frac {e^{\frac {\log (x) \log \left (x^2\right )}{x^3+e^{x^2} \log \left (x^2\right )+5 x}} \left (2 x^2 \log (x)-1\right ) \log \left (x^2\right )}{x \left (x^3+e^{x^2} \log \left (x^2\right )+5 x\right )}\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x^{\frac {\log \left (x^2\right )}{x^3+e^{x^2} \log \left (x^2\right )+5 x}} \log (x) \left (2 x^2+7 x^2 \log \left (x^2\right )-5 \log \left (x^2\right )+2 x^4 \log \left (x^2\right )+10\right )}{\left (x^3+e^{x^2} \log \left (x^2\right )+5 x\right )^2}-\frac {x^{\frac {\log \left (x^2\right )}{x^3+e^{x^2} \log \left (x^2\right )+5 x}-1} \left (2 x^2 \log (x)-1\right ) \log \left (x^2\right )}{x^3+e^{x^2} \log \left (x^2\right )+5 x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \int \frac {x^{\frac {\log \left (x^2\right )}{x^3+5 x+e^{x^2} \log \left (x^2\right )}+2} \log (x)}{\left (x^3+5 x+e^{x^2} \log \left (x^2\right )\right )^2}dx+7 \int \frac {x^{\frac {\log \left (x^2\right )}{x^3+5 x+e^{x^2} \log \left (x^2\right )}+2} \log (x) \log \left (x^2\right )}{\left (x^3+5 x+e^{x^2} \log \left (x^2\right )\right )^2}dx+2 \int \frac {x^{\frac {\log \left (x^2\right )}{x^3+5 x+e^{x^2} \log \left (x^2\right )}+4} \log (x) \log \left (x^2\right )}{\left (x^3+5 x+e^{x^2} \log \left (x^2\right )\right )^2}dx+\int \frac {x^{\frac {\log \left (x^2\right )}{x^3+5 x+e^{x^2} \log \left (x^2\right )}-1} \log \left (x^2\right )}{x^3+5 x+e^{x^2} \log \left (x^2\right )}dx-2 \int \frac {x^{\frac {\log \left (x^2\right )}{x^3+5 x+e^{x^2} \log \left (x^2\right )}+1} \log (x) \log \left (x^2\right )}{x^3+5 x+e^{x^2} \log \left (x^2\right )}dx+10 \int \frac {x^{\frac {\log \left (x^2\right )}{x^3+5 x+e^{x^2} \log \left (x^2\right )}} \log (x)}{\left (x^3+5 x+e^{x^2} \log \left (x^2\right )\right )^2}dx-5 \int \frac {x^{\frac {\log \left (x^2\right )}{x^3+5 x+e^{x^2} \log \left (x^2\right )}} \log (x) \log \left (x^2\right )}{\left (x^3+5 x+e^{x^2} \log \left (x^2\right )\right )^2}dx\)

Input:

Int[(E^((Log[x]*Log[x^2])/(5*x + x^3 + E^x^2*Log[x^2]))*((10*x + 2*x^3)*Lo 
g[x] + (5*x + x^3 + (-5*x - 3*x^3)*Log[x])*Log[x^2] + (E^x^2 - 2*E^x^2*x^2 
*Log[x])*Log[x^2]^2))/(25*x^3 + 10*x^5 + x^7 + E^x^2*(10*x^2 + 2*x^4)*Log[ 
x^2] + E^(2*x^2)*x*Log[x^2]^2),x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {\left (\left (-2 x^{2} {\mathrm e}^{x^{2}} \ln \left (x \right )+{\mathrm e}^{x^{2}}\right ) \ln \left (x^{2}\right )^{2}+\left (\left (-3 x^{3}-5 x \right ) \ln \left (x \right )+x^{3}+5 x \right ) \ln \left (x^{2}\right )+\left (2 x^{3}+10 x \right ) \ln \left (x \right )\right ) {\mathrm e}^{\frac {\ln \left (x \right ) \ln \left (x^{2}\right )}{{\mathrm e}^{x^{2}} \ln \left (x^{2}\right )+x^{3}+5 x}}}{x \,{\mathrm e}^{2 x^{2}} \ln \left (x^{2}\right )^{2}+\left (2 x^{4}+10 x^{2}\right ) {\mathrm e}^{x^{2}} \ln \left (x^{2}\right )+x^{7}+10 x^{5}+25 x^{3}}d x\]

Input:

int(((-2*x^2*exp(x^2)*ln(x)+exp(x^2))*ln(x^2)^2+((-3*x^3-5*x)*ln(x)+x^3+5* 
x)*ln(x^2)+(2*x^3+10*x)*ln(x))*exp(ln(x)*ln(x^2)/(exp(x^2)*ln(x^2)+x^3+5*x 
))/(x*exp(x^2)^2*ln(x^2)^2+(2*x^4+10*x^2)*exp(x^2)*ln(x^2)+x^7+10*x^5+25*x 
^3),x)
 

Output:

int(((-2*x^2*exp(x^2)*ln(x)+exp(x^2))*ln(x^2)^2+((-3*x^3-5*x)*ln(x)+x^3+5* 
x)*ln(x^2)+(2*x^3+10*x)*ln(x))*exp(ln(x)*ln(x^2)/(exp(x^2)*ln(x^2)+x^3+5*x 
))/(x*exp(x^2)^2*ln(x^2)^2+(2*x^4+10*x^2)*exp(x^2)*ln(x^2)+x^7+10*x^5+25*x 
^3),x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \left (\left (10 x+2 x^3\right ) \log (x)+\left (5 x+x^3+\left (-5 x-3 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (e^{x^2}-2 e^{x^2} x^2 \log (x)\right ) \log ^2\left (x^2\right )\right )}{25 x^3+10 x^5+x^7+e^{x^2} \left (10 x^2+2 x^4\right ) \log \left (x^2\right )+e^{2 x^2} x \log ^2\left (x^2\right )} \, dx=e^{\left (\frac {2 \, \log \left (x\right )^{2}}{x^{3} + 2 \, e^{\left (x^{2}\right )} \log \left (x\right ) + 5 \, x}\right )} \] Input:

integrate(((-2*x^2*exp(x^2)*log(x)+exp(x^2))*log(x^2)^2+((-3*x^3-5*x)*log( 
x)+x^3+5*x)*log(x^2)+(2*x^3+10*x)*log(x))*exp(log(x)*log(x^2)/(exp(x^2)*lo 
g(x^2)+x^3+5*x))/(x*exp(x^2)^2*log(x^2)^2+(2*x^4+10*x^2)*exp(x^2)*log(x^2) 
+x^7+10*x^5+25*x^3),x, algorithm="fricas")
 

Output:

e^(2*log(x)^2/(x^3 + 2*e^(x^2)*log(x) + 5*x))
 

Sympy [A] (verification not implemented)

Time = 0.83 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \left (\left (10 x+2 x^3\right ) \log (x)+\left (5 x+x^3+\left (-5 x-3 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (e^{x^2}-2 e^{x^2} x^2 \log (x)\right ) \log ^2\left (x^2\right )\right )}{25 x^3+10 x^5+x^7+e^{x^2} \left (10 x^2+2 x^4\right ) \log \left (x^2\right )+e^{2 x^2} x \log ^2\left (x^2\right )} \, dx=e^{\frac {2 \log {\left (x \right )}^{2}}{x^{3} + 5 x + 2 e^{x^{2}} \log {\left (x \right )}}} \] Input:

integrate(((-2*x**2*exp(x**2)*ln(x)+exp(x**2))*ln(x**2)**2+((-3*x**3-5*x)* 
ln(x)+x**3+5*x)*ln(x**2)+(2*x**3+10*x)*ln(x))*exp(ln(x)*ln(x**2)/(exp(x**2 
)*ln(x**2)+x**3+5*x))/(x*exp(x**2)**2*ln(x**2)**2+(2*x**4+10*x**2)*exp(x** 
2)*ln(x**2)+x**7+10*x**5+25*x**3),x)
 

Output:

exp(2*log(x)**2/(x**3 + 5*x + 2*exp(x**2)*log(x)))
 

Maxima [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \left (\left (10 x+2 x^3\right ) \log (x)+\left (5 x+x^3+\left (-5 x-3 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (e^{x^2}-2 e^{x^2} x^2 \log (x)\right ) \log ^2\left (x^2\right )\right )}{25 x^3+10 x^5+x^7+e^{x^2} \left (10 x^2+2 x^4\right ) \log \left (x^2\right )+e^{2 x^2} x \log ^2\left (x^2\right )} \, dx=e^{\left (\frac {2 \, \log \left (x\right )^{2}}{x^{3} + 2 \, e^{\left (x^{2}\right )} \log \left (x\right ) + 5 \, x}\right )} \] Input:

integrate(((-2*x^2*exp(x^2)*log(x)+exp(x^2))*log(x^2)^2+((-3*x^3-5*x)*log( 
x)+x^3+5*x)*log(x^2)+(2*x^3+10*x)*log(x))*exp(log(x)*log(x^2)/(exp(x^2)*lo 
g(x^2)+x^3+5*x))/(x*exp(x^2)^2*log(x^2)^2+(2*x^4+10*x^2)*exp(x^2)*log(x^2) 
+x^7+10*x^5+25*x^3),x, algorithm="maxima")
 

Output:

e^(2*log(x)^2/(x^3 + 2*e^(x^2)*log(x) + 5*x))
 

Giac [F]

\[ \int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \left (\left (10 x+2 x^3\right ) \log (x)+\left (5 x+x^3+\left (-5 x-3 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (e^{x^2}-2 e^{x^2} x^2 \log (x)\right ) \log ^2\left (x^2\right )\right )}{25 x^3+10 x^5+x^7+e^{x^2} \left (10 x^2+2 x^4\right ) \log \left (x^2\right )+e^{2 x^2} x \log ^2\left (x^2\right )} \, dx=\int { -\frac {{\left ({\left (2 \, x^{2} e^{\left (x^{2}\right )} \log \left (x\right ) - e^{\left (x^{2}\right )}\right )} \log \left (x^{2}\right )^{2} - {\left (x^{3} - {\left (3 \, x^{3} + 5 \, x\right )} \log \left (x\right ) + 5 \, x\right )} \log \left (x^{2}\right ) - 2 \, {\left (x^{3} + 5 \, x\right )} \log \left (x\right )\right )} e^{\left (\frac {\log \left (x^{2}\right ) \log \left (x\right )}{x^{3} + e^{\left (x^{2}\right )} \log \left (x^{2}\right ) + 5 \, x}\right )}}{x^{7} + 10 \, x^{5} + x e^{\left (2 \, x^{2}\right )} \log \left (x^{2}\right )^{2} + 25 \, x^{3} + 2 \, {\left (x^{4} + 5 \, x^{2}\right )} e^{\left (x^{2}\right )} \log \left (x^{2}\right )} \,d x } \] Input:

integrate(((-2*x^2*exp(x^2)*log(x)+exp(x^2))*log(x^2)^2+((-3*x^3-5*x)*log( 
x)+x^3+5*x)*log(x^2)+(2*x^3+10*x)*log(x))*exp(log(x)*log(x^2)/(exp(x^2)*lo 
g(x^2)+x^3+5*x))/(x*exp(x^2)^2*log(x^2)^2+(2*x^4+10*x^2)*exp(x^2)*log(x^2) 
+x^7+10*x^5+25*x^3),x, algorithm="giac")
 

Output:

integrate(-((2*x^2*e^(x^2)*log(x) - e^(x^2))*log(x^2)^2 - (x^3 - (3*x^3 + 
5*x)*log(x) + 5*x)*log(x^2) - 2*(x^3 + 5*x)*log(x))*e^(log(x^2)*log(x)/(x^ 
3 + e^(x^2)*log(x^2) + 5*x))/(x^7 + 10*x^5 + x*e^(2*x^2)*log(x^2)^2 + 25*x 
^3 + 2*(x^4 + 5*x^2)*e^(x^2)*log(x^2)), x)
 

Mupad [B] (verification not implemented)

Time = 3.61 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \left (\left (10 x+2 x^3\right ) \log (x)+\left (5 x+x^3+\left (-5 x-3 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (e^{x^2}-2 e^{x^2} x^2 \log (x)\right ) \log ^2\left (x^2\right )\right )}{25 x^3+10 x^5+x^7+e^{x^2} \left (10 x^2+2 x^4\right ) \log \left (x^2\right )+e^{2 x^2} x \log ^2\left (x^2\right )} \, dx={\mathrm {e}}^{\frac {\ln \left (x^2\right )\,\ln \left (x\right )}{5\,x+x^3+\ln \left (x^2\right )\,{\mathrm {e}}^{x^2}}} \] Input:

int((exp((log(x^2)*log(x))/(5*x + x^3 + log(x^2)*exp(x^2)))*(log(x^2)*(5*x 
 - log(x)*(5*x + 3*x^3) + x^3) + log(x)*(10*x + 2*x^3) + log(x^2)^2*(exp(x 
^2) - 2*x^2*exp(x^2)*log(x))))/(25*x^3 + 10*x^5 + x^7 + x*log(x^2)^2*exp(2 
*x^2) + log(x^2)*exp(x^2)*(10*x^2 + 2*x^4)),x)
 

Output:

exp((log(x^2)*log(x))/(5*x + x^3 + log(x^2)*exp(x^2)))
 

Reduce [F]

\[ \int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \left (\left (10 x+2 x^3\right ) \log (x)+\left (5 x+x^3+\left (-5 x-3 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (e^{x^2}-2 e^{x^2} x^2 \log (x)\right ) \log ^2\left (x^2\right )\right )}{25 x^3+10 x^5+x^7+e^{x^2} \left (10 x^2+2 x^4\right ) \log \left (x^2\right )+e^{2 x^2} x \log ^2\left (x^2\right )} \, dx=\int \frac {\left (\left (-2 x^{2} {\mathrm e}^{x^{2}} \mathrm {log}\left (x \right )+{\mathrm e}^{x^{2}}\right ) \mathrm {log}\left (x^{2}\right )^{2}+\left (\left (-3 x^{3}-5 x \right ) \mathrm {log}\left (x \right )+x^{3}+5 x \right ) \mathrm {log}\left (x^{2}\right )+\left (2 x^{3}+10 x \right ) \mathrm {log}\left (x \right )\right ) {\mathrm e}^{\frac {\mathrm {log}\left (x \right ) \mathrm {log}\left (x^{2}\right )}{{\mathrm e}^{x^{2}} \mathrm {log}\left (x^{2}\right )+x^{3}+5 x}}}{x \left ({\mathrm e}^{x^{2}}\right )^{2} \mathrm {log}\left (x^{2}\right )^{2}+\left (2 x^{4}+10 x^{2}\right ) {\mathrm e}^{x^{2}} \mathrm {log}\left (x^{2}\right )+x^{7}+10 x^{5}+25 x^{3}}d x \] Input:

int(((-2*x^2*exp(x^2)*log(x)+exp(x^2))*log(x^2)^2+((-3*x^3-5*x)*log(x)+x^3 
+5*x)*log(x^2)+(2*x^3+10*x)*log(x))*exp(log(x)*log(x^2)/(exp(x^2)*log(x^2) 
+x^3+5*x))/(x*exp(x^2)^2*log(x^2)^2+(2*x^4+10*x^2)*exp(x^2)*log(x^2)+x^7+1 
0*x^5+25*x^3),x)
 

Output:

int(((-2*x^2*exp(x^2)*log(x)+exp(x^2))*log(x^2)^2+((-3*x^3-5*x)*log(x)+x^3 
+5*x)*log(x^2)+(2*x^3+10*x)*log(x))*exp(log(x)*log(x^2)/(exp(x^2)*log(x^2) 
+x^3+5*x))/(x*exp(x^2)^2*log(x^2)^2+(2*x^4+10*x^2)*exp(x^2)*log(x^2)+x^7+1 
0*x^5+25*x^3),x)