\(\int \frac {e^{\frac {x^2+x \log (x)}{-x+\log (\frac {e^6 (9 x^2-6 x^3+x^4)+e^3 (-6 x+2 x^2) \log (x)+\log ^2(x)}{e^6 x^2})}} (-2 x+e^3 (3 x^2-x^4)+(-2+x-x^2-2 e^3 x^2) \log (x)+2 \log ^2(x)+(e^3 (-3 x-5 x^2+2 x^3)+(1+2 x+e^3 (-3 x+x^2)) \log (x)+\log ^2(x)) \log (\frac {e^6 (9 x^2-6 x^3+x^4)+e^3 (-6 x+2 x^2) \log (x)+\log ^2(x)}{e^6 x^2}))}{e^3 (-3 x^3+x^4)+x^2 \log (x)+(e^3 (6 x^2-2 x^3)-2 x \log (x)) \log (\frac {e^6 (9 x^2-6 x^3+x^4)+e^3 (-6 x+2 x^2) \log (x)+\log ^2(x)}{e^6 x^2})+(e^3 (-3 x+x^2)+\log (x)) \log ^2(\frac {e^6 (9 x^2-6 x^3+x^4)+e^3 (-6 x+2 x^2) \log (x)+\log ^2(x)}{e^6 x^2})} \, dx\) [2405]

Optimal result

Integrand size = 350, antiderivative size = 29 \[ \int \frac {e^{\frac {x^2+x \log (x)}{-x+\log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )}} \left (-2 x+e^3 \left (3 x^2-x^4\right )+\left (-2+x-x^2-2 e^3 x^2\right ) \log (x)+2 \log ^2(x)+\left (e^3 \left (-3 x-5 x^2+2 x^3\right )+\left (1+2 x+e^3 \left (-3 x+x^2\right )\right ) \log (x)+\log ^2(x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )\right )}{e^3 \left (-3 x^3+x^4\right )+x^2 \log (x)+\left (e^3 \left (6 x^2-2 x^3\right )-2 x \log (x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )+\left (e^3 \left (-3 x+x^2\right )+\log (x)\right ) \log ^2\left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )} \, dx=e^{\frac {x (x+\log (x))}{-x+\log \left (\left (-3+x+\frac {\log (x)}{e^3 x}\right )^2\right )}} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(65\) vs. \(2(29)=58\).

Time = 0.23 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.24 \[ \int \frac {e^{\frac {x^2+x \log (x)}{-x+\log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )}} \left (-2 x+e^3 \left (3 x^2-x^4\right )+\left (-2+x-x^2-2 e^3 x^2\right ) \log (x)+2 \log ^2(x)+\left (e^3 \left (-3 x-5 x^2+2 x^3\right )+\left (1+2 x+e^3 \left (-3 x+x^2\right )\right ) \log (x)+\log ^2(x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )\right )}{e^3 \left (-3 x^3+x^4\right )+x^2 \log (x)+\left (e^3 \left (6 x^2-2 x^3\right )-2 x \log (x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )+\left (e^3 \left (-3 x+x^2\right )+\log (x)\right ) \log ^2\left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )} \, dx=e^{\frac {x^2}{-x+\log \left (\frac {\left (e^3 (-3+x) x+\log (x)\right )^2}{e^6 x^2}\right )}} x^{\frac {x}{-x+\log \left (\frac {\left (e^3 (-3+x) x+\log (x)\right )^2}{e^6 x^2}\right )}} \] Input:

Integrate[(E^((x^2 + x*Log[x])/(-x + Log[(E^6*(9*x^2 - 6*x^3 + x^4) + E^3* 
(-6*x + 2*x^2)*Log[x] + Log[x]^2)/(E^6*x^2)]))*(-2*x + E^3*(3*x^2 - x^4) + 
 (-2 + x - x^2 - 2*E^3*x^2)*Log[x] + 2*Log[x]^2 + (E^3*(-3*x - 5*x^2 + 2*x 
^3) + (1 + 2*x + E^3*(-3*x + x^2))*Log[x] + Log[x]^2)*Log[(E^6*(9*x^2 - 6* 
x^3 + x^4) + E^3*(-6*x + 2*x^2)*Log[x] + Log[x]^2)/(E^6*x^2)]))/(E^3*(-3*x 
^3 + x^4) + x^2*Log[x] + (E^3*(6*x^2 - 2*x^3) - 2*x*Log[x])*Log[(E^6*(9*x^ 
2 - 6*x^3 + x^4) + E^3*(-6*x + 2*x^2)*Log[x] + Log[x]^2)/(E^6*x^2)] + (E^3 
*(-3*x + x^2) + Log[x])*Log[(E^6*(9*x^2 - 6*x^3 + x^4) + E^3*(-6*x + 2*x^2 
)*Log[x] + Log[x]^2)/(E^6*x^2)]^2),x]
 

Output:

E^(x^2/(-x + Log[(E^3*(-3 + x)*x + Log[x])^2/(E^6*x^2)]))*x^(x/(-x + Log[( 
E^3*(-3 + x)*x + Log[x])^2/(E^6*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.

\(\displaystyle \int \frac {\left (\left (-2 e^3 x^2-x^2+x-2\right ) \log (x)+e^3 \left (3 x^2-x^4\right )+\left (\left (e^3 \left (x^2-3 x\right )+2 x+1\right ) \log (x)+e^3 \left (2 x^3-5 x^2-3 x\right )+\log ^2(x)\right ) \log \left (\frac {e^3 \left (2 x^2-6 x\right ) \log (x)+e^6 \left (x^4-6 x^3+9 x^2\right )+\log ^2(x)}{e^6 x^2}\right )-2 x+2 \log ^2(x)\right ) \exp \left (\frac {x^2+x \log (x)}{\log \left (\frac {e^3 \left (2 x^2-6 x\right ) \log (x)+e^6 \left (x^4-6 x^3+9 x^2\right )+\log ^2(x)}{e^6 x^2}\right )-x}\right )}{x^2 \log (x)+e^3 \left (x^4-3 x^3\right )+\left (e^3 \left (x^2-3 x\right )+\log (x)\right ) \log ^2\left (\frac {e^3 \left (2 x^2-6 x\right ) \log (x)+e^6 \left (x^4-6 x^3+9 x^2\right )+\log ^2(x)}{e^6 x^2}\right )+\left (e^3 \left (6 x^2-2 x^3\right )-2 x \log (x)\right ) \log \left (\frac {e^3 \left (2 x^2-6 x\right ) \log (x)+e^6 \left (x^4-6 x^3+9 x^2\right )+\log ^2(x)}{e^6 x^2}\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (-\left (-2 e^3 x^2-x^2+x-2\right ) \log (x)-e^3 \left (3 x^2-x^4\right )-\left (\left (e^3 \left (x^2-3 x\right )+2 x+1\right ) \log (x)+e^3 \left (2 x^3-5 x^2-3 x\right )+\log ^2(x)\right ) \log \left (\frac {e^3 \left (2 x^2-6 x\right ) \log (x)+e^6 \left (x^4-6 x^3+9 x^2\right )+\log ^2(x)}{e^6 x^2}\right )+2 x-2 \log ^2(x)\right ) \exp \left (\frac {x^2+x \log (x)}{\log \left (\frac {e^3 \left (2 x^2-6 x\right ) \log (x)+e^6 \left (x^4-6 x^3+9 x^2\right )+\log ^2(x)}{e^6 x^2}\right )-x}\right )}{\left (-e^3 x^2+3 e^3 x-\log (x)\right ) \left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\left (x^2-3\right ) x^2 \exp \left (\frac {x^2+x \log (x)}{\log \left (\frac {e^3 \left (2 x^2-6 x\right ) \log (x)+e^6 \left (x^4-6 x^3+9 x^2\right )+\log ^2(x)}{e^6 x^2}\right )-x}+3\right )}{\left (e^3 x^2-3 e^3 x+\log (x)\right ) \left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}-\frac {2 x \exp \left (\frac {x^2+x \log (x)}{\log \left (\frac {e^3 \left (2 x^2-6 x\right ) \log (x)+e^6 \left (x^4-6 x^3+9 x^2\right )+\log ^2(x)}{e^6 x^2}\right )-x}\right )}{\left (e^3 x^2-3 e^3 x+\log (x)\right ) \left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}+\frac {(2 x+\log (x)+1) \log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right ) \exp \left (\frac {x^2+x \log (x)}{\log \left (\frac {e^3 \left (2 x^2-6 x\right ) \log (x)+e^6 \left (x^4-6 x^3+9 x^2\right )+\log ^2(x)}{e^6 x^2}\right )-x}\right )}{\left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}+\frac {\left (\left (1+2 e^3\right ) x^2-x+2\right ) \log (x) \exp \left (\frac {x^2+x \log (x)}{\log \left (\frac {e^3 \left (2 x^2-6 x\right ) \log (x)+e^6 \left (x^4-6 x^3+9 x^2\right )+\log ^2(x)}{e^6 x^2}\right )-x}\right )}{\left (-e^3 x^2+3 e^3 x-\log (x)\right ) \left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}+\frac {2 \log ^2(x) \exp \left (\frac {x^2+x \log (x)}{\log \left (\frac {e^3 \left (2 x^2-6 x\right ) \log (x)+e^6 \left (x^4-6 x^3+9 x^2\right )+\log ^2(x)}{e^6 x^2}\right )-x}\right )}{\left (e^3 x^2-3 e^3 x+\log (x)\right ) \left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^{-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} \left (\left (\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )+2\right ) \log ^2(x)+\left (-2 e^3 x^2-x^2+\left (e^3 x^2+\left (2-3 e^3\right ) x+1\right ) \log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )+x-2\right ) \log (x)-x \left (e^3 x \left (x^2-3\right )+e^3 \left (-2 x^2+5 x+3\right ) \log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )+2\right )\right ) \exp \left (-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}\right )}{\left (e^3 (x-3) x+\log (x)\right ) \left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {(-2 x-\log (x)-1) x^{-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} \exp \left (-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}\right )}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}+\frac {(x+\log (x)) \left (e^3 x^3-5 e^3 x^2+x \log (x)+2 \log (x)-2\right ) x^{-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} \exp \left (-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}\right )}{\left (e^3 x^2-3 e^3 x+\log (x)\right ) \left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x^{-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} (x+\log (x)) \left (e^3 x^3-5 e^3 x^2+x \log (x)+2 \log (x)-2\right ) \exp \left (-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}\right )}{\left (e^3 x^2-3 e^3 x+\log (x)\right ) \left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}-\frac {x^{-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} (2 x+\log (x)+1) \exp \left (-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}\right )}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \int \frac {e^{-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} x^{1-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}}}{\left (e^3 x^2-3 e^3 x+\log (x)\right ) \left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}dx-5 \int \frac {e^{3-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} x^{3-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}}}{\left (e^3 x^2-3 e^3 x+\log (x)\right ) \left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}dx+\int \frac {e^{3-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} x^{4-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}}}{\left (e^3 x^2-3 e^3 x+\log (x)\right ) \left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}dx+2 \int \frac {e^{-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} x^{1-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} \log (x)}{\left (e^3 x^2-3 e^3 x+\log (x)\right ) \left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}dx+\left (1-5 e^3\right ) \int \frac {e^{-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} x^{2-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} \log (x)}{\left (e^3 x^2-3 e^3 x+\log (x)\right ) \left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}dx+\int \frac {e^{3-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} x^{3-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} \log (x)}{\left (e^3 x^2-3 e^3 x+\log (x)\right ) \left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}dx-2 \int \frac {e^{-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} x^{-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} \log (x)}{\left (e^3 x^2-3 e^3 x+\log (x)\right ) \left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}dx+\int \frac {e^{-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} x^{1-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} \log ^2(x)}{\left (e^3 x^2-3 e^3 x+\log (x)\right ) \left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}dx+2 \int \frac {e^{-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} x^{-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} \log ^2(x)}{\left (e^3 x^2-3 e^3 x+\log (x)\right ) \left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}dx-2 \int \frac {e^{-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} x^{1-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}}}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}dx-\int \frac {e^{-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} x^{-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}}}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}dx-\int \frac {e^{-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} x^{-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} \log (x)}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}dx\)

Input:

Int[(E^((x^2 + x*Log[x])/(-x + Log[(E^6*(9*x^2 - 6*x^3 + x^4) + E^3*(-6*x 
+ 2*x^2)*Log[x] + Log[x]^2)/(E^6*x^2)]))*(-2*x + E^3*(3*x^2 - x^4) + (-2 + 
 x - x^2 - 2*E^3*x^2)*Log[x] + 2*Log[x]^2 + (E^3*(-3*x - 5*x^2 + 2*x^3) + 
(1 + 2*x + E^3*(-3*x + x^2))*Log[x] + Log[x]^2)*Log[(E^6*(9*x^2 - 6*x^3 + 
x^4) + E^3*(-6*x + 2*x^2)*Log[x] + Log[x]^2)/(E^6*x^2)]))/(E^3*(-3*x^3 + x 
^4) + x^2*Log[x] + (E^3*(6*x^2 - 2*x^3) - 2*x*Log[x])*Log[(E^6*(9*x^2 - 6* 
x^3 + x^4) + E^3*(-6*x + 2*x^2)*Log[x] + Log[x]^2)/(E^6*x^2)] + (E^3*(-3*x 
 + x^2) + Log[x])*Log[(E^6*(9*x^2 - 6*x^3 + x^4) + E^3*(-6*x + 2*x^2)*Log[ 
x] + Log[x]^2)/(E^6*x^2)]^2),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

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

Time = 0.20 (sec) , antiderivative size = 356, normalized size of antiderivative = 12.28

\[{\mathrm e}^{-\frac {2 x \left (x +\ln \left (x \right )\right )}{i {\operatorname {csgn}\left (i {\left (\ln \left (x \right )+\left (x^{2}-3 x \right ) {\mathrm e}^{3}\right )}^{2}\right )}^{3} \pi -2 i {\operatorname {csgn}\left (i {\left (\ln \left (x \right )+\left (x^{2}-3 x \right ) {\mathrm e}^{3}\right )}^{2}\right )}^{2} \operatorname {csgn}\left (i \left (\ln \left (x \right )+\left (x^{2}-3 x \right ) {\mathrm e}^{3}\right )\right ) \pi +i \operatorname {csgn}\left (i {\left (\ln \left (x \right )+\left (x^{2}-3 x \right ) {\mathrm e}^{3}\right )}^{2}\right ) {\operatorname {csgn}\left (i \left (\ln \left (x \right )+\left (x^{2}-3 x \right ) {\mathrm e}^{3}\right )\right )}^{2} \pi -i \operatorname {csgn}\left (i {\left (\ln \left (x \right )+\left (x^{2}-3 x \right ) {\mathrm e}^{3}\right )}^{2}\right ) \pi {\operatorname {csgn}\left (\frac {i {\left (\ln \left (x \right )+\left (x^{2}-3 x \right ) {\mathrm e}^{3}\right )}^{2}}{x^{2}}\right )}^{2}+i \operatorname {csgn}\left (i {\left (\ln \left (x \right )+\left (x^{2}-3 x \right ) {\mathrm e}^{3}\right )}^{2}\right ) \pi \,\operatorname {csgn}\left (\frac {i {\left (\ln \left (x \right )+\left (x^{2}-3 x \right ) {\mathrm e}^{3}\right )}^{2}}{x^{2}}\right ) \operatorname {csgn}\left (\frac {i}{x^{2}}\right )-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+i \pi {\operatorname {csgn}\left (\frac {i {\left (\ln \left (x \right )+\left (x^{2}-3 x \right ) {\mathrm e}^{3}\right )}^{2}}{x^{2}}\right )}^{3}-i \pi {\operatorname {csgn}\left (\frac {i {\left (\ln \left (x \right )+\left (x^{2}-3 x \right ) {\mathrm e}^{3}\right )}^{2}}{x^{2}}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x^{2}}\right )+4 \ln \left (x \right )-4 \ln \left (\ln \left (x \right )+\left (x^{2}-3 x \right ) {\mathrm e}^{3}\right )+2 x +12}}\]

Input:

int(((ln(x)^2+((x^2-3*x)*exp(3)+2*x+1)*ln(x)+(2*x^3-5*x^2-3*x)*exp(3))*ln( 
(ln(x)^2+(2*x^2-6*x)*exp(3)*ln(x)+(x^4-6*x^3+9*x^2)*exp(3)^2)/x^2/exp(3)^2 
)+2*ln(x)^2+(-2*x^2*exp(3)-x^2+x-2)*ln(x)+(-x^4+3*x^2)*exp(3)-2*x)*exp((x* 
ln(x)+x^2)/(ln((ln(x)^2+(2*x^2-6*x)*exp(3)*ln(x)+(x^4-6*x^3+9*x^2)*exp(3)^ 
2)/x^2/exp(3)^2)-x))/((ln(x)+(x^2-3*x)*exp(3))*ln((ln(x)^2+(2*x^2-6*x)*exp 
(3)*ln(x)+(x^4-6*x^3+9*x^2)*exp(3)^2)/x^2/exp(3)^2)^2+(-2*x*ln(x)+(-2*x^3+ 
6*x^2)*exp(3))*ln((ln(x)^2+(2*x^2-6*x)*exp(3)*ln(x)+(x^4-6*x^3+9*x^2)*exp( 
3)^2)/x^2/exp(3)^2)+x^2*ln(x)+(x^4-3*x^3)*exp(3)),x)
 

Output:

exp(-2*x*(x+ln(x))/(I*csgn(I*(ln(x)+(x^2-3*x)*exp(3))^2)^3*Pi-2*I*csgn(I*( 
ln(x)+(x^2-3*x)*exp(3))^2)^2*csgn(I*(ln(x)+(x^2-3*x)*exp(3)))*Pi+I*csgn(I* 
(ln(x)+(x^2-3*x)*exp(3))^2)*csgn(I*(ln(x)+(x^2-3*x)*exp(3)))^2*Pi-I*csgn(I 
*(ln(x)+(x^2-3*x)*exp(3))^2)*Pi*csgn(I/x^2*(ln(x)+(x^2-3*x)*exp(3))^2)^2+I 
*csgn(I*(ln(x)+(x^2-3*x)*exp(3))^2)*Pi*csgn(I/x^2*(ln(x)+(x^2-3*x)*exp(3)) 
^2)*csgn(I/x^2)-I*Pi*csgn(I*x^2)^3+2*I*Pi*csgn(I*x^2)^2*csgn(I*x)-I*Pi*csg 
n(I*x^2)*csgn(I*x)^2+I*Pi*csgn(I/x^2*(ln(x)+(x^2-3*x)*exp(3))^2)^3-I*Pi*cs 
gn(I/x^2*(ln(x)+(x^2-3*x)*exp(3))^2)^2*csgn(I/x^2)+4*ln(x)-4*ln(ln(x)+(x^2 
-3*x)*exp(3))+2*x+12))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (28) = 56\).

Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03 \[ \int \frac {e^{\frac {x^2+x \log (x)}{-x+\log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )}} \left (-2 x+e^3 \left (3 x^2-x^4\right )+\left (-2+x-x^2-2 e^3 x^2\right ) \log (x)+2 \log ^2(x)+\left (e^3 \left (-3 x-5 x^2+2 x^3\right )+\left (1+2 x+e^3 \left (-3 x+x^2\right )\right ) \log (x)+\log ^2(x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )\right )}{e^3 \left (-3 x^3+x^4\right )+x^2 \log (x)+\left (e^3 \left (6 x^2-2 x^3\right )-2 x \log (x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )+\left (e^3 \left (-3 x+x^2\right )+\log (x)\right ) \log ^2\left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )} \, dx=e^{\left (-\frac {x^{2} + x \log \left (x\right )}{x - \log \left (\frac {{\left (2 \, {\left (x^{2} - 3 \, x\right )} e^{3} \log \left (x\right ) + {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} e^{6} + \log \left (x\right )^{2}\right )} e^{\left (-6\right )}}{x^{2}}\right )}\right )} \] Input:

integrate(((log(x)^2+((x^2-3*x)*exp(3)+2*x+1)*log(x)+(2*x^3-5*x^2-3*x)*exp 
(3))*log((log(x)^2+(2*x^2-6*x)*exp(3)*log(x)+(x^4-6*x^3+9*x^2)*exp(3)^2)/x 
^2/exp(3)^2)+2*log(x)^2+(-2*x^2*exp(3)-x^2+x-2)*log(x)+(-x^4+3*x^2)*exp(3) 
-2*x)*exp((x*log(x)+x^2)/(log((log(x)^2+(2*x^2-6*x)*exp(3)*log(x)+(x^4-6*x 
^3+9*x^2)*exp(3)^2)/x^2/exp(3)^2)-x))/((log(x)+(x^2-3*x)*exp(3))*log((log( 
x)^2+(2*x^2-6*x)*exp(3)*log(x)+(x^4-6*x^3+9*x^2)*exp(3)^2)/x^2/exp(3)^2)^2 
+(-2*x*log(x)+(-2*x^3+6*x^2)*exp(3))*log((log(x)^2+(2*x^2-6*x)*exp(3)*log( 
x)+(x^4-6*x^3+9*x^2)*exp(3)^2)/x^2/exp(3)^2)+x^2*log(x)+(x^4-3*x^3)*exp(3) 
),x, algorithm="fricas")
 

Output:

e^(-(x^2 + x*log(x))/(x - log((2*(x^2 - 3*x)*e^3*log(x) + (x^4 - 6*x^3 + 9 
*x^2)*e^6 + log(x)^2)*e^(-6)/x^2)))
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {x^2+x \log (x)}{-x+\log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )}} \left (-2 x+e^3 \left (3 x^2-x^4\right )+\left (-2+x-x^2-2 e^3 x^2\right ) \log (x)+2 \log ^2(x)+\left (e^3 \left (-3 x-5 x^2+2 x^3\right )+\left (1+2 x+e^3 \left (-3 x+x^2\right )\right ) \log (x)+\log ^2(x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )\right )}{e^3 \left (-3 x^3+x^4\right )+x^2 \log (x)+\left (e^3 \left (6 x^2-2 x^3\right )-2 x \log (x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )+\left (e^3 \left (-3 x+x^2\right )+\log (x)\right ) \log ^2\left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(((ln(x)**2+((x**2-3*x)*exp(3)+2*x+1)*ln(x)+(2*x**3-5*x**2-3*x)*e 
xp(3))*ln((ln(x)**2+(2*x**2-6*x)*exp(3)*ln(x)+(x**4-6*x**3+9*x**2)*exp(3)* 
*2)/x**2/exp(3)**2)+2*ln(x)**2+(-2*x**2*exp(3)-x**2+x-2)*ln(x)+(-x**4+3*x* 
*2)*exp(3)-2*x)*exp((x*ln(x)+x**2)/(ln((ln(x)**2+(2*x**2-6*x)*exp(3)*ln(x) 
+(x**4-6*x**3+9*x**2)*exp(3)**2)/x**2/exp(3)**2)-x))/((ln(x)+(x**2-3*x)*ex 
p(3))*ln((ln(x)**2+(2*x**2-6*x)*exp(3)*ln(x)+(x**4-6*x**3+9*x**2)*exp(3)** 
2)/x**2/exp(3)**2)**2+(-2*x*ln(x)+(-2*x**3+6*x**2)*exp(3))*ln((ln(x)**2+(2 
*x**2-6*x)*exp(3)*ln(x)+(x**4-6*x**3+9*x**2)*exp(3)**2)/x**2/exp(3)**2)+x* 
*2*ln(x)+(x**4-3*x**3)*exp(3)),x)
 

Output:

Exception raised: TypeError >> '>' not supported between instances of 'Pol 
y' and 'int'
 

Maxima [F]

\[ \int \frac {e^{\frac {x^2+x \log (x)}{-x+\log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )}} \left (-2 x+e^3 \left (3 x^2-x^4\right )+\left (-2+x-x^2-2 e^3 x^2\right ) \log (x)+2 \log ^2(x)+\left (e^3 \left (-3 x-5 x^2+2 x^3\right )+\left (1+2 x+e^3 \left (-3 x+x^2\right )\right ) \log (x)+\log ^2(x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )\right )}{e^3 \left (-3 x^3+x^4\right )+x^2 \log (x)+\left (e^3 \left (6 x^2-2 x^3\right )-2 x \log (x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )+\left (e^3 \left (-3 x+x^2\right )+\log (x)\right ) \log ^2\left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )} \, dx=\int { -\frac {{\left ({\left (x^{4} - 3 \, x^{2}\right )} e^{3} + {\left (2 \, x^{2} e^{3} + x^{2} - x + 2\right )} \log \left (x\right ) - 2 \, \log \left (x\right )^{2} - {\left ({\left (2 \, x^{3} - 5 \, x^{2} - 3 \, x\right )} e^{3} + {\left ({\left (x^{2} - 3 \, x\right )} e^{3} + 2 \, x + 1\right )} \log \left (x\right ) + \log \left (x\right )^{2}\right )} \log \left (\frac {{\left (2 \, {\left (x^{2} - 3 \, x\right )} e^{3} \log \left (x\right ) + {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} e^{6} + \log \left (x\right )^{2}\right )} e^{\left (-6\right )}}{x^{2}}\right ) + 2 \, x\right )} e^{\left (-\frac {x^{2} + x \log \left (x\right )}{x - \log \left (\frac {{\left (2 \, {\left (x^{2} - 3 \, x\right )} e^{3} \log \left (x\right ) + {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} e^{6} + \log \left (x\right )^{2}\right )} e^{\left (-6\right )}}{x^{2}}\right )}\right )}}{x^{2} \log \left (x\right ) + {\left ({\left (x^{2} - 3 \, x\right )} e^{3} + \log \left (x\right )\right )} \log \left (\frac {{\left (2 \, {\left (x^{2} - 3 \, x\right )} e^{3} \log \left (x\right ) + {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} e^{6} + \log \left (x\right )^{2}\right )} e^{\left (-6\right )}}{x^{2}}\right )^{2} + {\left (x^{4} - 3 \, x^{3}\right )} e^{3} - 2 \, {\left ({\left (x^{3} - 3 \, x^{2}\right )} e^{3} + x \log \left (x\right )\right )} \log \left (\frac {{\left (2 \, {\left (x^{2} - 3 \, x\right )} e^{3} \log \left (x\right ) + {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} e^{6} + \log \left (x\right )^{2}\right )} e^{\left (-6\right )}}{x^{2}}\right )} \,d x } \] Input:

integrate(((log(x)^2+((x^2-3*x)*exp(3)+2*x+1)*log(x)+(2*x^3-5*x^2-3*x)*exp 
(3))*log((log(x)^2+(2*x^2-6*x)*exp(3)*log(x)+(x^4-6*x^3+9*x^2)*exp(3)^2)/x 
^2/exp(3)^2)+2*log(x)^2+(-2*x^2*exp(3)-x^2+x-2)*log(x)+(-x^4+3*x^2)*exp(3) 
-2*x)*exp((x*log(x)+x^2)/(log((log(x)^2+(2*x^2-6*x)*exp(3)*log(x)+(x^4-6*x 
^3+9*x^2)*exp(3)^2)/x^2/exp(3)^2)-x))/((log(x)+(x^2-3*x)*exp(3))*log((log( 
x)^2+(2*x^2-6*x)*exp(3)*log(x)+(x^4-6*x^3+9*x^2)*exp(3)^2)/x^2/exp(3)^2)^2 
+(-2*x*log(x)+(-2*x^3+6*x^2)*exp(3))*log((log(x)^2+(2*x^2-6*x)*exp(3)*log( 
x)+(x^4-6*x^3+9*x^2)*exp(3)^2)/x^2/exp(3)^2)+x^2*log(x)+(x^4-3*x^3)*exp(3) 
),x, algorithm="maxima")
 

Output:

-integrate(((x^4 - 3*x^2)*e^3 + (2*x^2*e^3 + x^2 - x + 2)*log(x) - 2*log(x 
)^2 - ((2*x^3 - 5*x^2 - 3*x)*e^3 + ((x^2 - 3*x)*e^3 + 2*x + 1)*log(x) + lo 
g(x)^2)*log((2*(x^2 - 3*x)*e^3*log(x) + (x^4 - 6*x^3 + 9*x^2)*e^6 + log(x) 
^2)*e^(-6)/x^2) + 2*x)*e^(-(x^2 + x*log(x))/(x - log((2*(x^2 - 3*x)*e^3*lo 
g(x) + (x^4 - 6*x^3 + 9*x^2)*e^6 + log(x)^2)*e^(-6)/x^2)))/(x^2*log(x) + ( 
(x^2 - 3*x)*e^3 + log(x))*log((2*(x^2 - 3*x)*e^3*log(x) + (x^4 - 6*x^3 + 9 
*x^2)*e^6 + log(x)^2)*e^(-6)/x^2)^2 + (x^4 - 3*x^3)*e^3 - 2*((x^3 - 3*x^2) 
*e^3 + x*log(x))*log((2*(x^2 - 3*x)*e^3*log(x) + (x^4 - 6*x^3 + 9*x^2)*e^6 
 + log(x)^2)*e^(-6)/x^2)), x)
 

Giac [F]

\[ \int \frac {e^{\frac {x^2+x \log (x)}{-x+\log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )}} \left (-2 x+e^3 \left (3 x^2-x^4\right )+\left (-2+x-x^2-2 e^3 x^2\right ) \log (x)+2 \log ^2(x)+\left (e^3 \left (-3 x-5 x^2+2 x^3\right )+\left (1+2 x+e^3 \left (-3 x+x^2\right )\right ) \log (x)+\log ^2(x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )\right )}{e^3 \left (-3 x^3+x^4\right )+x^2 \log (x)+\left (e^3 \left (6 x^2-2 x^3\right )-2 x \log (x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )+\left (e^3 \left (-3 x+x^2\right )+\log (x)\right ) \log ^2\left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )} \, dx=\int { -\frac {{\left ({\left (x^{4} - 3 \, x^{2}\right )} e^{3} + {\left (2 \, x^{2} e^{3} + x^{2} - x + 2\right )} \log \left (x\right ) - 2 \, \log \left (x\right )^{2} - {\left ({\left (2 \, x^{3} - 5 \, x^{2} - 3 \, x\right )} e^{3} + {\left ({\left (x^{2} - 3 \, x\right )} e^{3} + 2 \, x + 1\right )} \log \left (x\right ) + \log \left (x\right )^{2}\right )} \log \left (\frac {{\left (2 \, {\left (x^{2} - 3 \, x\right )} e^{3} \log \left (x\right ) + {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} e^{6} + \log \left (x\right )^{2}\right )} e^{\left (-6\right )}}{x^{2}}\right ) + 2 \, x\right )} e^{\left (-\frac {x^{2} + x \log \left (x\right )}{x - \log \left (\frac {{\left (2 \, {\left (x^{2} - 3 \, x\right )} e^{3} \log \left (x\right ) + {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} e^{6} + \log \left (x\right )^{2}\right )} e^{\left (-6\right )}}{x^{2}}\right )}\right )}}{x^{2} \log \left (x\right ) + {\left ({\left (x^{2} - 3 \, x\right )} e^{3} + \log \left (x\right )\right )} \log \left (\frac {{\left (2 \, {\left (x^{2} - 3 \, x\right )} e^{3} \log \left (x\right ) + {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} e^{6} + \log \left (x\right )^{2}\right )} e^{\left (-6\right )}}{x^{2}}\right )^{2} + {\left (x^{4} - 3 \, x^{3}\right )} e^{3} - 2 \, {\left ({\left (x^{3} - 3 \, x^{2}\right )} e^{3} + x \log \left (x\right )\right )} \log \left (\frac {{\left (2 \, {\left (x^{2} - 3 \, x\right )} e^{3} \log \left (x\right ) + {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} e^{6} + \log \left (x\right )^{2}\right )} e^{\left (-6\right )}}{x^{2}}\right )} \,d x } \] Input:

integrate(((log(x)^2+((x^2-3*x)*exp(3)+2*x+1)*log(x)+(2*x^3-5*x^2-3*x)*exp 
(3))*log((log(x)^2+(2*x^2-6*x)*exp(3)*log(x)+(x^4-6*x^3+9*x^2)*exp(3)^2)/x 
^2/exp(3)^2)+2*log(x)^2+(-2*x^2*exp(3)-x^2+x-2)*log(x)+(-x^4+3*x^2)*exp(3) 
-2*x)*exp((x*log(x)+x^2)/(log((log(x)^2+(2*x^2-6*x)*exp(3)*log(x)+(x^4-6*x 
^3+9*x^2)*exp(3)^2)/x^2/exp(3)^2)-x))/((log(x)+(x^2-3*x)*exp(3))*log((log( 
x)^2+(2*x^2-6*x)*exp(3)*log(x)+(x^4-6*x^3+9*x^2)*exp(3)^2)/x^2/exp(3)^2)^2 
+(-2*x*log(x)+(-2*x^3+6*x^2)*exp(3))*log((log(x)^2+(2*x^2-6*x)*exp(3)*log( 
x)+(x^4-6*x^3+9*x^2)*exp(3)^2)/x^2/exp(3)^2)+x^2*log(x)+(x^4-3*x^3)*exp(3) 
),x, algorithm="giac")
 

Output:

integrate(-((x^4 - 3*x^2)*e^3 + (2*x^2*e^3 + x^2 - x + 2)*log(x) - 2*log(x 
)^2 - ((2*x^3 - 5*x^2 - 3*x)*e^3 + ((x^2 - 3*x)*e^3 + 2*x + 1)*log(x) + lo 
g(x)^2)*log((2*(x^2 - 3*x)*e^3*log(x) + (x^4 - 6*x^3 + 9*x^2)*e^6 + log(x) 
^2)*e^(-6)/x^2) + 2*x)*e^(-(x^2 + x*log(x))/(x - log((2*(x^2 - 3*x)*e^3*lo 
g(x) + (x^4 - 6*x^3 + 9*x^2)*e^6 + log(x)^2)*e^(-6)/x^2)))/(x^2*log(x) + ( 
(x^2 - 3*x)*e^3 + log(x))*log((2*(x^2 - 3*x)*e^3*log(x) + (x^4 - 6*x^3 + 9 
*x^2)*e^6 + log(x)^2)*e^(-6)/x^2)^2 + (x^4 - 3*x^3)*e^3 - 2*((x^3 - 3*x^2) 
*e^3 + x*log(x))*log((2*(x^2 - 3*x)*e^3*log(x) + (x^4 - 6*x^3 + 9*x^2)*e^6 
 + log(x)^2)*e^(-6)/x^2)), x)
 

Mupad [B] (verification not implemented)

Time = 4.22 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.76 \[ \int \frac {e^{\frac {x^2+x \log (x)}{-x+\log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )}} \left (-2 x+e^3 \left (3 x^2-x^4\right )+\left (-2+x-x^2-2 e^3 x^2\right ) \log (x)+2 \log ^2(x)+\left (e^3 \left (-3 x-5 x^2+2 x^3\right )+\left (1+2 x+e^3 \left (-3 x+x^2\right )\right ) \log (x)+\log ^2(x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )\right )}{e^3 \left (-3 x^3+x^4\right )+x^2 \log (x)+\left (e^3 \left (6 x^2-2 x^3\right )-2 x \log (x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )+\left (e^3 \left (-3 x+x^2\right )+\log (x)\right ) \log ^2\left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )} \, dx={\mathrm {e}}^{-\frac {x^2}{x-\ln \left (\frac {x^4-6\,x^3+2\,{\mathrm {e}}^{-3}\,x^2\,\ln \left (x\right )+9\,x^2-6\,{\mathrm {e}}^{-3}\,x\,\ln \left (x\right )+{\mathrm {e}}^{-6}\,{\ln \left (x\right )}^2}{x^2}\right )}}\,{\mathrm {e}}^{-\frac {x\,\ln \left (x\right )}{x-\ln \left (\frac {x^4-6\,x^3+2\,{\mathrm {e}}^{-3}\,x^2\,\ln \left (x\right )+9\,x^2-6\,{\mathrm {e}}^{-3}\,x\,\ln \left (x\right )+{\mathrm {e}}^{-6}\,{\ln \left (x\right )}^2}{x^2}\right )}} \] Input:

int((exp(-(x*log(x) + x^2)/(x - log((exp(-6)*(log(x)^2 + exp(6)*(9*x^2 - 6 
*x^3 + x^4) - exp(3)*log(x)*(6*x - 2*x^2)))/x^2)))*(2*log(x)^2 - 2*x + log 
((exp(-6)*(log(x)^2 + exp(6)*(9*x^2 - 6*x^3 + x^4) - exp(3)*log(x)*(6*x - 
2*x^2)))/x^2)*(log(x)^2 - exp(3)*(3*x + 5*x^2 - 2*x^3) + log(x)*(2*x - exp 
(3)*(3*x - x^2) + 1)) + exp(3)*(3*x^2 - x^4) - log(x)*(2*x^2*exp(3) - x + 
x^2 + 2)))/(x^2*log(x) + log((exp(-6)*(log(x)^2 + exp(6)*(9*x^2 - 6*x^3 + 
x^4) - exp(3)*log(x)*(6*x - 2*x^2)))/x^2)*(exp(3)*(6*x^2 - 2*x^3) - 2*x*lo 
g(x)) + log((exp(-6)*(log(x)^2 + exp(6)*(9*x^2 - 6*x^3 + x^4) - exp(3)*log 
(x)*(6*x - 2*x^2)))/x^2)^2*(log(x) - exp(3)*(3*x - x^2)) - exp(3)*(3*x^3 - 
 x^4)),x)
 

Output:

exp(-x^2/(x - log((exp(-6)*log(x)^2 + 9*x^2 - 6*x^3 + x^4 - 6*x*exp(-3)*lo 
g(x) + 2*x^2*exp(-3)*log(x))/x^2)))*exp(-(x*log(x))/(x - log((exp(-6)*log( 
x)^2 + 9*x^2 - 6*x^3 + x^4 - 6*x*exp(-3)*log(x) + 2*x^2*exp(-3)*log(x))/x^ 
2)))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.45 \[ \int \frac {e^{\frac {x^2+x \log (x)}{-x+\log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )}} \left (-2 x+e^3 \left (3 x^2-x^4\right )+\left (-2+x-x^2-2 e^3 x^2\right ) \log (x)+2 \log ^2(x)+\left (e^3 \left (-3 x-5 x^2+2 x^3\right )+\left (1+2 x+e^3 \left (-3 x+x^2\right )\right ) \log (x)+\log ^2(x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )\right )}{e^3 \left (-3 x^3+x^4\right )+x^2 \log (x)+\left (e^3 \left (6 x^2-2 x^3\right )-2 x \log (x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )+\left (e^3 \left (-3 x+x^2\right )+\log (x)\right ) \log ^2\left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )} \, dx=e^{\frac {\mathrm {log}\left (x \right ) x +x^{2}}{\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )^{2}+2 \,\mathrm {log}\left (x \right ) e^{3} x^{2}-6 \,\mathrm {log}\left (x \right ) e^{3} x +e^{6} x^{4}-6 e^{6} x^{3}+9 e^{6} x^{2}}{e^{6} x^{2}}\right )-x}} \] Input:

int(((log(x)^2+((x^2-3*x)*exp(3)+2*x+1)*log(x)+(2*x^3-5*x^2-3*x)*exp(3))*l 
og((log(x)^2+(2*x^2-6*x)*exp(3)*log(x)+(x^4-6*x^3+9*x^2)*exp(3)^2)/x^2/exp 
(3)^2)+2*log(x)^2+(-2*x^2*exp(3)-x^2+x-2)*log(x)+(-x^4+3*x^2)*exp(3)-2*x)* 
exp((x*log(x)+x^2)/(log((log(x)^2+(2*x^2-6*x)*exp(3)*log(x)+(x^4-6*x^3+9*x 
^2)*exp(3)^2)/x^2/exp(3)^2)-x))/((log(x)+(x^2-3*x)*exp(3))*log((log(x)^2+( 
2*x^2-6*x)*exp(3)*log(x)+(x^4-6*x^3+9*x^2)*exp(3)^2)/x^2/exp(3)^2)^2+(-2*x 
*log(x)+(-2*x^3+6*x^2)*exp(3))*log((log(x)^2+(2*x^2-6*x)*exp(3)*log(x)+(x^ 
4-6*x^3+9*x^2)*exp(3)^2)/x^2/exp(3)^2)+x^2*log(x)+(x^4-3*x^3)*exp(3)),x)
 

Output:

e**((log(x)*x + x**2)/(log((log(x)**2 + 2*log(x)*e**3*x**2 - 6*log(x)*e**3 
*x + e**6*x**4 - 6*e**6*x**3 + 9*e**6*x**2)/(e**6*x**2)) - x))