Integrand size = 350, antiderivative size = 29 \[ \int \frac {e^{\frac {x^2+x \log (x)}{-x+\log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )}} \left (-2 x+e^3 \left (3 x^2-x^4\right )+\left (-2+x-x^2-2 e^3 x^2\right ) \log (x)+2 \log ^2(x)+\left (e^3 \left (-3 x-5 x^2+2 x^3\right )+\left (1+2 x+e^3 \left (-3 x+x^2\right )\right ) \log (x)+\log ^2(x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )\right )}{e^3 \left (-3 x^3+x^4\right )+x^2 \log (x)+\left (e^3 \left (6 x^2-2 x^3\right )-2 x \log (x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )+\left (e^3 \left (-3 x+x^2\right )+\log (x)\right ) \log ^2\left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )} \, dx=e^{\frac {x (x+\log (x))}{-x+\log \left (\left (-3+x+\frac {\log (x)}{e^3 x}\right )^2\right )}} \]
Leaf count is larger than twice the leaf count of optimal. \(65\) vs. \(2(29)=58\).
Time = 0.35 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.24 \[ \int \frac {e^{\frac {x^2+x \log (x)}{-x+\log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )}} \left (-2 x+e^3 \left (3 x^2-x^4\right )+\left (-2+x-x^2-2 e^3 x^2\right ) \log (x)+2 \log ^2(x)+\left (e^3 \left (-3 x-5 x^2+2 x^3\right )+\left (1+2 x+e^3 \left (-3 x+x^2\right )\right ) \log (x)+\log ^2(x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )\right )}{e^3 \left (-3 x^3+x^4\right )+x^2 \log (x)+\left (e^3 \left (6 x^2-2 x^3\right )-2 x \log (x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )+\left (e^3 \left (-3 x+x^2\right )+\log (x)\right ) \log ^2\left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )} \, dx=e^{\frac {x^2}{-x+\log \left (\frac {\left (e^3 (-3+x) x+\log (x)\right )^2}{e^6 x^2}\right )}} x^{\frac {x}{-x+\log \left (\frac {\left (e^3 (-3+x) x+\log (x)\right )^2}{e^6 x^2}\right )}} \]
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]
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)]))
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 |
| \(\displaystyle \int \left (\frac {x^{-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} (x+\log (x)) \left (e^3 x^3-5 e^3 x^2+x \log (x)+2 \log (x)-2\right ) \exp \left (-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}\right )}{\left (e^3 x^2-3 e^3 x+\log (x)\right ) \left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}-\frac {x^{-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} (2 x+\log (x)+1) \exp \left (-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}\right )}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {x^{-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} (x+\log (x)) \left (e^3 x^3-5 e^3 x^2+x \log (x)+2 \log (x)-2\right ) \exp \left (-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}\right )}{\left (e^3 x^2-3 e^3 x+\log (x)\right ) \left (x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )\right )^2}-\frac {x^{-\frac {x}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}} (2 x+\log (x)+1) \exp \left (-\frac {x^2}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}\right )}{x-\log \left (\frac {\left (e^3 (x-3) x+\log (x)\right )^2}{e^6 x^2}\right )}\right )dx\) |
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]
3.25.5.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(60\) vs. \(2(29)=58\).
Time = 297.77 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.10
| method | result | size |
| parallelrisch | \({\mathrm e}^{\frac {x \left (x +\ln \left (x \right )\right )}{\ln \left (\frac {\left (\ln \left (x \right )^{2}+\left (2 x^{2}-6 x \right ) {\mathrm e}^{3} \ln \left (x \right )+\left (x^{4}-6 x^{3}+9 x^{2}\right ) {\mathrm e}^{6}\right ) {\mathrm e}^{-6}}{x^{2}}\right )-x}}\) | \(61\) |
| risch | \({\mathrm e}^{-\frac {2 x \left (x +\ln \left (x \right )\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 )-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 (i {\left (\ln \left (x \right )+\left (x^{2}-3 x \right ) {\mathrm e}^{3}\right )}^{2}\right )+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 ) \operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i {\left (\ln \left (x \right )+\left (x^{2}-3 x \right ) {\mathrm e}^{3}\right )}^{2}\right )+i \pi {\operatorname {csgn}\left (i {\left (\ln \left (x \right )+\left (x^{2}-3 x \right ) {\mathrm e}^{3}\right )}^{2}\right )}^{3}-2 i \pi {\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 )+i \pi \,\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}+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}}\) | \(356\) |
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,method=_RETURNVERBOSE)
exp(1/(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)*(x+ln(x))*x)
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (28) = 56\).
Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03 \[ \int \frac {e^{\frac {x^2+x \log (x)}{-x+\log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )}} \left (-2 x+e^3 \left (3 x^2-x^4\right )+\left (-2+x-x^2-2 e^3 x^2\right ) \log (x)+2 \log ^2(x)+\left (e^3 \left (-3 x-5 x^2+2 x^3\right )+\left (1+2 x+e^3 \left (-3 x+x^2\right )\right ) \log (x)+\log ^2(x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )\right )}{e^3 \left (-3 x^3+x^4\right )+x^2 \log (x)+\left (e^3 \left (6 x^2-2 x^3\right )-2 x \log (x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )+\left (e^3 \left (-3 x+x^2\right )+\log (x)\right ) \log ^2\left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )} \, dx=e^{\left (-\frac {x^{2} + x \log \left (x\right )}{x - \log \left (\frac {{\left (2 \, {\left (x^{2} - 3 \, x\right )} e^{3} \log \left (x\right ) + {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} e^{6} + \log \left (x\right )^{2}\right )} e^{\left (-6\right )}}{x^{2}}\right )}\right )} \]
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=\
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)))
Exception generated. \[ \int \frac {e^{\frac {x^2+x \log (x)}{-x+\log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )}} \left (-2 x+e^3 \left (3 x^2-x^4\right )+\left (-2+x-x^2-2 e^3 x^2\right ) \log (x)+2 \log ^2(x)+\left (e^3 \left (-3 x-5 x^2+2 x^3\right )+\left (1+2 x+e^3 \left (-3 x+x^2\right )\right ) \log (x)+\log ^2(x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )\right )}{e^3 \left (-3 x^3+x^4\right )+x^2 \log (x)+\left (e^3 \left (6 x^2-2 x^3\right )-2 x \log (x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )+\left (e^3 \left (-3 x+x^2\right )+\log (x)\right ) \log ^2\left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )} \, dx=\text {Exception raised: TypeError} \]
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)
\[ \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 } \]
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=\
-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)
Exception generated. \[ \int \frac {e^{\frac {x^2+x \log (x)}{-x+\log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )}} \left (-2 x+e^3 \left (3 x^2-x^4\right )+\left (-2+x-x^2-2 e^3 x^2\right ) \log (x)+2 \log ^2(x)+\left (e^3 \left (-3 x-5 x^2+2 x^3\right )+\left (1+2 x+e^3 \left (-3 x+x^2\right )\right ) \log (x)+\log ^2(x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )\right )}{e^3 \left (-3 x^3+x^4\right )+x^2 \log (x)+\left (e^3 \left (6 x^2-2 x^3\right )-2 x \log (x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )+\left (e^3 \left (-3 x+x^2\right )+\log (x)\right ) \log ^2\left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )} \, dx=\text {Exception raised: TypeError} \]
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=\
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{1,[0,11,8,8,0]%%%}+%%%{5,[0,11,8,7,0]%%%}+%%%{-40,[0,11,8, 5,0]%%%}+
Time = 12.73 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.76 \[ \int \frac {e^{\frac {x^2+x \log (x)}{-x+\log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )}} \left (-2 x+e^3 \left (3 x^2-x^4\right )+\left (-2+x-x^2-2 e^3 x^2\right ) \log (x)+2 \log ^2(x)+\left (e^3 \left (-3 x-5 x^2+2 x^3\right )+\left (1+2 x+e^3 \left (-3 x+x^2\right )\right ) \log (x)+\log ^2(x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )\right )}{e^3 \left (-3 x^3+x^4\right )+x^2 \log (x)+\left (e^3 \left (6 x^2-2 x^3\right )-2 x \log (x)\right ) \log \left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )+\left (e^3 \left (-3 x+x^2\right )+\log (x)\right ) \log ^2\left (\frac {e^6 \left (9 x^2-6 x^3+x^4\right )+e^3 \left (-6 x+2 x^2\right ) \log (x)+\log ^2(x)}{e^6 x^2}\right )} \, dx={\mathrm {e}}^{-\frac {x^2}{x-\ln \left (\frac {x^4-6\,x^3+2\,{\mathrm {e}}^{-3}\,x^2\,\ln \left (x\right )+9\,x^2-6\,{\mathrm {e}}^{-3}\,x\,\ln \left (x\right )+{\mathrm {e}}^{-6}\,{\ln \left (x\right )}^2}{x^2}\right )}}\,{\mathrm {e}}^{-\frac {x\,\ln \left (x\right )}{x-\ln \left (\frac {x^4-6\,x^3+2\,{\mathrm {e}}^{-3}\,x^2\,\ln \left (x\right )+9\,x^2-6\,{\mathrm {e}}^{-3}\,x\,\ln \left (x\right )+{\mathrm {e}}^{-6}\,{\ln \left (x\right )}^2}{x^2}\right )}} \]
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)