Integrand size = 286, antiderivative size = 27 \[ \int \frac {e^{x^2} \left (-20 x+30 x^2-10 x^3+10 x^4\right )+e^{x^2} \left (-30 x+40 x^2-20 x^3+20 x^4\right ) \log \left (-x+x^2\right )+e^{x^2} \left (-10 x+10 x^2-10 x^3+10 x^4\right ) \log ^2\left (-x+x^2\right )}{-4+4 x+e^{x^2} \left (4 x^2-4 x^3\right )+e^{2 x^2} \left (-x^4+x^5\right )+\left (e^{x^2} \left (8 x^2-8 x^3\right )+e^{2 x^2} \left (-4 x^4+4 x^5\right )\right ) \log \left (-x+x^2\right )+\left (e^{x^2} \left (4 x^2-4 x^3\right )+e^{2 x^2} \left (-6 x^4+6 x^5\right )\right ) \log ^2\left (-x+x^2\right )+e^{2 x^2} \left (-4 x^4+4 x^5\right ) \log ^3\left (-x+x^2\right )+e^{2 x^2} \left (-x^4+x^5\right ) \log ^4\left (-x+x^2\right )} \, dx=\frac {5}{2-e^{x^2} \left (x+x \log \left (-x+x^2\right )\right )^2} \] Output:
5/(2-(x+x*ln(x^2-x))^2*exp(x^2))
Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {e^{x^2} \left (-20 x+30 x^2-10 x^3+10 x^4\right )+e^{x^2} \left (-30 x+40 x^2-20 x^3+20 x^4\right ) \log \left (-x+x^2\right )+e^{x^2} \left (-10 x+10 x^2-10 x^3+10 x^4\right ) \log ^2\left (-x+x^2\right )}{-4+4 x+e^{x^2} \left (4 x^2-4 x^3\right )+e^{2 x^2} \left (-x^4+x^5\right )+\left (e^{x^2} \left (8 x^2-8 x^3\right )+e^{2 x^2} \left (-4 x^4+4 x^5\right )\right ) \log \left (-x+x^2\right )+\left (e^{x^2} \left (4 x^2-4 x^3\right )+e^{2 x^2} \left (-6 x^4+6 x^5\right )\right ) \log ^2\left (-x+x^2\right )+e^{2 x^2} \left (-4 x^4+4 x^5\right ) \log ^3\left (-x+x^2\right )+e^{2 x^2} \left (-x^4+x^5\right ) \log ^4\left (-x+x^2\right )} \, dx=-\frac {5}{-2+e^{x^2} x^2+2 e^{x^2} x^2 \log ((-1+x) x)+e^{x^2} x^2 \log ^2((-1+x) x)} \] Input:
Integrate[(E^x^2*(-20*x + 30*x^2 - 10*x^3 + 10*x^4) + E^x^2*(-30*x + 40*x^ 2 - 20*x^3 + 20*x^4)*Log[-x + x^2] + E^x^2*(-10*x + 10*x^2 - 10*x^3 + 10*x ^4)*Log[-x + x^2]^2)/(-4 + 4*x + E^x^2*(4*x^2 - 4*x^3) + E^(2*x^2)*(-x^4 + x^5) + (E^x^2*(8*x^2 - 8*x^3) + E^(2*x^2)*(-4*x^4 + 4*x^5))*Log[-x + x^2] + (E^x^2*(4*x^2 - 4*x^3) + E^(2*x^2)*(-6*x^4 + 6*x^5))*Log[-x + x^2]^2 + E^(2*x^2)*(-4*x^4 + 4*x^5)*Log[-x + x^2]^3 + E^(2*x^2)*(-x^4 + x^5)*Log[-x + x^2]^4),x]
Output:
-5/(-2 + E^x^2*x^2 + 2*E^x^2*x^2*Log[(-1 + x)*x] + E^x^2*x^2*Log[(-1 + x)* 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.
| \(\displaystyle \int \frac {e^{x^2} \left (10 x^4-10 x^3+30 x^2-20 x\right )+e^{x^2} \left (10 x^4-10 x^3+10 x^2-10 x\right ) \log ^2\left (x^2-x\right )+e^{x^2} \left (20 x^4-20 x^3+40 x^2-30 x\right ) \log \left (x^2-x\right )}{e^{x^2} \left (4 x^2-4 x^3\right )+e^{2 x^2} \left (x^5-x^4\right )+e^{2 x^2} \left (x^5-x^4\right ) \log ^4\left (x^2-x\right )+e^{2 x^2} \left (4 x^5-4 x^4\right ) \log ^3\left (x^2-x\right )+\left (e^{x^2} \left (4 x^2-4 x^3\right )+e^{2 x^2} \left (6 x^5-6 x^4\right )\right ) \log ^2\left (x^2-x\right )+\left (e^{x^2} \left (8 x^2-8 x^3\right )+e^{2 x^2} \left (4 x^5-4 x^4\right )\right ) \log \left (x^2-x\right )+4 x-4} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {10 e^{x^2} x (\log ((x-1) x)+1) \left (-x^3+x^2-\left (x^3-x^2+x-1\right ) \log ((x-1) x)-3 x+2\right )}{(1-x) \left (-e^{x^2} x^2-e^{x^2} x^2 \log ^2((x-1) x)-2 e^{x^2} x^2 \log ((x-1) x)+2\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 10 \int \frac {e^{x^2} x (\log (-((1-x) x))+1) \left (-x^3+x^2-3 x+\left (-x^3+x^2-x+1\right ) \log (-((1-x) x))+2\right )}{(1-x) \left (-e^{x^2} x^2-e^{x^2} \log ^2(-((1-x) x)) x^2-2 e^{x^2} \log (-((1-x) x)) x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{\left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}+\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{(x-1) \left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {e^{x^2} x (\log ((x-1) x)+1) \left (-x^3+x^2-3 x-\left (x^3-x^2+x-1\right ) \log ((x-1) x)+2\right )}{(1-x) \left (-e^{x^2} x^2-e^{x^2} \log ^2((x-1) x) x^2-2 e^{x^2} \log ((x-1) x) x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{\left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}+\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{(x-1) \left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {e^{x^2} x (\log ((x-1) x)+1) \left (-x^3+x^2-3 x-\left (x^3-x^2+x-1\right ) \log ((x-1) x)+2\right )}{(1-x) \left (-e^{x^2} x^2-e^{x^2} \log ^2((x-1) x) x^2-2 e^{x^2} \log ((x-1) x) x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{\left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}+\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{(x-1) \left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {e^{x^2} x (\log ((x-1) x)+1) \left (-x^3+x^2-3 x-\left (x^3-x^2+x-1\right ) \log ((x-1) x)+2\right )}{(1-x) \left (-e^{x^2} x^2-e^{x^2} \log ^2((x-1) x) x^2-2 e^{x^2} \log ((x-1) x) x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{\left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}+\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{(x-1) \left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {e^{x^2} x (\log ((x-1) x)+1) \left (-x^3+x^2-3 x-\left (x^3-x^2+x-1\right ) \log ((x-1) x)+2\right )}{(1-x) \left (-e^{x^2} x^2-e^{x^2} \log ^2((x-1) x) x^2-2 e^{x^2} \log ((x-1) x) x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{\left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}+\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{(x-1) \left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {e^{x^2} x (\log ((x-1) x)+1) \left (-x^3+x^2-3 x-\left (x^3-x^2+x-1\right ) \log ((x-1) x)+2\right )}{(1-x) \left (-e^{x^2} x^2-e^{x^2} \log ^2((x-1) x) x^2-2 e^{x^2} \log ((x-1) x) x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{\left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}+\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{(x-1) \left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {e^{x^2} x (\log ((x-1) x)+1) \left (-x^3+x^2-3 x-\left (x^3-x^2+x-1\right ) \log ((x-1) x)+2\right )}{(1-x) \left (-e^{x^2} x^2-e^{x^2} \log ^2((x-1) x) x^2-2 e^{x^2} \log ((x-1) x) x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{\left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}+\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{(x-1) \left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {e^{x^2} x (\log ((x-1) x)+1) \left (-x^3+x^2-3 x-\left (x^3-x^2+x-1\right ) \log ((x-1) x)+2\right )}{(1-x) \left (-e^{x^2} x^2-e^{x^2} \log ^2((x-1) x) x^2-2 e^{x^2} \log ((x-1) x) x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{\left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}+\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{(x-1) \left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {e^{x^2} x (\log ((x-1) x)+1) \left (-x^3+x^2-3 x-\left (x^3-x^2+x-1\right ) \log ((x-1) x)+2\right )}{(1-x) \left (-e^{x^2} x^2-e^{x^2} \log ^2((x-1) x) x^2-2 e^{x^2} \log ((x-1) x) x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{\left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}+\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{(x-1) \left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {e^{x^2} x (\log ((x-1) x)+1) \left (-x^3+x^2-3 x-\left (x^3-x^2+x-1\right ) \log ((x-1) x)+2\right )}{(1-x) \left (-e^{x^2} x^2-e^{x^2} \log ^2((x-1) x) x^2-2 e^{x^2} \log ((x-1) x) x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{\left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}+\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{(x-1) \left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {e^{x^2} x (\log ((x-1) x)+1) \left (-x^3+x^2-3 x-\left (x^3-x^2+x-1\right ) \log ((x-1) x)+2\right )}{(1-x) \left (-e^{x^2} x^2-e^{x^2} \log ^2((x-1) x) x^2-2 e^{x^2} \log ((x-1) x) x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{\left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}+\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{(x-1) \left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {e^{x^2} x (\log ((x-1) x)+1) \left (-x^3+x^2-3 x-\left (x^3-x^2+x-1\right ) \log ((x-1) x)+2\right )}{(1-x) \left (-e^{x^2} x^2-e^{x^2} \log ^2((x-1) x) x^2-2 e^{x^2} \log ((x-1) x) x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{\left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}+\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{(x-1) \left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {e^{x^2} x (\log ((x-1) x)+1) \left (-x^3+x^2-3 x-\left (x^3-x^2+x-1\right ) \log ((x-1) x)+2\right )}{(1-x) \left (-e^{x^2} x^2-e^{x^2} \log ^2((x-1) x) x^2-2 e^{x^2} \log ((x-1) x) x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{\left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}+\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{(x-1) \left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {e^{x^2} x (\log ((x-1) x)+1) \left (-x^3+x^2-3 x-\left (x^3-x^2+x-1\right ) \log ((x-1) x)+2\right )}{(1-x) \left (-e^{x^2} x^2-e^{x^2} \log ^2((x-1) x) x^2-2 e^{x^2} \log ((x-1) x) x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{\left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}+\frac {e^{x^2} (\log ((x-1) x)+1) \left (\log ((x-1) x) x^3+x^3-\log ((x-1) x) x^2-x^2+\log ((x-1) x) x+3 x-\log ((x-1) x)-2\right )}{(x-1) \left (e^{x^2} x^2+e^{x^2} \log ^2((x-1) x) x^2+2 e^{x^2} \log ((x-1) x) x^2-2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {e^{x^2} x (\log ((x-1) x)+1) \left (-x^3+x^2-3 x-\left (x^3-x^2+x-1\right ) \log ((x-1) x)+2\right )}{(1-x) \left (-e^{x^2} x^2-e^{x^2} \log ^2((x-1) x) x^2-2 e^{x^2} \log ((x-1) x) x^2+2\right )^2}dx\) |
Input:
Int[(E^x^2*(-20*x + 30*x^2 - 10*x^3 + 10*x^4) + E^x^2*(-30*x + 40*x^2 - 20 *x^3 + 20*x^4)*Log[-x + x^2] + E^x^2*(-10*x + 10*x^2 - 10*x^3 + 10*x^4)*Lo g[-x + x^2]^2)/(-4 + 4*x + E^x^2*(4*x^2 - 4*x^3) + E^(2*x^2)*(-x^4 + x^5) + (E^x^2*(8*x^2 - 8*x^3) + E^(2*x^2)*(-4*x^4 + 4*x^5))*Log[-x + x^2] + (E^ x^2*(4*x^2 - 4*x^3) + E^(2*x^2)*(-6*x^4 + 6*x^5))*Log[-x + x^2]^2 + E^(2*x ^2)*(-4*x^4 + 4*x^5)*Log[-x + x^2]^3 + E^(2*x^2)*(-x^4 + x^5)*Log[-x + x^2 ]^4),x]
Output:
$Aborted
Time = 7.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.85
| method | result | size |
| parallelrisch | \(-\frac {5}{{\mathrm e}^{x^{2}} \ln \left (x^{2}-x \right )^{2} x^{2}+2 \,{\mathrm e}^{x^{2}} \ln \left (x^{2}-x \right ) x^{2}+x^{2} {\mathrm e}^{x^{2}}-2}\) | \(50\) |
| risch | \(\text {Expression too large to display}\) | \(703\) |
Input:
int(((10*x^4-10*x^3+10*x^2-10*x)*exp(x^2)*ln(x^2-x)^2+(20*x^4-20*x^3+40*x^ 2-30*x)*exp(x^2)*ln(x^2-x)+(10*x^4-10*x^3+30*x^2-20*x)*exp(x^2))/((x^5-x^4 )*exp(x^2)^2*ln(x^2-x)^4+(4*x^5-4*x^4)*exp(x^2)^2*ln(x^2-x)^3+((6*x^5-6*x^ 4)*exp(x^2)^2+(-4*x^3+4*x^2)*exp(x^2))*ln(x^2-x)^2+((4*x^5-4*x^4)*exp(x^2) ^2+(-8*x^3+8*x^2)*exp(x^2))*ln(x^2-x)+(x^5-x^4)*exp(x^2)^2+(-4*x^3+4*x^2)* exp(x^2)+4*x-4),x,method=_RETURNVERBOSE)
Output:
-5/(exp(x^2)*ln(x^2-x)^2*x^2+2*exp(x^2)*ln(x^2-x)*x^2+x^2*exp(x^2)-2)
Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {e^{x^2} \left (-20 x+30 x^2-10 x^3+10 x^4\right )+e^{x^2} \left (-30 x+40 x^2-20 x^3+20 x^4\right ) \log \left (-x+x^2\right )+e^{x^2} \left (-10 x+10 x^2-10 x^3+10 x^4\right ) \log ^2\left (-x+x^2\right )}{-4+4 x+e^{x^2} \left (4 x^2-4 x^3\right )+e^{2 x^2} \left (-x^4+x^5\right )+\left (e^{x^2} \left (8 x^2-8 x^3\right )+e^{2 x^2} \left (-4 x^4+4 x^5\right )\right ) \log \left (-x+x^2\right )+\left (e^{x^2} \left (4 x^2-4 x^3\right )+e^{2 x^2} \left (-6 x^4+6 x^5\right )\right ) \log ^2\left (-x+x^2\right )+e^{2 x^2} \left (-4 x^4+4 x^5\right ) \log ^3\left (-x+x^2\right )+e^{2 x^2} \left (-x^4+x^5\right ) \log ^4\left (-x+x^2\right )} \, dx=-\frac {5}{x^{2} e^{\left (x^{2}\right )} \log \left (x^{2} - x\right )^{2} + 2 \, x^{2} e^{\left (x^{2}\right )} \log \left (x^{2} - x\right ) + x^{2} e^{\left (x^{2}\right )} - 2} \] Input:
integrate(((10*x^4-10*x^3+10*x^2-10*x)*exp(x^2)*log(x^2-x)^2+(20*x^4-20*x^ 3+40*x^2-30*x)*exp(x^2)*log(x^2-x)+(10*x^4-10*x^3+30*x^2-20*x)*exp(x^2))/( (x^5-x^4)*exp(x^2)^2*log(x^2-x)^4+(4*x^5-4*x^4)*exp(x^2)^2*log(x^2-x)^3+(( 6*x^5-6*x^4)*exp(x^2)^2+(-4*x^3+4*x^2)*exp(x^2))*log(x^2-x)^2+((4*x^5-4*x^ 4)*exp(x^2)^2+(-8*x^3+8*x^2)*exp(x^2))*log(x^2-x)+(x^5-x^4)*exp(x^2)^2+(-4 *x^3+4*x^2)*exp(x^2)+4*x-4),x, algorithm="fricas")
Output:
-5/(x^2*e^(x^2)*log(x^2 - x)^2 + 2*x^2*e^(x^2)*log(x^2 - x) + x^2*e^(x^2) - 2)
Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {e^{x^2} \left (-20 x+30 x^2-10 x^3+10 x^4\right )+e^{x^2} \left (-30 x+40 x^2-20 x^3+20 x^4\right ) \log \left (-x+x^2\right )+e^{x^2} \left (-10 x+10 x^2-10 x^3+10 x^4\right ) \log ^2\left (-x+x^2\right )}{-4+4 x+e^{x^2} \left (4 x^2-4 x^3\right )+e^{2 x^2} \left (-x^4+x^5\right )+\left (e^{x^2} \left (8 x^2-8 x^3\right )+e^{2 x^2} \left (-4 x^4+4 x^5\right )\right ) \log \left (-x+x^2\right )+\left (e^{x^2} \left (4 x^2-4 x^3\right )+e^{2 x^2} \left (-6 x^4+6 x^5\right )\right ) \log ^2\left (-x+x^2\right )+e^{2 x^2} \left (-4 x^4+4 x^5\right ) \log ^3\left (-x+x^2\right )+e^{2 x^2} \left (-x^4+x^5\right ) \log ^4\left (-x+x^2\right )} \, dx=- \frac {5}{\left (x^{2} \log {\left (x^{2} - x \right )}^{2} + 2 x^{2} \log {\left (x^{2} - x \right )} + x^{2}\right ) e^{x^{2}} - 2} \] Input:
integrate(((10*x**4-10*x**3+10*x**2-10*x)*exp(x**2)*ln(x**2-x)**2+(20*x**4 -20*x**3+40*x**2-30*x)*exp(x**2)*ln(x**2-x)+(10*x**4-10*x**3+30*x**2-20*x) *exp(x**2))/((x**5-x**4)*exp(x**2)**2*ln(x**2-x)**4+(4*x**5-4*x**4)*exp(x* *2)**2*ln(x**2-x)**3+((6*x**5-6*x**4)*exp(x**2)**2+(-4*x**3+4*x**2)*exp(x* *2))*ln(x**2-x)**2+((4*x**5-4*x**4)*exp(x**2)**2+(-8*x**3+8*x**2)*exp(x**2 ))*ln(x**2-x)+(x**5-x**4)*exp(x**2)**2+(-4*x**3+4*x**2)*exp(x**2)+4*x-4),x )
Output:
-5/((x**2*log(x**2 - x)**2 + 2*x**2*log(x**2 - x) + x**2)*exp(x**2) - 2)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (25) = 50\).
Time = 0.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.07 \[ \int \frac {e^{x^2} \left (-20 x+30 x^2-10 x^3+10 x^4\right )+e^{x^2} \left (-30 x+40 x^2-20 x^3+20 x^4\right ) \log \left (-x+x^2\right )+e^{x^2} \left (-10 x+10 x^2-10 x^3+10 x^4\right ) \log ^2\left (-x+x^2\right )}{-4+4 x+e^{x^2} \left (4 x^2-4 x^3\right )+e^{2 x^2} \left (-x^4+x^5\right )+\left (e^{x^2} \left (8 x^2-8 x^3\right )+e^{2 x^2} \left (-4 x^4+4 x^5\right )\right ) \log \left (-x+x^2\right )+\left (e^{x^2} \left (4 x^2-4 x^3\right )+e^{2 x^2} \left (-6 x^4+6 x^5\right )\right ) \log ^2\left (-x+x^2\right )+e^{2 x^2} \left (-4 x^4+4 x^5\right ) \log ^3\left (-x+x^2\right )+e^{2 x^2} \left (-x^4+x^5\right ) \log ^4\left (-x+x^2\right )} \, dx=-\frac {5}{{\left (x^{2} \log \left (x - 1\right )^{2} + x^{2} \log \left (x\right )^{2} + 2 \, x^{2} \log \left (x\right ) + x^{2} + 2 \, {\left (x^{2} \log \left (x\right ) + x^{2}\right )} \log \left (x - 1\right )\right )} e^{\left (x^{2}\right )} - 2} \] Input:
integrate(((10*x^4-10*x^3+10*x^2-10*x)*exp(x^2)*log(x^2-x)^2+(20*x^4-20*x^ 3+40*x^2-30*x)*exp(x^2)*log(x^2-x)+(10*x^4-10*x^3+30*x^2-20*x)*exp(x^2))/( (x^5-x^4)*exp(x^2)^2*log(x^2-x)^4+(4*x^5-4*x^4)*exp(x^2)^2*log(x^2-x)^3+(( 6*x^5-6*x^4)*exp(x^2)^2+(-4*x^3+4*x^2)*exp(x^2))*log(x^2-x)^2+((4*x^5-4*x^ 4)*exp(x^2)^2+(-8*x^3+8*x^2)*exp(x^2))*log(x^2-x)+(x^5-x^4)*exp(x^2)^2+(-4 *x^3+4*x^2)*exp(x^2)+4*x-4),x, algorithm="maxima")
Output:
-5/((x^2*log(x - 1)^2 + x^2*log(x)^2 + 2*x^2*log(x) + x^2 + 2*(x^2*log(x) + x^2)*log(x - 1))*e^(x^2) - 2)
Time = 0.46 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {e^{x^2} \left (-20 x+30 x^2-10 x^3+10 x^4\right )+e^{x^2} \left (-30 x+40 x^2-20 x^3+20 x^4\right ) \log \left (-x+x^2\right )+e^{x^2} \left (-10 x+10 x^2-10 x^3+10 x^4\right ) \log ^2\left (-x+x^2\right )}{-4+4 x+e^{x^2} \left (4 x^2-4 x^3\right )+e^{2 x^2} \left (-x^4+x^5\right )+\left (e^{x^2} \left (8 x^2-8 x^3\right )+e^{2 x^2} \left (-4 x^4+4 x^5\right )\right ) \log \left (-x+x^2\right )+\left (e^{x^2} \left (4 x^2-4 x^3\right )+e^{2 x^2} \left (-6 x^4+6 x^5\right )\right ) \log ^2\left (-x+x^2\right )+e^{2 x^2} \left (-4 x^4+4 x^5\right ) \log ^3\left (-x+x^2\right )+e^{2 x^2} \left (-x^4+x^5\right ) \log ^4\left (-x+x^2\right )} \, dx=-\frac {5}{x^{2} e^{\left (x^{2}\right )} \log \left (x^{2} - x\right )^{2} + 2 \, x^{2} e^{\left (x^{2}\right )} \log \left (x^{2} - x\right ) + x^{2} e^{\left (x^{2}\right )} - 2} \] Input:
integrate(((10*x^4-10*x^3+10*x^2-10*x)*exp(x^2)*log(x^2-x)^2+(20*x^4-20*x^ 3+40*x^2-30*x)*exp(x^2)*log(x^2-x)+(10*x^4-10*x^3+30*x^2-20*x)*exp(x^2))/( (x^5-x^4)*exp(x^2)^2*log(x^2-x)^4+(4*x^5-4*x^4)*exp(x^2)^2*log(x^2-x)^3+(( 6*x^5-6*x^4)*exp(x^2)^2+(-4*x^3+4*x^2)*exp(x^2))*log(x^2-x)^2+((4*x^5-4*x^ 4)*exp(x^2)^2+(-8*x^3+8*x^2)*exp(x^2))*log(x^2-x)+(x^5-x^4)*exp(x^2)^2+(-4 *x^3+4*x^2)*exp(x^2)+4*x-4),x, algorithm="giac")
Output:
-5/(x^2*e^(x^2)*log(x^2 - x)^2 + 2*x^2*e^(x^2)*log(x^2 - x) + x^2*e^(x^2) - 2)
Timed out. \[ \int \frac {e^{x^2} \left (-20 x+30 x^2-10 x^3+10 x^4\right )+e^{x^2} \left (-30 x+40 x^2-20 x^3+20 x^4\right ) \log \left (-x+x^2\right )+e^{x^2} \left (-10 x+10 x^2-10 x^3+10 x^4\right ) \log ^2\left (-x+x^2\right )}{-4+4 x+e^{x^2} \left (4 x^2-4 x^3\right )+e^{2 x^2} \left (-x^4+x^5\right )+\left (e^{x^2} \left (8 x^2-8 x^3\right )+e^{2 x^2} \left (-4 x^4+4 x^5\right )\right ) \log \left (-x+x^2\right )+\left (e^{x^2} \left (4 x^2-4 x^3\right )+e^{2 x^2} \left (-6 x^4+6 x^5\right )\right ) \log ^2\left (-x+x^2\right )+e^{2 x^2} \left (-4 x^4+4 x^5\right ) \log ^3\left (-x+x^2\right )+e^{2 x^2} \left (-x^4+x^5\right ) \log ^4\left (-x+x^2\right )} \, dx=-\int \frac {{\mathrm {e}}^{x^2}\,\left (-10\,x^4+10\,x^3-10\,x^2+10\,x\right )\,{\ln \left (x^2-x\right )}^2+{\mathrm {e}}^{x^2}\,\left (-20\,x^4+20\,x^3-40\,x^2+30\,x\right )\,\ln \left (x^2-x\right )+{\mathrm {e}}^{x^2}\,\left (-10\,x^4+10\,x^3-30\,x^2+20\,x\right )}{-{\mathrm {e}}^{2\,x^2}\,\left (x^4-x^5\right )\,{\ln \left (x^2-x\right )}^4-{\mathrm {e}}^{2\,x^2}\,\left (4\,x^4-4\,x^5\right )\,{\ln \left (x^2-x\right )}^3+\left ({\mathrm {e}}^{x^2}\,\left (4\,x^2-4\,x^3\right )-{\mathrm {e}}^{2\,x^2}\,\left (6\,x^4-6\,x^5\right )\right )\,{\ln \left (x^2-x\right )}^2+\left ({\mathrm {e}}^{x^2}\,\left (8\,x^2-8\,x^3\right )-{\mathrm {e}}^{2\,x^2}\,\left (4\,x^4-4\,x^5\right )\right )\,\ln \left (x^2-x\right )+4\,x-{\mathrm {e}}^{2\,x^2}\,\left (x^4-x^5\right )+{\mathrm {e}}^{x^2}\,\left (4\,x^2-4\,x^3\right )-4} \,d x \] Input:
int(-(exp(x^2)*(20*x - 30*x^2 + 10*x^3 - 10*x^4) + exp(x^2)*log(x^2 - x)^2 *(10*x - 10*x^2 + 10*x^3 - 10*x^4) + exp(x^2)*log(x^2 - x)*(30*x - 40*x^2 + 20*x^3 - 20*x^4))/(4*x - exp(2*x^2)*(x^4 - x^5) + exp(x^2)*(4*x^2 - 4*x^ 3) + log(x^2 - x)*(exp(x^2)*(8*x^2 - 8*x^3) - exp(2*x^2)*(4*x^4 - 4*x^5)) + log(x^2 - x)^2*(exp(x^2)*(4*x^2 - 4*x^3) - exp(2*x^2)*(6*x^4 - 6*x^5)) - exp(2*x^2)*log(x^2 - x)^4*(x^4 - x^5) - exp(2*x^2)*log(x^2 - x)^3*(4*x^4 - 4*x^5) - 4),x)
Output:
-int((exp(x^2)*(20*x - 30*x^2 + 10*x^3 - 10*x^4) + exp(x^2)*log(x^2 - x)^2 *(10*x - 10*x^2 + 10*x^3 - 10*x^4) + exp(x^2)*log(x^2 - x)*(30*x - 40*x^2 + 20*x^3 - 20*x^4))/(4*x - exp(2*x^2)*(x^4 - x^5) + exp(x^2)*(4*x^2 - 4*x^ 3) + log(x^2 - x)*(exp(x^2)*(8*x^2 - 8*x^3) - exp(2*x^2)*(4*x^4 - 4*x^5)) + log(x^2 - x)^2*(exp(x^2)*(4*x^2 - 4*x^3) - exp(2*x^2)*(6*x^4 - 6*x^5)) - exp(2*x^2)*log(x^2 - x)^4*(x^4 - x^5) - exp(2*x^2)*log(x^2 - x)^3*(4*x^4 - 4*x^5) - 4), x)
\[ \int \frac {e^{x^2} \left (-20 x+30 x^2-10 x^3+10 x^4\right )+e^{x^2} \left (-30 x+40 x^2-20 x^3+20 x^4\right ) \log \left (-x+x^2\right )+e^{x^2} \left (-10 x+10 x^2-10 x^3+10 x^4\right ) \log ^2\left (-x+x^2\right )}{-4+4 x+e^{x^2} \left (4 x^2-4 x^3\right )+e^{2 x^2} \left (-x^4+x^5\right )+\left (e^{x^2} \left (8 x^2-8 x^3\right )+e^{2 x^2} \left (-4 x^4+4 x^5\right )\right ) \log \left (-x+x^2\right )+\left (e^{x^2} \left (4 x^2-4 x^3\right )+e^{2 x^2} \left (-6 x^4+6 x^5\right )\right ) \log ^2\left (-x+x^2\right )+e^{2 x^2} \left (-4 x^4+4 x^5\right ) \log ^3\left (-x+x^2\right )+e^{2 x^2} \left (-x^4+x^5\right ) \log ^4\left (-x+x^2\right )} \, dx=\int \frac {\left (10 x^{4}-10 x^{3}+10 x^{2}-10 x \right ) {\mathrm e}^{x^{2}} \mathrm {log}\left (x^{2}-x \right )^{2}+\left (20 x^{4}-20 x^{3}+40 x^{2}-30 x \right ) {\mathrm e}^{x^{2}} \mathrm {log}\left (x^{2}-x \right )+\left (10 x^{4}-10 x^{3}+30 x^{2}-20 x \right ) {\mathrm e}^{x^{2}}}{\left (x^{5}-x^{4}\right ) \left ({\mathrm e}^{x^{2}}\right )^{2} \mathrm {log}\left (x^{2}-x \right )^{4}+\left (4 x^{5}-4 x^{4}\right ) \left ({\mathrm e}^{x^{2}}\right )^{2} \mathrm {log}\left (x^{2}-x \right )^{3}+\left (\left (6 x^{5}-6 x^{4}\right ) \left ({\mathrm e}^{x^{2}}\right )^{2}+\left (-4 x^{3}+4 x^{2}\right ) {\mathrm e}^{x^{2}}\right ) \mathrm {log}\left (x^{2}-x \right )^{2}+\left (\left (4 x^{5}-4 x^{4}\right ) \left ({\mathrm e}^{x^{2}}\right )^{2}+\left (-8 x^{3}+8 x^{2}\right ) {\mathrm e}^{x^{2}}\right ) \mathrm {log}\left (x^{2}-x \right )+\left (x^{5}-x^{4}\right ) \left ({\mathrm e}^{x^{2}}\right )^{2}+\left (-4 x^{3}+4 x^{2}\right ) {\mathrm e}^{x^{2}}+4 x -4}d x \] Input:
int(((10*x^4-10*x^3+10*x^2-10*x)*exp(x^2)*log(x^2-x)^2+(20*x^4-20*x^3+40*x ^2-30*x)*exp(x^2)*log(x^2-x)+(10*x^4-10*x^3+30*x^2-20*x)*exp(x^2))/((x^5-x ^4)*exp(x^2)^2*log(x^2-x)^4+(4*x^5-4*x^4)*exp(x^2)^2*log(x^2-x)^3+((6*x^5- 6*x^4)*exp(x^2)^2+(-4*x^3+4*x^2)*exp(x^2))*log(x^2-x)^2+((4*x^5-4*x^4)*exp (x^2)^2+(-8*x^3+8*x^2)*exp(x^2))*log(x^2-x)+(x^5-x^4)*exp(x^2)^2+(-4*x^3+4 *x^2)*exp(x^2)+4*x-4),x)
Output:
int(((10*x^4-10*x^3+10*x^2-10*x)*exp(x^2)*log(x^2-x)^2+(20*x^4-20*x^3+40*x ^2-30*x)*exp(x^2)*log(x^2-x)+(10*x^4-10*x^3+30*x^2-20*x)*exp(x^2))/((x^5-x ^4)*exp(x^2)^2*log(x^2-x)^4+(4*x^5-4*x^4)*exp(x^2)^2*log(x^2-x)^3+((6*x^5- 6*x^4)*exp(x^2)^2+(-4*x^3+4*x^2)*exp(x^2))*log(x^2-x)^2+((4*x^5-4*x^4)*exp (x^2)^2+(-8*x^3+8*x^2)*exp(x^2))*log(x^2-x)+(x^5-x^4)*exp(x^2)^2+(-4*x^3+4 *x^2)*exp(x^2)+4*x-4),x)