Integrand size = 210, antiderivative size = 35 \[ \int \frac {-8 x+10 x^2-2 x^3+e^{2+2 x} \left (-20-6 x+2 x^2\right )+e^{1+x} \left (16-2 x-8 x^2+2 x^3\right )+e^{\frac {x^2}{4}} \left (-2 x+2 x^2+e^{1+x} \left (4-3 x^2\right )+e^{2+2 x} \left (-4+x^2\right )\right )+\left (e^{2+2 x} (10-2 x)+e^{1+x} \left (-8+10 x-2 x^2\right )+e^{\frac {x^2}{4}} \left (e^{2+2 x} (2-x)+e^{1+x} (-2+2 x)\right )\right ) \log \left (-4-e^{\frac {x^2}{4}}+x\right )}{e^{2+2 x+\frac {x^2}{4}}+e^{2+2 x} (4-x)} \, dx=8-\left (-2-x+e^{-1-x} x+\log \left (-4-e^{\frac {x^2}{4}}+x\right )\right )^2 \]
\[ \int \frac {-8 x+10 x^2-2 x^3+e^{2+2 x} \left (-20-6 x+2 x^2\right )+e^{1+x} \left (16-2 x-8 x^2+2 x^3\right )+e^{\frac {x^2}{4}} \left (-2 x+2 x^2+e^{1+x} \left (4-3 x^2\right )+e^{2+2 x} \left (-4+x^2\right )\right )+\left (e^{2+2 x} (10-2 x)+e^{1+x} \left (-8+10 x-2 x^2\right )+e^{\frac {x^2}{4}} \left (e^{2+2 x} (2-x)+e^{1+x} (-2+2 x)\right )\right ) \log \left (-4-e^{\frac {x^2}{4}}+x\right )}{e^{2+2 x+\frac {x^2}{4}}+e^{2+2 x} (4-x)} \, dx=\int \frac {-8 x+10 x^2-2 x^3+e^{2+2 x} \left (-20-6 x+2 x^2\right )+e^{1+x} \left (16-2 x-8 x^2+2 x^3\right )+e^{\frac {x^2}{4}} \left (-2 x+2 x^2+e^{1+x} \left (4-3 x^2\right )+e^{2+2 x} \left (-4+x^2\right )\right )+\left (e^{2+2 x} (10-2 x)+e^{1+x} \left (-8+10 x-2 x^2\right )+e^{\frac {x^2}{4}} \left (e^{2+2 x} (2-x)+e^{1+x} (-2+2 x)\right )\right ) \log \left (-4-e^{\frac {x^2}{4}}+x\right )}{e^{2+2 x+\frac {x^2}{4}}+e^{2+2 x} (4-x)} \, dx \]
Integrate[(-8*x + 10*x^2 - 2*x^3 + E^(2 + 2*x)*(-20 - 6*x + 2*x^2) + E^(1 + x)*(16 - 2*x - 8*x^2 + 2*x^3) + E^(x^2/4)*(-2*x + 2*x^2 + E^(1 + x)*(4 - 3*x^2) + E^(2 + 2*x)*(-4 + x^2)) + (E^(2 + 2*x)*(10 - 2*x) + E^(1 + x)*(- 8 + 10*x - 2*x^2) + E^(x^2/4)*(E^(2 + 2*x)*(2 - x) + E^(1 + x)*(-2 + 2*x)) )*Log[-4 - E^(x^2/4) + x])/(E^(2 + 2*x + x^2/4) + E^(2 + 2*x)*(4 - x)),x]
Integrate[(-8*x + 10*x^2 - 2*x^3 + E^(2 + 2*x)*(-20 - 6*x + 2*x^2) + E^(1 + x)*(16 - 2*x - 8*x^2 + 2*x^3) + E^(x^2/4)*(-2*x + 2*x^2 + E^(1 + x)*(4 - 3*x^2) + E^(2 + 2*x)*(-4 + x^2)) + (E^(2 + 2*x)*(10 - 2*x) + E^(1 + x)*(- 8 + 10*x - 2*x^2) + E^(x^2/4)*(E^(2 + 2*x)*(2 - x) + E^(1 + x)*(-2 + 2*x)) )*Log[-4 - E^(x^2/4) + x])/(E^(2 + 2*x + x^2/4) + E^(2 + 2*x)*(4 - x)), x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-2 x^3+10 x^2+e^{2 x+2} \left (2 x^2-6 x-20\right )+e^{\frac {x^2}{4}} \left (2 x^2+e^{x+1} \left (4-3 x^2\right )+e^{2 x+2} \left (x^2-4\right )-2 x\right )+\left (e^{x+1} \left (-2 x^2+10 x-8\right )+e^{\frac {x^2}{4}} \left (e^{2 x+2} (2-x)+e^{x+1} (2 x-2)\right )+e^{2 x+2} (10-2 x)\right ) \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} \left (2 x^3-8 x^2-2 x+16\right )-8 x}{e^{\frac {x^2}{4}+2 x+2}+e^{2 x+2} (4-x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 x-2} \left (-2 e^{\frac {x^2}{4}} (x-1)+2 \left (x^2-5 x+4\right )+2 e^{x+1} (x-5)+e^{\frac {1}{4} (x+2)^2} (x-2)\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )-x+e^{x+1} (x+2)\right )}{e^{\frac {x^2}{4}}-x+4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{\frac {1}{4} (x+2)^2-2 x-2} (x-2) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}-\frac {2 e^{-2 x-2} \left (-x^2+e^{\frac {x^2}{4}} x-e^{\frac {x^2}{4}}-e^{x+1} x+5 x+5 e^{x+1}-4\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 x-2} \left (-2 e^{\frac {x^2}{4}} (x-1)+2 \left (x^2-5 x+4\right )+2 e^{x+1} (x-5)+e^{\frac {1}{4} (x+2)^2} (x-2)\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )-x+e^{x+1} (x+2)\right )}{e^{\frac {x^2}{4}}-x+4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{\frac {1}{4} (x+2)^2-2 x-2} (x-2) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}-\frac {2 e^{-2 x-2} \left (-x^2+e^{\frac {x^2}{4}} x-e^{\frac {x^2}{4}}-e^{x+1} x+5 x+5 e^{x+1}-4\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 x-2} \left (-2 e^{\frac {x^2}{4}} (x-1)+2 \left (x^2-5 x+4\right )+2 e^{x+1} (x-5)+e^{\frac {1}{4} (x+2)^2} (x-2)\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )-x+e^{x+1} (x+2)\right )}{e^{\frac {x^2}{4}}-x+4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{\frac {1}{4} (x+2)^2-2 x-2} (x-2) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}-\frac {2 e^{-2 x-2} \left (-x^2+e^{\frac {x^2}{4}} x-e^{\frac {x^2}{4}}-e^{x+1} x+5 x+5 e^{x+1}-4\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 x-2} \left (-2 e^{\frac {x^2}{4}} (x-1)+2 \left (x^2-5 x+4\right )+2 e^{x+1} (x-5)+e^{\frac {1}{4} (x+2)^2} (x-2)\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )-x+e^{x+1} (x+2)\right )}{e^{\frac {x^2}{4}}-x+4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{\frac {1}{4} (x+2)^2-2 x-2} (x-2) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}-\frac {2 e^{-2 x-2} \left (-x^2+e^{\frac {x^2}{4}} x-e^{\frac {x^2}{4}}-e^{x+1} x+5 x+5 e^{x+1}-4\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 x-2} \left (-2 e^{\frac {x^2}{4}} (x-1)+2 \left (x^2-5 x+4\right )+2 e^{x+1} (x-5)+e^{\frac {1}{4} (x+2)^2} (x-2)\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )-x+e^{x+1} (x+2)\right )}{e^{\frac {x^2}{4}}-x+4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{\frac {1}{4} (x+2)^2-2 x-2} (x-2) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}-\frac {2 e^{-2 x-2} \left (-x^2+e^{\frac {x^2}{4}} x-e^{\frac {x^2}{4}}-e^{x+1} x+5 x+5 e^{x+1}-4\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 x-2} \left (-2 e^{\frac {x^2}{4}} (x-1)+2 \left (x^2-5 x+4\right )+2 e^{x+1} (x-5)+e^{\frac {1}{4} (x+2)^2} (x-2)\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )-x+e^{x+1} (x+2)\right )}{e^{\frac {x^2}{4}}-x+4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{\frac {1}{4} (x+2)^2-2 x-2} (x-2) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}-\frac {2 e^{-2 x-2} \left (-x^2+e^{\frac {x^2}{4}} x-e^{\frac {x^2}{4}}-e^{x+1} x+5 x+5 e^{x+1}-4\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 x-2} \left (-2 e^{\frac {x^2}{4}} (x-1)+2 \left (x^2-5 x+4\right )+2 e^{x+1} (x-5)+e^{\frac {1}{4} (x+2)^2} (x-2)\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )-x+e^{x+1} (x+2)\right )}{e^{\frac {x^2}{4}}-x+4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{\frac {1}{4} (x+2)^2-2 x-2} (x-2) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}-\frac {2 e^{-2 x-2} \left (-x^2+e^{\frac {x^2}{4}} x-e^{\frac {x^2}{4}}-e^{x+1} x+5 x+5 e^{x+1}-4\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 x-2} \left (-2 e^{\frac {x^2}{4}} (x-1)+2 \left (x^2-5 x+4\right )+2 e^{x+1} (x-5)+e^{\frac {1}{4} (x+2)^2} (x-2)\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )-x+e^{x+1} (x+2)\right )}{e^{\frac {x^2}{4}}-x+4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{\frac {1}{4} (x+2)^2-2 x-2} (x-2) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}-\frac {2 e^{-2 x-2} \left (-x^2+e^{\frac {x^2}{4}} x-e^{\frac {x^2}{4}}-e^{x+1} x+5 x+5 e^{x+1}-4\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 x-2} \left (-2 e^{\frac {x^2}{4}} (x-1)+2 \left (x^2-5 x+4\right )+2 e^{x+1} (x-5)+e^{\frac {1}{4} (x+2)^2} (x-2)\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )-x+e^{x+1} (x+2)\right )}{e^{\frac {x^2}{4}}-x+4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{\frac {1}{4} (x+2)^2-2 x-2} (x-2) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}-\frac {2 e^{-2 x-2} \left (-x^2+e^{\frac {x^2}{4}} x-e^{\frac {x^2}{4}}-e^{x+1} x+5 x+5 e^{x+1}-4\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 x-2} \left (-2 e^{\frac {x^2}{4}} (x-1)+2 \left (x^2-5 x+4\right )+2 e^{x+1} (x-5)+e^{\frac {1}{4} (x+2)^2} (x-2)\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )-x+e^{x+1} (x+2)\right )}{e^{\frac {x^2}{4}}-x+4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{\frac {1}{4} (x+2)^2-2 x-2} (x-2) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}-\frac {2 e^{-2 x-2} \left (-x^2+e^{\frac {x^2}{4}} x-e^{\frac {x^2}{4}}-e^{x+1} x+5 x+5 e^{x+1}-4\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 x-2} \left (-2 e^{\frac {x^2}{4}} (x-1)+2 \left (x^2-5 x+4\right )+2 e^{x+1} (x-5)+e^{\frac {1}{4} (x+2)^2} (x-2)\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )-x+e^{x+1} (x+2)\right )}{e^{\frac {x^2}{4}}-x+4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{\frac {1}{4} (x+2)^2-2 x-2} (x-2) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}-\frac {2 e^{-2 x-2} \left (-x^2+e^{\frac {x^2}{4}} x-e^{\frac {x^2}{4}}-e^{x+1} x+5 x+5 e^{x+1}-4\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 x-2} \left (-2 e^{\frac {x^2}{4}} (x-1)+2 \left (x^2-5 x+4\right )+2 e^{x+1} (x-5)+e^{\frac {1}{4} (x+2)^2} (x-2)\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )-x+e^{x+1} (x+2)\right )}{e^{\frac {x^2}{4}}-x+4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{\frac {1}{4} (x+2)^2-2 x-2} (x-2) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}-\frac {2 e^{-2 x-2} \left (-x^2+e^{\frac {x^2}{4}} x-e^{\frac {x^2}{4}}-e^{x+1} x+5 x+5 e^{x+1}-4\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 x-2} \left (-2 e^{\frac {x^2}{4}} (x-1)+2 \left (x^2-5 x+4\right )+2 e^{x+1} (x-5)+e^{\frac {1}{4} (x+2)^2} (x-2)\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )-x+e^{x+1} (x+2)\right )}{e^{\frac {x^2}{4}}-x+4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{\frac {1}{4} (x+2)^2-2 x-2} (x-2) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}-\frac {2 e^{-2 x-2} \left (-x^2+e^{\frac {x^2}{4}} x-e^{\frac {x^2}{4}}-e^{x+1} x+5 x+5 e^{x+1}-4\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 x-2} \left (-2 e^{\frac {x^2}{4}} (x-1)+2 \left (x^2-5 x+4\right )+2 e^{x+1} (x-5)+e^{\frac {1}{4} (x+2)^2} (x-2)\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )-x+e^{x+1} (x+2)\right )}{e^{\frac {x^2}{4}}-x+4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{\frac {1}{4} (x+2)^2-2 x-2} (x-2) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}-\frac {2 e^{-2 x-2} \left (-x^2+e^{\frac {x^2}{4}} x-e^{\frac {x^2}{4}}-e^{x+1} x+5 x+5 e^{x+1}-4\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 x-2} \left (-2 e^{\frac {x^2}{4}} (x-1)+2 \left (x^2-5 x+4\right )+2 e^{x+1} (x-5)+e^{\frac {1}{4} (x+2)^2} (x-2)\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )-x+e^{x+1} (x+2)\right )}{e^{\frac {x^2}{4}}-x+4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{\frac {1}{4} (x+2)^2-2 x-2} (x-2) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}-\frac {2 e^{-2 x-2} \left (-x^2+e^{\frac {x^2}{4}} x-e^{\frac {x^2}{4}}-e^{x+1} x+5 x+5 e^{x+1}-4\right ) \left (-e^{x+1} \log \left (-e^{\frac {x^2}{4}}+x-4\right )+e^{x+1} x-x+2 e^{x+1}\right )}{e^{\frac {x^2}{4}}-x+4}\right )dx\) |
Int[(-8*x + 10*x^2 - 2*x^3 + E^(2 + 2*x)*(-20 - 6*x + 2*x^2) + E^(1 + x)*( 16 - 2*x - 8*x^2 + 2*x^3) + E^(x^2/4)*(-2*x + 2*x^2 + E^(1 + x)*(4 - 3*x^2 ) + E^(2 + 2*x)*(-4 + x^2)) + (E^(2 + 2*x)*(10 - 2*x) + E^(1 + x)*(-8 + 10 *x - 2*x^2) + E^(x^2/4)*(E^(2 + 2*x)*(2 - x) + E^(1 + x)*(-2 + 2*x)))*Log[ -4 - E^(x^2/4) + x])/(E^(2 + 2*x + x^2/4) + E^(2 + 2*x)*(4 - x)),x]
3.3.66.3.1 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. \(106\) vs. \(2(31)=62\).
Time = 0.48 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.06
| method | result | size |
| risch | \(2 x \ln \left (-{\mathrm e}^{\frac {x^{2}}{4}}+x -4\right )+2 x^{2} {\mathrm e}^{-1-x}-x^{2}+4 x \,{\mathrm e}^{-1-x}-2 x \,{\mathrm e}^{-1-x} \ln \left (-{\mathrm e}^{\frac {x^{2}}{4}}+x -4\right )+4 \ln \left ({\mathrm e}^{\frac {x^{2}}{4}}-x +4\right )-4 x -{\mathrm e}^{-2-2 x} x^{2}-\ln \left (-{\mathrm e}^{\frac {x^{2}}{4}}+x -4\right )^{2}\) | \(107\) |
| parallelrisch | \(\frac {\left (-8 x^{2} {\mathrm e}^{2+2 x}+16 \ln \left (-{\mathrm e}^{\frac {x^{2}}{4}}+x -4\right ) {\mathrm e}^{2+2 x} x -8 \,{\mathrm e}^{2+2 x} \ln \left (-{\mathrm e}^{\frac {x^{2}}{4}}+x -4\right )^{2}+32 \ln \left (-{\mathrm e}^{\frac {x^{2}}{4}}+x -4\right ) {\mathrm e}^{2+2 x}+16 x^{2} {\mathrm e}^{1+x}-32 x \,{\mathrm e}^{2+2 x}-16 \,{\mathrm e}^{1+x} \ln \left (-{\mathrm e}^{\frac {x^{2}}{4}}+x -4\right ) x -8 x^{2}+32 x \,{\mathrm e}^{1+x}\right ) {\mathrm e}^{-2-2 x}}{8}\) | \(133\) |
int(((((2-x)*exp(1+x)^2+(-2+2*x)*exp(1+x))*exp(1/4*x^2)+(-2*x+10)*exp(1+x) ^2+(-2*x^2+10*x-8)*exp(1+x))*ln(-exp(1/4*x^2)+x-4)+((x^2-4)*exp(1+x)^2+(-3 *x^2+4)*exp(1+x)+2*x^2-2*x)*exp(1/4*x^2)+(2*x^2-6*x-20)*exp(1+x)^2+(2*x^3- 8*x^2-2*x+16)*exp(1+x)-2*x^3+10*x^2-8*x)/(exp(1+x)^2*exp(1/4*x^2)+(-x+4)*e xp(1+x)^2),x,method=_RETURNVERBOSE)
2*x*ln(-exp(1/4*x^2)+x-4)+2*x^2*exp(-1-x)-x^2+4*x*exp(-1-x)-2*x*exp(-1-x)* ln(-exp(1/4*x^2)+x-4)+4*ln(exp(1/4*x^2)-x+4)-4*x-exp(-2-2*x)*x^2-ln(-exp(1 /4*x^2)+x-4)^2
Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (31) = 62\).
Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 3.77 \[ \int \frac {-8 x+10 x^2-2 x^3+e^{2+2 x} \left (-20-6 x+2 x^2\right )+e^{1+x} \left (16-2 x-8 x^2+2 x^3\right )+e^{\frac {x^2}{4}} \left (-2 x+2 x^2+e^{1+x} \left (4-3 x^2\right )+e^{2+2 x} \left (-4+x^2\right )\right )+\left (e^{2+2 x} (10-2 x)+e^{1+x} \left (-8+10 x-2 x^2\right )+e^{\frac {x^2}{4}} \left (e^{2+2 x} (2-x)+e^{1+x} (-2+2 x)\right )\right ) \log \left (-4-e^{\frac {x^2}{4}}+x\right )}{e^{2+2 x+\frac {x^2}{4}}+e^{2+2 x} (4-x)} \, dx=-{\left (e^{\left (2 \, x + 2\right )} \log \left ({\left ({\left (x - 4\right )} e^{\left (2 \, x + 2\right )} - e^{\left (\frac {1}{4} \, x^{2} + 2 \, x + 2\right )}\right )} e^{\left (-2 \, x - 2\right )}\right )^{2} + x^{2} + {\left (x^{2} + 4 \, x\right )} e^{\left (2 \, x + 2\right )} - 2 \, {\left (x^{2} + 2 \, x\right )} e^{\left (x + 1\right )} - 2 \, {\left ({\left (x + 2\right )} e^{\left (2 \, x + 2\right )} - x e^{\left (x + 1\right )}\right )} \log \left ({\left ({\left (x - 4\right )} e^{\left (2 \, x + 2\right )} - e^{\left (\frac {1}{4} \, x^{2} + 2 \, x + 2\right )}\right )} e^{\left (-2 \, x - 2\right )}\right )\right )} e^{\left (-2 \, x - 2\right )} \]
integrate(((((2-x)*exp(1+x)^2+(-2+2*x)*exp(1+x))*exp(1/4*x^2)+(-2*x+10)*ex p(1+x)^2+(-2*x^2+10*x-8)*exp(1+x))*log(-exp(1/4*x^2)+x-4)+((x^2-4)*exp(1+x )^2+(-3*x^2+4)*exp(1+x)+2*x^2-2*x)*exp(1/4*x^2)+(2*x^2-6*x-20)*exp(1+x)^2+ (2*x^3-8*x^2-2*x+16)*exp(1+x)-2*x^3+10*x^2-8*x)/(exp(1+x)^2*exp(1/4*x^2)+( -x+4)*exp(1+x)^2),x, algorithm=\
-(e^(2*x + 2)*log(((x - 4)*e^(2*x + 2) - e^(1/4*x^2 + 2*x + 2))*e^(-2*x - 2))^2 + x^2 + (x^2 + 4*x)*e^(2*x + 2) - 2*(x^2 + 2*x)*e^(x + 1) - 2*((x + 2)*e^(2*x + 2) - x*e^(x + 1))*log(((x - 4)*e^(2*x + 2) - e^(1/4*x^2 + 2*x + 2))*e^(-2*x - 2)))*e^(-2*x - 2)
Timed out. \[ \int \frac {-8 x+10 x^2-2 x^3+e^{2+2 x} \left (-20-6 x+2 x^2\right )+e^{1+x} \left (16-2 x-8 x^2+2 x^3\right )+e^{\frac {x^2}{4}} \left (-2 x+2 x^2+e^{1+x} \left (4-3 x^2\right )+e^{2+2 x} \left (-4+x^2\right )\right )+\left (e^{2+2 x} (10-2 x)+e^{1+x} \left (-8+10 x-2 x^2\right )+e^{\frac {x^2}{4}} \left (e^{2+2 x} (2-x)+e^{1+x} (-2+2 x)\right )\right ) \log \left (-4-e^{\frac {x^2}{4}}+x\right )}{e^{2+2 x+\frac {x^2}{4}}+e^{2+2 x} (4-x)} \, dx=\text {Timed out} \]
integrate(((((2-x)*exp(1+x)**2+(-2+2*x)*exp(1+x))*exp(1/4*x**2)+(-2*x+10)* exp(1+x)**2+(-2*x**2+10*x-8)*exp(1+x))*ln(-exp(1/4*x**2)+x-4)+((x**2-4)*ex p(1+x)**2+(-3*x**2+4)*exp(1+x)+2*x**2-2*x)*exp(1/4*x**2)+(2*x**2-6*x-20)*e xp(1+x)**2+(2*x**3-8*x**2-2*x+16)*exp(1+x)-2*x**3+10*x**2-8*x)/(exp(1+x)** 2*exp(1/4*x**2)+(-x+4)*exp(1+x)**2),x)
Exception generated. \[ \int \frac {-8 x+10 x^2-2 x^3+e^{2+2 x} \left (-20-6 x+2 x^2\right )+e^{1+x} \left (16-2 x-8 x^2+2 x^3\right )+e^{\frac {x^2}{4}} \left (-2 x+2 x^2+e^{1+x} \left (4-3 x^2\right )+e^{2+2 x} \left (-4+x^2\right )\right )+\left (e^{2+2 x} (10-2 x)+e^{1+x} \left (-8+10 x-2 x^2\right )+e^{\frac {x^2}{4}} \left (e^{2+2 x} (2-x)+e^{1+x} (-2+2 x)\right )\right ) \log \left (-4-e^{\frac {x^2}{4}}+x\right )}{e^{2+2 x+\frac {x^2}{4}}+e^{2+2 x} (4-x)} \, dx=\text {Exception raised: RuntimeError} \]
integrate(((((2-x)*exp(1+x)^2+(-2+2*x)*exp(1+x))*exp(1/4*x^2)+(-2*x+10)*ex p(1+x)^2+(-2*x^2+10*x-8)*exp(1+x))*log(-exp(1/4*x^2)+x-4)+((x^2-4)*exp(1+x )^2+(-3*x^2+4)*exp(1+x)+2*x^2-2*x)*exp(1/4*x^2)+(2*x^2-6*x-20)*exp(1+x)^2+ (2*x^3-8*x^2-2*x+16)*exp(1+x)-2*x^3+10*x^2-8*x)/(exp(1+x)^2*exp(1/4*x^2)+( -x+4)*exp(1+x)^2),x, algorithm=\
Exception raised: RuntimeError >> ECL says: Memory limit reached. Please j ump to an outer pointer, quit program and enlarge thememory limits before executing the program again.
\[ \int \frac {-8 x+10 x^2-2 x^3+e^{2+2 x} \left (-20-6 x+2 x^2\right )+e^{1+x} \left (16-2 x-8 x^2+2 x^3\right )+e^{\frac {x^2}{4}} \left (-2 x+2 x^2+e^{1+x} \left (4-3 x^2\right )+e^{2+2 x} \left (-4+x^2\right )\right )+\left (e^{2+2 x} (10-2 x)+e^{1+x} \left (-8+10 x-2 x^2\right )+e^{\frac {x^2}{4}} \left (e^{2+2 x} (2-x)+e^{1+x} (-2+2 x)\right )\right ) \log \left (-4-e^{\frac {x^2}{4}}+x\right )}{e^{2+2 x+\frac {x^2}{4}}+e^{2+2 x} (4-x)} \, dx=\int { \frac {2 \, x^{3} - 10 \, x^{2} - {\left (2 \, x^{2} + {\left (x^{2} - 4\right )} e^{\left (2 \, x + 2\right )} - {\left (3 \, x^{2} - 4\right )} e^{\left (x + 1\right )} - 2 \, x\right )} e^{\left (\frac {1}{4} \, x^{2}\right )} - 2 \, {\left (x^{2} - 3 \, x - 10\right )} e^{\left (2 \, x + 2\right )} - 2 \, {\left (x^{3} - 4 \, x^{2} - x + 8\right )} e^{\left (x + 1\right )} + {\left ({\left ({\left (x - 2\right )} e^{\left (2 \, x + 2\right )} - 2 \, {\left (x - 1\right )} e^{\left (x + 1\right )}\right )} e^{\left (\frac {1}{4} \, x^{2}\right )} + 2 \, {\left (x - 5\right )} e^{\left (2 \, x + 2\right )} + 2 \, {\left (x^{2} - 5 \, x + 4\right )} e^{\left (x + 1\right )}\right )} \log \left (x - e^{\left (\frac {1}{4} \, x^{2}\right )} - 4\right ) + 8 \, x}{{\left (x - 4\right )} e^{\left (2 \, x + 2\right )} - e^{\left (\frac {1}{4} \, x^{2} + 2 \, x + 2\right )}} \,d x } \]
integrate(((((2-x)*exp(1+x)^2+(-2+2*x)*exp(1+x))*exp(1/4*x^2)+(-2*x+10)*ex p(1+x)^2+(-2*x^2+10*x-8)*exp(1+x))*log(-exp(1/4*x^2)+x-4)+((x^2-4)*exp(1+x )^2+(-3*x^2+4)*exp(1+x)+2*x^2-2*x)*exp(1/4*x^2)+(2*x^2-6*x-20)*exp(1+x)^2+ (2*x^3-8*x^2-2*x+16)*exp(1+x)-2*x^3+10*x^2-8*x)/(exp(1+x)^2*exp(1/4*x^2)+( -x+4)*exp(1+x)^2),x, algorithm=\
integrate((2*x^3 - 10*x^2 - (2*x^2 + (x^2 - 4)*e^(2*x + 2) - (3*x^2 - 4)*e ^(x + 1) - 2*x)*e^(1/4*x^2) - 2*(x^2 - 3*x - 10)*e^(2*x + 2) - 2*(x^3 - 4* x^2 - x + 8)*e^(x + 1) + (((x - 2)*e^(2*x + 2) - 2*(x - 1)*e^(x + 1))*e^(1 /4*x^2) + 2*(x - 5)*e^(2*x + 2) + 2*(x^2 - 5*x + 4)*e^(x + 1))*log(x - e^( 1/4*x^2) - 4) + 8*x)/((x - 4)*e^(2*x + 2) - e^(1/4*x^2 + 2*x + 2)), x)
Timed out. \[ \int \frac {-8 x+10 x^2-2 x^3+e^{2+2 x} \left (-20-6 x+2 x^2\right )+e^{1+x} \left (16-2 x-8 x^2+2 x^3\right )+e^{\frac {x^2}{4}} \left (-2 x+2 x^2+e^{1+x} \left (4-3 x^2\right )+e^{2+2 x} \left (-4+x^2\right )\right )+\left (e^{2+2 x} (10-2 x)+e^{1+x} \left (-8+10 x-2 x^2\right )+e^{\frac {x^2}{4}} \left (e^{2+2 x} (2-x)+e^{1+x} (-2+2 x)\right )\right ) \log \left (-4-e^{\frac {x^2}{4}}+x\right )}{e^{2+2 x+\frac {x^2}{4}}+e^{2+2 x} (4-x)} \, dx=\int -\frac {8\,x+\ln \left (x-{\mathrm {e}}^{\frac {x^2}{4}}-4\right )\,\left ({\mathrm {e}}^{x+1}\,\left (2\,x^2-10\,x+8\right )-{\mathrm {e}}^{\frac {x^2}{4}}\,\left ({\mathrm {e}}^{x+1}\,\left (2\,x-2\right )-{\mathrm {e}}^{2\,x+2}\,\left (x-2\right )\right )+{\mathrm {e}}^{2\,x+2}\,\left (2\,x-10\right )\right )+{\mathrm {e}}^{x+1}\,\left (-2\,x^3+8\,x^2+2\,x-16\right )+{\mathrm {e}}^{2\,x+2}\,\left (-2\,x^2+6\,x+20\right )+{\mathrm {e}}^{\frac {x^2}{4}}\,\left (2\,x+{\mathrm {e}}^{x+1}\,\left (3\,x^2-4\right )-{\mathrm {e}}^{2\,x+2}\,\left (x^2-4\right )-2\,x^2\right )-10\,x^2+2\,x^3}{{\mathrm {e}}^{2\,x+2}\,{\mathrm {e}}^{\frac {x^2}{4}}-{\mathrm {e}}^{2\,x+2}\,\left (x-4\right )} \,d x \]
int(-(8*x + log(x - exp(x^2/4) - 4)*(exp(x + 1)*(2*x^2 - 10*x + 8) - exp(x ^2/4)*(exp(x + 1)*(2*x - 2) - exp(2*x + 2)*(x - 2)) + exp(2*x + 2)*(2*x - 10)) + exp(x + 1)*(2*x + 8*x^2 - 2*x^3 - 16) + exp(2*x + 2)*(6*x - 2*x^2 + 20) + exp(x^2/4)*(2*x + exp(x + 1)*(3*x^2 - 4) - exp(2*x + 2)*(x^2 - 4) - 2*x^2) - 10*x^2 + 2*x^3)/(exp(2*x + 2)*exp(x^2/4) - exp(2*x + 2)*(x - 4)) ,x)
int(-(8*x + log(x - exp(x^2/4) - 4)*(exp(x + 1)*(2*x^2 - 10*x + 8) - exp(x ^2/4)*(exp(x + 1)*(2*x - 2) - exp(2*x + 2)*(x - 2)) + exp(2*x + 2)*(2*x - 10)) + exp(x + 1)*(2*x + 8*x^2 - 2*x^3 - 16) + exp(2*x + 2)*(6*x - 2*x^2 + 20) + exp(x^2/4)*(2*x + exp(x + 1)*(3*x^2 - 4) - exp(2*x + 2)*(x^2 - 4) - 2*x^2) - 10*x^2 + 2*x^3)/(exp(2*x + 2)*exp(x^2/4) - exp(2*x + 2)*(x - 4)) , x)