Integrand size = 218, antiderivative size = 34 \[ \int \frac {-384 x+96 x^2-12 x^3+e^{\frac {1}{4} \left (4 x+e^{-5+x} x-x^2\right )} \left (-48 x^2+60 x^3-36 x^4+6 x^5+e^{-5+x} \left (12 x^3+9 x^4-3 x^5\right )\right )+\left (-48 x^2+12 x^3\right ) \log (-4+x)}{-256+64 x+e^{\frac {1}{2} \left (4 x+e^{-5+x} x-x^2\right )} \left (-16 x^2+4 x^3\right )+\left (-128 x+32 x^2\right ) \log (-4+x)+\left (-16 x^2+4 x^3\right ) \log ^2(-4+x)+e^{\frac {1}{4} \left (4 x+e^{-5+x} x-x^2\right )} \left (-128 x+32 x^2+\left (-32 x^2+8 x^3\right ) \log (-4+x)\right )} \, dx=-1+\frac {3 x}{e^{\frac {1}{4} \left (4+e^{-5+x}-x\right ) x}+\frac {4}{x}+\log (-4+x)} \] Output:
3*x/(4/x+exp(1/4*(4+exp(-5+x)-x)*x)+ln(-4+x))-1
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.74 \[ \int \frac {-384 x+96 x^2-12 x^3+e^{\frac {1}{4} \left (4 x+e^{-5+x} x-x^2\right )} \left (-48 x^2+60 x^3-36 x^4+6 x^5+e^{-5+x} \left (12 x^3+9 x^4-3 x^5\right )\right )+\left (-48 x^2+12 x^3\right ) \log (-4+x)}{-256+64 x+e^{\frac {1}{2} \left (4 x+e^{-5+x} x-x^2\right )} \left (-16 x^2+4 x^3\right )+\left (-128 x+32 x^2\right ) \log (-4+x)+\left (-16 x^2+4 x^3\right ) \log ^2(-4+x)+e^{\frac {1}{4} \left (4 x+e^{-5+x} x-x^2\right )} \left (-128 x+32 x^2+\left (-32 x^2+8 x^3\right ) \log (-4+x)\right )} \, dx=\frac {3 e^{\frac {x^2}{4}} x^2}{4 e^{\frac {x^2}{4}}+e^{x+\frac {1}{4} e^{-5+x} x} x+e^{\frac {x^2}{4}} x \log (-4+x)} \] Input:
Integrate[(-384*x + 96*x^2 - 12*x^3 + E^((4*x + E^(-5 + x)*x - x^2)/4)*(-4 8*x^2 + 60*x^3 - 36*x^4 + 6*x^5 + E^(-5 + x)*(12*x^3 + 9*x^4 - 3*x^5)) + ( -48*x^2 + 12*x^3)*Log[-4 + x])/(-256 + 64*x + E^((4*x + E^(-5 + x)*x - x^2 )/2)*(-16*x^2 + 4*x^3) + (-128*x + 32*x^2)*Log[-4 + x] + (-16*x^2 + 4*x^3) *Log[-4 + x]^2 + E^((4*x + E^(-5 + x)*x - x^2)/4)*(-128*x + 32*x^2 + (-32* x^2 + 8*x^3)*Log[-4 + x])),x]
Output:
(3*E^(x^2/4)*x^2)/(4*E^(x^2/4) + E^(x + (E^(-5 + x)*x)/4)*x + E^(x^2/4)*x* Log[-4 + 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.
| \(\displaystyle \int \frac {-12 x^3+96 x^2+\left (12 x^3-48 x^2\right ) \log (x-4)+e^{\frac {1}{4} \left (-x^2+e^{x-5} x+4 x\right )} \left (6 x^5-36 x^4+60 x^3-48 x^2+e^{x-5} \left (-3 x^5+9 x^4+12 x^3\right )\right )-384 x}{\left (32 x^2-128 x\right ) \log (x-4)+e^{\frac {1}{2} \left (-x^2+e^{x-5} x+4 x\right )} \left (4 x^3-16 x^2\right )+\left (4 x^3-16 x^2\right ) \log ^2(x-4)+e^{\frac {1}{4} \left (-x^2+e^{x-5} x+4 x\right )} \left (32 x^2+\left (8 x^3-32 x^2\right ) \log (x-4)-128 x\right )+64 x-256} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{\frac {x^2}{2}} \left (12 x^3-96 x^2-\left (12 x^3-48 x^2\right ) \log (x-4)-e^{\frac {1}{4} \left (-x^2+e^{x-5} x+4 x\right )} \left (6 x^5-36 x^4+60 x^3-48 x^2+e^{x-5} \left (-3 x^5+9 x^4+12 x^3\right )\right )+384 x\right )}{4 (4-x) \left (4 e^{\frac {x^2}{4}}+e^{\frac {x^2}{4}} x \log (x-4)+e^{\frac {1}{4} e^{x-5} x+x} x\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \frac {3 e^{\frac {x^2}{2}} \left (4 x^3-32 x^2+128 x+e^{\frac {1}{4} \left (-x^2+e^{x-5} x+4 x\right )} \left (-2 x^5+12 x^4-20 x^3+16 x^2-e^{x-5} \left (-x^5+3 x^4+4 x^3\right )\right )+4 \left (4 x^2-x^3\right ) \log (x-4)\right )}{(4-x) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{4} \int \frac {e^{\frac {x^2}{2}} \left (4 x^3-32 x^2+128 x+e^{\frac {1}{4} \left (-x^2+e^{x-5} x+4 x\right )} \left (-2 x^5+12 x^4-20 x^3+16 x^2-e^{x-5} \left (-x^5+3 x^4+4 x^3\right )\right )+4 \left (4 x^2-x^3\right ) \log (x-4)\right )}{(4-x) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{4} \int \left (-e^{\frac {x^2}{4}-\frac {1}{4} e^{x-5} x-x-5} \left (e^x x^2-2 e^5 x^2+e^x x+4 e^5 x-4 e^5\right )+\frac {e^{\frac {x^2}{2}-\frac {1}{4} e^{x-5} x-x-5} (x \log (x-4)+4) \left (e^x x^2-2 e^5 x^2+e^x x+4 e^5 x-4 e^5\right )}{e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}}-\frac {e^{\frac {x^2}{2}-5} x \left (-e^x \log (x-4) x^4+2 e^5 \log (x-4) x^4-4 e^x x^3+3 e^x \log (x-4) x^3-12 e^5 \log (x-4) x^3+8 e^5 x^3+12 e^x x^2+4 e^x \log (x-4) x^2+16 e^5 \log (x-4) x^2-44 e^5 x^2+16 e^x x+48 e^5 x+64 e^5\right )}{(x-4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{4} \int \frac {e^{\frac {x^2}{4}-5} x \left (e^{\frac {1}{4} \left (8+e^{x-5}\right ) x} \left (x^2-3 x-4\right ) x^2-2 e^{\frac {1}{4} e^{x-5} x+x+5} \left (x^3-6 x^2+10 x-8\right ) x-4 e^{\frac {x^2}{4}+5} (x-4) \log (x-4) x+4 e^{\frac {x^2}{4}+5} \left (x^2-8 x+32\right )\right )}{(4-x) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{4} \int \left (\frac {4 e^{\frac {x^2}{4}} x \left (\log (x-4) x^2-x^2-4 \log (x-4) x+8 x-32\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )}-\frac {e^{\frac {x^2}{4}+\frac {1}{4} e^{x-5} x+x-5} x^2 \left (e^x \log (x-4) x^4-2 e^5 \log (x-4) x^4+4 e^x x^3-3 e^x \log (x-4) x^3+12 e^5 \log (x-4) x^3-8 e^5 x^3-12 e^x x^2-4 e^x \log (x-4) x^2-16 e^5 \log (x-4) x^2+44 e^5 x^2-16 e^x x-48 e^5 x-64 e^5\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{4} \int \frac {e^{\frac {x^2}{4}-5} x \left (e^{\frac {1}{4} \left (8+e^{x-5}\right ) x} \left (x^2-3 x-4\right ) x^2-2 e^{\frac {1}{4} e^{x-5} x+x+5} \left (x^3-6 x^2+10 x-8\right ) x-4 e^{\frac {x^2}{4}+5} (x-4) \log (x-4) x+4 e^{\frac {x^2}{4}+5} \left (x^2-8 x+32\right )\right )}{(4-x) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{4} \int \left (\frac {4 e^{\frac {x^2}{4}} x \left (\log (x-4) x^2-x^2-4 \log (x-4) x+8 x-32\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )}-\frac {e^{\frac {x^2}{4}+\frac {1}{4} e^{x-5} x+x-5} x^2 \left (e^x \log (x-4) x^4-2 e^5 \log (x-4) x^4+4 e^x x^3-3 e^x \log (x-4) x^3+12 e^5 \log (x-4) x^3-8 e^5 x^3-12 e^x x^2-4 e^x \log (x-4) x^2-16 e^5 \log (x-4) x^2+44 e^5 x^2-16 e^x x-48 e^5 x-64 e^5\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{4} \int \frac {e^{\frac {x^2}{4}-5} x \left (e^{\frac {1}{4} \left (8+e^{x-5}\right ) x} \left (x^2-3 x-4\right ) x^2-2 e^{\frac {1}{4} e^{x-5} x+x+5} \left (x^3-6 x^2+10 x-8\right ) x-4 e^{\frac {x^2}{4}+5} (x-4) \log (x-4) x+4 e^{\frac {x^2}{4}+5} \left (x^2-8 x+32\right )\right )}{(4-x) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{4} \int \left (\frac {4 e^{\frac {x^2}{4}} x \left (\log (x-4) x^2-x^2-4 \log (x-4) x+8 x-32\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )}-\frac {e^{\frac {x^2}{4}+\frac {1}{4} e^{x-5} x+x-5} x^2 \left (e^x \log (x-4) x^4-2 e^5 \log (x-4) x^4+4 e^x x^3-3 e^x \log (x-4) x^3+12 e^5 \log (x-4) x^3-8 e^5 x^3-12 e^x x^2-4 e^x \log (x-4) x^2-16 e^5 \log (x-4) x^2+44 e^5 x^2-16 e^x x-48 e^5 x-64 e^5\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{4} \int \frac {e^{\frac {x^2}{4}-5} x \left (e^{\frac {1}{4} \left (8+e^{x-5}\right ) x} \left (x^2-3 x-4\right ) x^2-2 e^{\frac {1}{4} e^{x-5} x+x+5} \left (x^3-6 x^2+10 x-8\right ) x-4 e^{\frac {x^2}{4}+5} (x-4) \log (x-4) x+4 e^{\frac {x^2}{4}+5} \left (x^2-8 x+32\right )\right )}{(4-x) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{4} \int \left (\frac {4 e^{\frac {x^2}{4}} x \left (\log (x-4) x^2-x^2-4 \log (x-4) x+8 x-32\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )}-\frac {e^{\frac {x^2}{4}+\frac {1}{4} e^{x-5} x+x-5} x^2 \left (e^x \log (x-4) x^4-2 e^5 \log (x-4) x^4+4 e^x x^3-3 e^x \log (x-4) x^3+12 e^5 \log (x-4) x^3-8 e^5 x^3-12 e^x x^2-4 e^x \log (x-4) x^2-16 e^5 \log (x-4) x^2+44 e^5 x^2-16 e^x x-48 e^5 x-64 e^5\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{4} \int \frac {e^{\frac {x^2}{4}-5} x \left (e^{\frac {1}{4} \left (8+e^{x-5}\right ) x} \left (x^2-3 x-4\right ) x^2-2 e^{\frac {1}{4} e^{x-5} x+x+5} \left (x^3-6 x^2+10 x-8\right ) x-4 e^{\frac {x^2}{4}+5} (x-4) \log (x-4) x+4 e^{\frac {x^2}{4}+5} \left (x^2-8 x+32\right )\right )}{(4-x) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{4} \int \left (\frac {4 e^{\frac {x^2}{4}} x \left (\log (x-4) x^2-x^2-4 \log (x-4) x+8 x-32\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )}-\frac {e^{\frac {x^2}{4}+\frac {1}{4} e^{x-5} x+x-5} x^2 \left (e^x \log (x-4) x^4-2 e^5 \log (x-4) x^4+4 e^x x^3-3 e^x \log (x-4) x^3+12 e^5 \log (x-4) x^3-8 e^5 x^3-12 e^x x^2-4 e^x \log (x-4) x^2-16 e^5 \log (x-4) x^2+44 e^5 x^2-16 e^x x-48 e^5 x-64 e^5\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{4} \int \frac {e^{\frac {x^2}{4}-5} x \left (e^{\frac {1}{4} \left (8+e^{x-5}\right ) x} \left (x^2-3 x-4\right ) x^2-2 e^{\frac {1}{4} e^{x-5} x+x+5} \left (x^3-6 x^2+10 x-8\right ) x-4 e^{\frac {x^2}{4}+5} (x-4) \log (x-4) x+4 e^{\frac {x^2}{4}+5} \left (x^2-8 x+32\right )\right )}{(4-x) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{4} \int \left (\frac {4 e^{\frac {x^2}{4}} x \left (\log (x-4) x^2-x^2-4 \log (x-4) x+8 x-32\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )}-\frac {e^{\frac {x^2}{4}+\frac {1}{4} e^{x-5} x+x-5} x^2 \left (e^x \log (x-4) x^4-2 e^5 \log (x-4) x^4+4 e^x x^3-3 e^x \log (x-4) x^3+12 e^5 \log (x-4) x^3-8 e^5 x^3-12 e^x x^2-4 e^x \log (x-4) x^2-16 e^5 \log (x-4) x^2+44 e^5 x^2-16 e^x x-48 e^5 x-64 e^5\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{4} \int \frac {e^{\frac {x^2}{4}-5} x \left (e^{\frac {1}{4} \left (8+e^{x-5}\right ) x} \left (x^2-3 x-4\right ) x^2-2 e^{\frac {1}{4} e^{x-5} x+x+5} \left (x^3-6 x^2+10 x-8\right ) x-4 e^{\frac {x^2}{4}+5} (x-4) \log (x-4) x+4 e^{\frac {x^2}{4}+5} \left (x^2-8 x+32\right )\right )}{(4-x) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{4} \int \left (\frac {4 e^{\frac {x^2}{4}} x \left (\log (x-4) x^2-x^2-4 \log (x-4) x+8 x-32\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )}-\frac {e^{\frac {x^2}{4}+\frac {1}{4} e^{x-5} x+x-5} x^2 \left (e^x \log (x-4) x^4-2 e^5 \log (x-4) x^4+4 e^x x^3-3 e^x \log (x-4) x^3+12 e^5 \log (x-4) x^3-8 e^5 x^3-12 e^x x^2-4 e^x \log (x-4) x^2-16 e^5 \log (x-4) x^2+44 e^5 x^2-16 e^x x-48 e^5 x-64 e^5\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{4} \int \frac {e^{\frac {x^2}{4}-5} x \left (e^{\frac {1}{4} \left (8+e^{x-5}\right ) x} \left (x^2-3 x-4\right ) x^2-2 e^{\frac {1}{4} e^{x-5} x+x+5} \left (x^3-6 x^2+10 x-8\right ) x-4 e^{\frac {x^2}{4}+5} (x-4) \log (x-4) x+4 e^{\frac {x^2}{4}+5} \left (x^2-8 x+32\right )\right )}{(4-x) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{4} \int \left (\frac {4 e^{\frac {x^2}{4}} x \left (\log (x-4) x^2-x^2-4 \log (x-4) x+8 x-32\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )}-\frac {e^{\frac {x^2}{4}+\frac {1}{4} e^{x-5} x+x-5} x^2 \left (e^x \log (x-4) x^4-2 e^5 \log (x-4) x^4+4 e^x x^3-3 e^x \log (x-4) x^3+12 e^5 \log (x-4) x^3-8 e^5 x^3-12 e^x x^2-4 e^x \log (x-4) x^2-16 e^5 \log (x-4) x^2+44 e^5 x^2-16 e^x x-48 e^5 x-64 e^5\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{4} \int \frac {e^{\frac {x^2}{4}-5} x \left (e^{\frac {1}{4} \left (8+e^{x-5}\right ) x} \left (x^2-3 x-4\right ) x^2-2 e^{\frac {1}{4} e^{x-5} x+x+5} \left (x^3-6 x^2+10 x-8\right ) x-4 e^{\frac {x^2}{4}+5} (x-4) \log (x-4) x+4 e^{\frac {x^2}{4}+5} \left (x^2-8 x+32\right )\right )}{(4-x) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{4} \int \left (\frac {4 e^{\frac {x^2}{4}} x \left (\log (x-4) x^2-x^2-4 \log (x-4) x+8 x-32\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )}-\frac {e^{\frac {x^2}{4}+\frac {1}{4} e^{x-5} x+x-5} x^2 \left (e^x \log (x-4) x^4-2 e^5 \log (x-4) x^4+4 e^x x^3-3 e^x \log (x-4) x^3+12 e^5 \log (x-4) x^3-8 e^5 x^3-12 e^x x^2-4 e^x \log (x-4) x^2-16 e^5 \log (x-4) x^2+44 e^5 x^2-16 e^x x-48 e^5 x-64 e^5\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{4} \int \frac {e^{\frac {x^2}{4}-5} x \left (e^{\frac {1}{4} \left (8+e^{x-5}\right ) x} \left (x^2-3 x-4\right ) x^2-2 e^{\frac {1}{4} e^{x-5} x+x+5} \left (x^3-6 x^2+10 x-8\right ) x-4 e^{\frac {x^2}{4}+5} (x-4) \log (x-4) x+4 e^{\frac {x^2}{4}+5} \left (x^2-8 x+32\right )\right )}{(4-x) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{4} \int \left (\frac {4 e^{\frac {x^2}{4}} x \left (\log (x-4) x^2-x^2-4 \log (x-4) x+8 x-32\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )}-\frac {e^{\frac {x^2}{4}+\frac {1}{4} e^{x-5} x+x-5} x^2 \left (e^x \log (x-4) x^4-2 e^5 \log (x-4) x^4+4 e^x x^3-3 e^x \log (x-4) x^3+12 e^5 \log (x-4) x^3-8 e^5 x^3-12 e^x x^2-4 e^x \log (x-4) x^2-16 e^5 \log (x-4) x^2+44 e^5 x^2-16 e^x x-48 e^5 x-64 e^5\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{4} \int \frac {e^{\frac {x^2}{4}-5} x \left (e^{\frac {1}{4} \left (8+e^{x-5}\right ) x} \left (x^2-3 x-4\right ) x^2-2 e^{\frac {1}{4} e^{x-5} x+x+5} \left (x^3-6 x^2+10 x-8\right ) x-4 e^{\frac {x^2}{4}+5} (x-4) \log (x-4) x+4 e^{\frac {x^2}{4}+5} \left (x^2-8 x+32\right )\right )}{(4-x) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{4} \int \left (\frac {4 e^{\frac {x^2}{4}} x \left (\log (x-4) x^2-x^2-4 \log (x-4) x+8 x-32\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )}-\frac {e^{\frac {x^2}{4}+\frac {1}{4} e^{x-5} x+x-5} x^2 \left (e^x \log (x-4) x^4-2 e^5 \log (x-4) x^4+4 e^x x^3-3 e^x \log (x-4) x^3+12 e^5 \log (x-4) x^3-8 e^5 x^3-12 e^x x^2-4 e^x \log (x-4) x^2-16 e^5 \log (x-4) x^2+44 e^5 x^2-16 e^x x-48 e^5 x-64 e^5\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{4} \int \frac {e^{\frac {x^2}{4}-5} x \left (e^{\frac {1}{4} \left (8+e^{x-5}\right ) x} \left (x^2-3 x-4\right ) x^2-2 e^{\frac {1}{4} e^{x-5} x+x+5} \left (x^3-6 x^2+10 x-8\right ) x-4 e^{\frac {x^2}{4}+5} (x-4) \log (x-4) x+4 e^{\frac {x^2}{4}+5} \left (x^2-8 x+32\right )\right )}{(4-x) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{4} \int \left (\frac {4 e^{\frac {x^2}{4}} x \left (\log (x-4) x^2-x^2-4 \log (x-4) x+8 x-32\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )}-\frac {e^{\frac {x^2}{4}+\frac {1}{4} e^{x-5} x+x-5} x^2 \left (e^x \log (x-4) x^4-2 e^5 \log (x-4) x^4+4 e^x x^3-3 e^x \log (x-4) x^3+12 e^5 \log (x-4) x^3-8 e^5 x^3-12 e^x x^2-4 e^x \log (x-4) x^2-16 e^5 \log (x-4) x^2+44 e^5 x^2-16 e^x x-48 e^5 x-64 e^5\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{4} \int \frac {e^{\frac {x^2}{4}-5} x \left (e^{\frac {1}{4} \left (8+e^{x-5}\right ) x} \left (x^2-3 x-4\right ) x^2-2 e^{\frac {1}{4} e^{x-5} x+x+5} \left (x^3-6 x^2+10 x-8\right ) x-4 e^{\frac {x^2}{4}+5} (x-4) \log (x-4) x+4 e^{\frac {x^2}{4}+5} \left (x^2-8 x+32\right )\right )}{(4-x) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{4} \int \left (\frac {4 e^{\frac {x^2}{4}} x \left (\log (x-4) x^2-x^2-4 \log (x-4) x+8 x-32\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )}-\frac {e^{\frac {x^2}{4}+\frac {1}{4} e^{x-5} x+x-5} x^2 \left (e^x \log (x-4) x^4-2 e^5 \log (x-4) x^4+4 e^x x^3-3 e^x \log (x-4) x^3+12 e^5 \log (x-4) x^3-8 e^5 x^3-12 e^x x^2-4 e^x \log (x-4) x^2-16 e^5 \log (x-4) x^2+44 e^5 x^2-16 e^x x-48 e^5 x-64 e^5\right )}{(x-4) (x \log (x-4)+4) \left (e^{\frac {1}{4} e^{x-5} x+x} x+e^{\frac {x^2}{4}} \log (x-4) x+4 e^{\frac {x^2}{4}}\right )^2}\right )dx\) |
Input:
Int[(-384*x + 96*x^2 - 12*x^3 + E^((4*x + E^(-5 + x)*x - x^2)/4)*(-48*x^2 + 60*x^3 - 36*x^4 + 6*x^5 + E^(-5 + x)*(12*x^3 + 9*x^4 - 3*x^5)) + (-48*x^ 2 + 12*x^3)*Log[-4 + x])/(-256 + 64*x + E^((4*x + E^(-5 + x)*x - x^2)/2)*( -16*x^2 + 4*x^3) + (-128*x + 32*x^2)*Log[-4 + x] + (-16*x^2 + 4*x^3)*Log[- 4 + x]^2 + E^((4*x + E^(-5 + x)*x - x^2)/4)*(-128*x + 32*x^2 + (-32*x^2 + 8*x^3)*Log[-4 + x])),x]
Output:
$Aborted
Time = 15.49 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91
| method | result | size |
| risch | \(\frac {3 x^{2}}{{\mathrm e}^{-\frac {\left (-4-{\mathrm e}^{-5+x}+x \right ) x}{4}} x +x \ln \left (x -4\right )+4}\) | \(31\) |
| parallelrisch | \(\frac {3 x^{2}}{x \ln \left (x -4\right )+{\mathrm e}^{\frac {\left (4+{\mathrm e}^{-5+x}-x \right ) x}{4}} x +4}\) | \(31\) |
Input:
int((((-3*x^5+9*x^4+12*x^3)*exp(-5+x)+6*x^5-36*x^4+60*x^3-48*x^2)*exp(1/4* x*exp(-5+x)-1/4*x^2+x)+(12*x^3-48*x^2)*ln(x-4)-12*x^3+96*x^2-384*x)/((4*x^ 3-16*x^2)*exp(1/4*x*exp(-5+x)-1/4*x^2+x)^2+((8*x^3-32*x^2)*ln(x-4)+32*x^2- 128*x)*exp(1/4*x*exp(-5+x)-1/4*x^2+x)+(4*x^3-16*x^2)*ln(x-4)^2+(32*x^2-128 *x)*ln(x-4)+64*x-256),x,method=_RETURNVERBOSE)
Output:
3*x^2/(exp(-1/4*(-4-exp(-5+x)+x)*x)*x+x*ln(x-4)+4)
Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {-384 x+96 x^2-12 x^3+e^{\frac {1}{4} \left (4 x+e^{-5+x} x-x^2\right )} \left (-48 x^2+60 x^3-36 x^4+6 x^5+e^{-5+x} \left (12 x^3+9 x^4-3 x^5\right )\right )+\left (-48 x^2+12 x^3\right ) \log (-4+x)}{-256+64 x+e^{\frac {1}{2} \left (4 x+e^{-5+x} x-x^2\right )} \left (-16 x^2+4 x^3\right )+\left (-128 x+32 x^2\right ) \log (-4+x)+\left (-16 x^2+4 x^3\right ) \log ^2(-4+x)+e^{\frac {1}{4} \left (4 x+e^{-5+x} x-x^2\right )} \left (-128 x+32 x^2+\left (-32 x^2+8 x^3\right ) \log (-4+x)\right )} \, dx=\frac {3 \, x^{2}}{x e^{\left (-\frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{\left (x - 5\right )} + x\right )} + x \log \left (x - 4\right ) + 4} \] Input:
integrate((((-3*x^5+9*x^4+12*x^3)*exp(-5+x)+6*x^5-36*x^4+60*x^3-48*x^2)*ex p(1/4*x*exp(-5+x)-1/4*x^2+x)+(12*x^3-48*x^2)*log(-4+x)-12*x^3+96*x^2-384*x )/((4*x^3-16*x^2)*exp(1/4*x*exp(-5+x)-1/4*x^2+x)^2+((8*x^3-32*x^2)*log(-4+ x)+32*x^2-128*x)*exp(1/4*x*exp(-5+x)-1/4*x^2+x)+(4*x^3-16*x^2)*log(-4+x)^2 +(32*x^2-128*x)*log(-4+x)+64*x-256),x, algorithm="fricas")
Output:
3*x^2/(x*e^(-1/4*x^2 + 1/4*x*e^(x - 5) + x) + x*log(x - 4) + 4)
Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {-384 x+96 x^2-12 x^3+e^{\frac {1}{4} \left (4 x+e^{-5+x} x-x^2\right )} \left (-48 x^2+60 x^3-36 x^4+6 x^5+e^{-5+x} \left (12 x^3+9 x^4-3 x^5\right )\right )+\left (-48 x^2+12 x^3\right ) \log (-4+x)}{-256+64 x+e^{\frac {1}{2} \left (4 x+e^{-5+x} x-x^2\right )} \left (-16 x^2+4 x^3\right )+\left (-128 x+32 x^2\right ) \log (-4+x)+\left (-16 x^2+4 x^3\right ) \log ^2(-4+x)+e^{\frac {1}{4} \left (4 x+e^{-5+x} x-x^2\right )} \left (-128 x+32 x^2+\left (-32 x^2+8 x^3\right ) \log (-4+x)\right )} \, dx=\frac {3 x^{2}}{x e^{- \frac {x^{2}}{4} + \frac {x e^{x - 5}}{4} + x} + x \log {\left (x - 4 \right )} + 4} \] Input:
integrate((((-3*x**5+9*x**4+12*x**3)*exp(-5+x)+6*x**5-36*x**4+60*x**3-48*x **2)*exp(1/4*x*exp(-5+x)-1/4*x**2+x)+(12*x**3-48*x**2)*ln(-4+x)-12*x**3+96 *x**2-384*x)/((4*x**3-16*x**2)*exp(1/4*x*exp(-5+x)-1/4*x**2+x)**2+((8*x**3 -32*x**2)*ln(-4+x)+32*x**2-128*x)*exp(1/4*x*exp(-5+x)-1/4*x**2+x)+(4*x**3- 16*x**2)*ln(-4+x)**2+(32*x**2-128*x)*ln(-4+x)+64*x-256),x)
Output:
3*x**2/(x*exp(-x**2/4 + x*exp(x - 5)/4 + x) + x*log(x - 4) + 4)
\[ \int \frac {-384 x+96 x^2-12 x^3+e^{\frac {1}{4} \left (4 x+e^{-5+x} x-x^2\right )} \left (-48 x^2+60 x^3-36 x^4+6 x^5+e^{-5+x} \left (12 x^3+9 x^4-3 x^5\right )\right )+\left (-48 x^2+12 x^3\right ) \log (-4+x)}{-256+64 x+e^{\frac {1}{2} \left (4 x+e^{-5+x} x-x^2\right )} \left (-16 x^2+4 x^3\right )+\left (-128 x+32 x^2\right ) \log (-4+x)+\left (-16 x^2+4 x^3\right ) \log ^2(-4+x)+e^{\frac {1}{4} \left (4 x+e^{-5+x} x-x^2\right )} \left (-128 x+32 x^2+\left (-32 x^2+8 x^3\right ) \log (-4+x)\right )} \, dx=\int { -\frac {3 \, {\left (4 \, x^{3} - 32 \, x^{2} - {\left (2 \, x^{5} - 12 \, x^{4} + 20 \, x^{3} - 16 \, x^{2} - {\left (x^{5} - 3 \, x^{4} - 4 \, x^{3}\right )} e^{\left (x - 5\right )}\right )} e^{\left (-\frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{\left (x - 5\right )} + x\right )} - 4 \, {\left (x^{3} - 4 \, x^{2}\right )} \log \left (x - 4\right ) + 128 \, x\right )}}{4 \, {\left ({\left (x^{3} - 4 \, x^{2}\right )} \log \left (x - 4\right )^{2} + 2 \, {\left (4 \, x^{2} + {\left (x^{3} - 4 \, x^{2}\right )} \log \left (x - 4\right ) - 16 \, x\right )} e^{\left (-\frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{\left (x - 5\right )} + x\right )} + {\left (x^{3} - 4 \, x^{2}\right )} e^{\left (-\frac {1}{2} \, x^{2} + \frac {1}{2} \, x e^{\left (x - 5\right )} + 2 \, x\right )} + 8 \, {\left (x^{2} - 4 \, x\right )} \log \left (x - 4\right ) + 16 \, x - 64\right )}} \,d x } \] Input:
integrate((((-3*x^5+9*x^4+12*x^3)*exp(-5+x)+6*x^5-36*x^4+60*x^3-48*x^2)*ex p(1/4*x*exp(-5+x)-1/4*x^2+x)+(12*x^3-48*x^2)*log(-4+x)-12*x^3+96*x^2-384*x )/((4*x^3-16*x^2)*exp(1/4*x*exp(-5+x)-1/4*x^2+x)^2+((8*x^3-32*x^2)*log(-4+ x)+32*x^2-128*x)*exp(1/4*x*exp(-5+x)-1/4*x^2+x)+(4*x^3-16*x^2)*log(-4+x)^2 +(32*x^2-128*x)*log(-4+x)+64*x-256),x, algorithm="maxima")
Output:
-3/4*integrate((4*x^3 - 32*x^2 - (2*x^5 - 12*x^4 + 20*x^3 - 16*x^2 - (x^5 - 3*x^4 - 4*x^3)*e^(x - 5))*e^(-1/4*x^2 + 1/4*x*e^(x - 5) + x) - 4*(x^3 - 4*x^2)*log(x - 4) + 128*x)/((x^3 - 4*x^2)*log(x - 4)^2 + 2*(4*x^2 + (x^3 - 4*x^2)*log(x - 4) - 16*x)*e^(-1/4*x^2 + 1/4*x*e^(x - 5) + x) + (x^3 - 4*x ^2)*e^(-1/2*x^2 + 1/2*x*e^(x - 5) + 2*x) + 8*(x^2 - 4*x)*log(x - 4) + 16*x - 64), x)
Timed out. \[ \int \frac {-384 x+96 x^2-12 x^3+e^{\frac {1}{4} \left (4 x+e^{-5+x} x-x^2\right )} \left (-48 x^2+60 x^3-36 x^4+6 x^5+e^{-5+x} \left (12 x^3+9 x^4-3 x^5\right )\right )+\left (-48 x^2+12 x^3\right ) \log (-4+x)}{-256+64 x+e^{\frac {1}{2} \left (4 x+e^{-5+x} x-x^2\right )} \left (-16 x^2+4 x^3\right )+\left (-128 x+32 x^2\right ) \log (-4+x)+\left (-16 x^2+4 x^3\right ) \log ^2(-4+x)+e^{\frac {1}{4} \left (4 x+e^{-5+x} x-x^2\right )} \left (-128 x+32 x^2+\left (-32 x^2+8 x^3\right ) \log (-4+x)\right )} \, dx=\text {Timed out} \] Input:
integrate((((-3*x^5+9*x^4+12*x^3)*exp(-5+x)+6*x^5-36*x^4+60*x^3-48*x^2)*ex p(1/4*x*exp(-5+x)-1/4*x^2+x)+(12*x^3-48*x^2)*log(-4+x)-12*x^3+96*x^2-384*x )/((4*x^3-16*x^2)*exp(1/4*x*exp(-5+x)-1/4*x^2+x)^2+((8*x^3-32*x^2)*log(-4+ x)+32*x^2-128*x)*exp(1/4*x*exp(-5+x)-1/4*x^2+x)+(4*x^3-16*x^2)*log(-4+x)^2 +(32*x^2-128*x)*log(-4+x)+64*x-256),x, algorithm="giac")
Output:
Timed out
Time = 3.28 (sec) , antiderivative size = 295, normalized size of antiderivative = 8.68 \[ \int \frac {-384 x+96 x^2-12 x^3+e^{\frac {1}{4} \left (4 x+e^{-5+x} x-x^2\right )} \left (-48 x^2+60 x^3-36 x^4+6 x^5+e^{-5+x} \left (12 x^3+9 x^4-3 x^5\right )\right )+\left (-48 x^2+12 x^3\right ) \log (-4+x)}{-256+64 x+e^{\frac {1}{2} \left (4 x+e^{-5+x} x-x^2\right )} \left (-16 x^2+4 x^3\right )+\left (-128 x+32 x^2\right ) \log (-4+x)+\left (-16 x^2+4 x^3\right ) \log ^2(-4+x)+e^{\frac {1}{4} \left (4 x+e^{-5+x} x-x^2\right )} \left (-128 x+32 x^2+\left (-32 x^2+8 x^3\right ) \log (-4+x)\right )} \, dx=\frac {192\,x^3\,{\mathrm {e}}^{x-5}+96\,x^4\,{\mathrm {e}}^{x-5}-84\,x^5\,{\mathrm {e}}^{x-5}+12\,x^6\,{\mathrm {e}}^{x-5}+\ln \left (x-4\right )\,\left (48\,x^4\,{\mathrm {e}}^{x-5}+24\,x^5\,{\mathrm {e}}^{x-5}-21\,x^6\,{\mathrm {e}}^{x-5}+3\,x^7\,{\mathrm {e}}^{x-5}+192\,x^4-192\,x^5+60\,x^6-6\,x^7\right )+768\,x^2+384\,x^3-672\,x^4+228\,x^5-24\,x^6}{\left (x\,\ln \left (x-4\right )+x\,{\mathrm {e}}^{x-\frac {x^2}{4}+\frac {x\,{\mathrm {e}}^{-5}\,{\mathrm {e}}^x}{4}}+4\right )\,\left (128\,x+64\,x\,{\mathrm {e}}^{x-5}+32\,x^2\,{\mathrm {e}}^{x-5}-28\,x^3\,{\mathrm {e}}^{x-5}+4\,x^4\,{\mathrm {e}}^{x-5}+64\,x^2\,\ln \left (x-4\right )-64\,x^3\,\ln \left (x-4\right )+20\,x^4\,\ln \left (x-4\right )-2\,x^5\,\ln \left (x-4\right )-224\,x^2+76\,x^3-8\,x^4+16\,x^2\,\ln \left (x-4\right )\,{\mathrm {e}}^{x-5}+8\,x^3\,\ln \left (x-4\right )\,{\mathrm {e}}^{x-5}-7\,x^4\,\ln \left (x-4\right )\,{\mathrm {e}}^{x-5}+x^5\,\ln \left (x-4\right )\,{\mathrm {e}}^{x-5}+256\right )} \] Input:
int((384*x + log(x - 4)*(48*x^2 - 12*x^3) - exp(x + (x*exp(x - 5))/4 - x^2 /4)*(exp(x - 5)*(12*x^3 + 9*x^4 - 3*x^5) - 48*x^2 + 60*x^3 - 36*x^4 + 6*x^ 5) - 96*x^2 + 12*x^3)/(log(x - 4)*(128*x - 32*x^2) - 64*x + exp(2*x + (x*e xp(x - 5))/2 - x^2/2)*(16*x^2 - 4*x^3) + exp(x + (x*exp(x - 5))/4 - x^2/4) *(128*x + log(x - 4)*(32*x^2 - 8*x^3) - 32*x^2) + log(x - 4)^2*(16*x^2 - 4 *x^3) + 256),x)
Output:
(192*x^3*exp(x - 5) + 96*x^4*exp(x - 5) - 84*x^5*exp(x - 5) + 12*x^6*exp(x - 5) + log(x - 4)*(48*x^4*exp(x - 5) + 24*x^5*exp(x - 5) - 21*x^6*exp(x - 5) + 3*x^7*exp(x - 5) + 192*x^4 - 192*x^5 + 60*x^6 - 6*x^7) + 768*x^2 + 3 84*x^3 - 672*x^4 + 228*x^5 - 24*x^6)/((x*log(x - 4) + x*exp(x - x^2/4 + (x *exp(-5)*exp(x))/4) + 4)*(128*x + 64*x*exp(x - 5) + 32*x^2*exp(x - 5) - 28 *x^3*exp(x - 5) + 4*x^4*exp(x - 5) + 64*x^2*log(x - 4) - 64*x^3*log(x - 4) + 20*x^4*log(x - 4) - 2*x^5*log(x - 4) - 224*x^2 + 76*x^3 - 8*x^4 + 16*x^ 2*log(x - 4)*exp(x - 5) + 8*x^3*log(x - 4)*exp(x - 5) - 7*x^4*log(x - 4)*e xp(x - 5) + x^5*log(x - 4)*exp(x - 5) + 256))
Time = 0.18 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.71 \[ \int \frac {-384 x+96 x^2-12 x^3+e^{\frac {1}{4} \left (4 x+e^{-5+x} x-x^2\right )} \left (-48 x^2+60 x^3-36 x^4+6 x^5+e^{-5+x} \left (12 x^3+9 x^4-3 x^5\right )\right )+\left (-48 x^2+12 x^3\right ) \log (-4+x)}{-256+64 x+e^{\frac {1}{2} \left (4 x+e^{-5+x} x-x^2\right )} \left (-16 x^2+4 x^3\right )+\left (-128 x+32 x^2\right ) \log (-4+x)+\left (-16 x^2+4 x^3\right ) \log ^2(-4+x)+e^{\frac {1}{4} \left (4 x+e^{-5+x} x-x^2\right )} \left (-128 x+32 x^2+\left (-32 x^2+8 x^3\right ) \log (-4+x)\right )} \, dx=\frac {3 e^{\frac {x^{2}}{4}} x^{2}}{e^{\frac {x^{2}}{4}} \mathrm {log}\left (x -4\right ) x +4 e^{\frac {x^{2}}{4}}+e^{\frac {e^{x} x +4 e^{5} x}{4 e^{5}}} x} \] Input:
int((((-3*x^5+9*x^4+12*x^3)*exp(-5+x)+6*x^5-36*x^4+60*x^3-48*x^2)*exp(1/4* x*exp(-5+x)-1/4*x^2+x)+(12*x^3-48*x^2)*log(-4+x)-12*x^3+96*x^2-384*x)/((4* x^3-16*x^2)*exp(1/4*x*exp(-5+x)-1/4*x^2+x)^2+((8*x^3-32*x^2)*log(-4+x)+32* x^2-128*x)*exp(1/4*x*exp(-5+x)-1/4*x^2+x)+(4*x^3-16*x^2)*log(-4+x)^2+(32*x ^2-128*x)*log(-4+x)+64*x-256),x)
Output:
(3*e**(x**2/4)*x**2)/(e**(x**2/4)*log(x - 4)*x + 4*e**(x**2/4) + e**((e**x *x + 4*e**5*x)/(4*e**5))*x)