Integrand size = 262, antiderivative size = 27 \[ \int \frac {x^{-\frac {25}{-84+8 x+x^2+e^{-4-2 x+2 x^2} x^2+e^{-2-x+x^2} x (8+2 x)}} \left (2100-200 x-25 x^2+\left (200 x+50 x^2\right ) \log (x)+e^{-4-2 x+2 x^2} x^2 \left (-25+\left (50-50 x+100 x^2\right ) \log (x)\right )+e^{-2-x+x^2} x \left (-200-50 x+\left (200-100 x+350 x^2+100 x^3\right ) \log (x)\right )\right )}{7056 x-1344 x^2-104 x^3+16 x^4+x^5+e^{-8-4 x+4 x^2} x^5+e^{-6-3 x+3 x^2} x^3 \left (16 x+4 x^2\right )+e^{-4-2 x+2 x^2} x^2 \left (-104 x+48 x^2+6 x^3\right )+e^{-2-x+x^2} x \left (-1344 x-208 x^2+48 x^3+4 x^4\right )} \, dx=x^{\frac {1}{4-\frac {1}{25} \left (4+x+e^{-2-x+x^2} x\right )^2}} \] Output:
exp(ln(x)/(4-1/5*(exp(ln(x)+x^2-x-2)+4+x)*(1/5*exp(ln(x)+x^2-x-2)+4/5+1/5* x)))
\[ \int \frac {x^{-\frac {25}{-84+8 x+x^2+e^{-4-2 x+2 x^2} x^2+e^{-2-x+x^2} x (8+2 x)}} \left (2100-200 x-25 x^2+\left (200 x+50 x^2\right ) \log (x)+e^{-4-2 x+2 x^2} x^2 \left (-25+\left (50-50 x+100 x^2\right ) \log (x)\right )+e^{-2-x+x^2} x \left (-200-50 x+\left (200-100 x+350 x^2+100 x^3\right ) \log (x)\right )\right )}{7056 x-1344 x^2-104 x^3+16 x^4+x^5+e^{-8-4 x+4 x^2} x^5+e^{-6-3 x+3 x^2} x^3 \left (16 x+4 x^2\right )+e^{-4-2 x+2 x^2} x^2 \left (-104 x+48 x^2+6 x^3\right )+e^{-2-x+x^2} x \left (-1344 x-208 x^2+48 x^3+4 x^4\right )} \, dx=\int \frac {x^{-\frac {25}{-84+8 x+x^2+e^{-4-2 x+2 x^2} x^2+e^{-2-x+x^2} x (8+2 x)}} \left (2100-200 x-25 x^2+\left (200 x+50 x^2\right ) \log (x)+e^{-4-2 x+2 x^2} x^2 \left (-25+\left (50-50 x+100 x^2\right ) \log (x)\right )+e^{-2-x+x^2} x \left (-200-50 x+\left (200-100 x+350 x^2+100 x^3\right ) \log (x)\right )\right )}{7056 x-1344 x^2-104 x^3+16 x^4+x^5+e^{-8-4 x+4 x^2} x^5+e^{-6-3 x+3 x^2} x^3 \left (16 x+4 x^2\right )+e^{-4-2 x+2 x^2} x^2 \left (-104 x+48 x^2+6 x^3\right )+e^{-2-x+x^2} x \left (-1344 x-208 x^2+48 x^3+4 x^4\right )} \, dx \] Input:
Integrate[(2100 - 200*x - 25*x^2 + (200*x + 50*x^2)*Log[x] + E^(-4 - 2*x + 2*x^2)*x^2*(-25 + (50 - 50*x + 100*x^2)*Log[x]) + E^(-2 - x + x^2)*x*(-20 0 - 50*x + (200 - 100*x + 350*x^2 + 100*x^3)*Log[x]))/(x^(25/(-84 + 8*x + x^2 + E^(-4 - 2*x + 2*x^2)*x^2 + E^(-2 - x + x^2)*x*(8 + 2*x)))*(7056*x - 1344*x^2 - 104*x^3 + 16*x^4 + x^5 + E^(-8 - 4*x + 4*x^2)*x^5 + E^(-6 - 3*x + 3*x^2)*x^3*(16*x + 4*x^2) + E^(-4 - 2*x + 2*x^2)*x^2*(-104*x + 48*x^2 + 6*x^3) + E^(-2 - x + x^2)*x*(-1344*x - 208*x^2 + 48*x^3 + 4*x^4))),x]
Output:
Integrate[(2100 - 200*x - 25*x^2 + (200*x + 50*x^2)*Log[x] + E^(-4 - 2*x + 2*x^2)*x^2*(-25 + (50 - 50*x + 100*x^2)*Log[x]) + E^(-2 - x + x^2)*x*(-20 0 - 50*x + (200 - 100*x + 350*x^2 + 100*x^3)*Log[x]))/(x^(25/(-84 + 8*x + x^2 + E^(-4 - 2*x + 2*x^2)*x^2 + E^(-2 - x + x^2)*x*(8 + 2*x)))*(7056*x - 1344*x^2 - 104*x^3 + 16*x^4 + x^5 + E^(-8 - 4*x + 4*x^2)*x^5 + E^(-6 - 3*x + 3*x^2)*x^3*(16*x + 4*x^2) + E^(-4 - 2*x + 2*x^2)*x^2*(-104*x + 48*x^2 + 6*x^3) + E^(-2 - x + x^2)*x*(-1344*x - 208*x^2 + 48*x^3 + 4*x^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.
| \(\displaystyle \int \frac {x^{-\frac {25}{e^{2 x^2-2 x-4} x^2+x^2+e^{x^2-x-2} (2 x+8) x+8 x-84}} \left (-25 x^2+e^{2 x^2-2 x-4} x^2 \left (\left (100 x^2-50 x+50\right ) \log (x)-25\right )+\left (50 x^2+200 x\right ) \log (x)+e^{x^2-x-2} x \left (\left (100 x^3+350 x^2-100 x+200\right ) \log (x)-50 x-200\right )-200 x+2100\right )}{x^5+16 x^4-104 x^3-1344 x^2+e^{4 x^2-4 x-8} x^5+e^{3 x^2-3 x-6} \left (4 x^2+16 x\right ) x^3+e^{2 x^2-2 x-4} \left (6 x^3+48 x^2-104 x\right ) x^2+e^{x^2-x-2} \left (4 x^4+48 x^3-208 x^2-1344 x\right ) x+7056 x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{4 x+8} x^{-\frac {25}{e^{2 x^2-2 x-4} x^2+x^2+e^{x^2-x-2} (2 x+8) x+8 x-84}-1} \left (-25 x^2+e^{2 x^2-2 x-4} x^2 \left (\left (100 x^2-50 x+50\right ) \log (x)-25\right )+\left (50 x^2+200 x\right ) \log (x)+e^{x^2-x-2} x \left (\left (100 x^3+350 x^2-100 x+200\right ) \log (x)-50 x-200\right )-200 x+2100\right )}{\left (-e^{2 x^2} x^2-e^{2 x+4} x^2-2 e^{x^2+x+2} x^2-8 e^{x^2+x+2} x-8 e^{2 x+4} x+84 e^{2 x+4}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 e^{x+2} x^{-\frac {25}{e^{2 x^2-2 x-4} x^2+x^2+e^{x^2-x-2} (2 x+8) x+8 x-84}-1} \left (2 x^2 \log (x)-x \log (x)+\log (x)-1\right )}{4 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )}-\frac {5 e^{x+2} x^{-\frac {25}{e^{2 x^2-2 x-4} x^2+x^2+e^{x^2-x-2} (2 x+8) x+8 x-84}-1} \left (2 x^2 \log (x)-x \log (x)+\log (x)-1\right )}{4 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )}-\frac {5 e^{2 x+4} \left (2 x^3-13 x^2+6 x-6\right ) x^{-\frac {25}{e^{2 x^2-2 x-4} x^2+x^2+e^{x^2-x-2} (2 x+8) x+8 x-84}-1} \log (x)}{4 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )^2}+\frac {5 e^{2 x+4} \left (2 x^3+27 x^2-14 x+14\right ) x^{-\frac {25}{e^{2 x^2-2 x-4} x^2+x^2+e^{x^2-x-2} (2 x+8) x+8 x-84}-1} \log (x)}{4 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {25 e^{2 x+4} x^{-\frac {25}{e^{2 x^2-2 x-4} x^2+x^2+2 e^{x^2-x-2} (x+4) x+8 x-84}-1} \left (-e^{2 x^2} x^2-2 e^{x^2+x+2} (x+4) x-e^{2 x+4} \left (x^2+8 x-84\right )+2 \left (e^{2 x^2} x \left (2 x^2-x+1\right )+e^{x^2+x+2} \left (2 x^3+7 x^2-2 x+4\right )+e^{2 x+4} (x+4)\right ) x \log (x)\right )}{\left (e^{x^2} x+e^{x+2} (x-6)\right )^2 \left (e^{x^2} x+e^{x+2} (x+14)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 25 \int -\frac {e^{2 x+4} x^{\frac {25}{-e^{2 x^2-2 x-4} x^2-x^2-2 e^{x^2-x-2} (x+4) x-8 x+84}-1} \left (e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x-2 \left (e^{2 x+4} (x+4)+e^{2 x^2} x \left (2 x^2-x+1\right )+e^{x^2+x+2} \left (2 x^3+7 x^2-2 x+4\right )\right ) \log (x) x-e^{2 x+4} \left (-x^2-8 x+84\right )\right )}{\left (e^{x+2} (6-x)-e^{x^2} x\right )^2 \left (e^{x^2} x+e^{x+2} (x+14)\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -25 \int \frac {e^{2 x+4} x^{\frac {25}{-e^{2 x^2-2 x-4} x^2-x^2-2 e^{x^2-x-2} (x+4) x-8 x+84}-1} \left (e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x-2 \left (e^{2 x+4} (x+4)+e^{2 x^2} x \left (2 x^2-x+1\right )+e^{x^2+x+2} \left (2 x^3+7 x^2-2 x+4\right )\right ) \log (x) x-e^{2 x+4} \left (-x^2-8 x+84\right )\right )}{\left (e^{x+2} (6-x)-e^{x^2} x\right )^2 \left (e^{x^2} x+e^{x+2} (x+14)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -25 \int \left (\frac {e^{2 x+4} \left (2 x^3-13 x^2+6 x-6\right ) \log (x) x^{\frac {25}{-e^{2 x^2-2 x-4} x^2-x^2-2 e^{x^2-x-2} (x+4) x-8 x+84}-1}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )^2}-\frac {e^{2 x+4} \left (2 x^3+27 x^2-14 x+14\right ) \log (x) x^{\frac {25}{-e^{2 x^2-2 x-4} x^2-x^2-2 e^{x^2-x-2} (x+4) x-8 x+84}-1}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )^2}-\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{\frac {25}{-e^{2 x^2-2 x-4} x^2-x^2-2 e^{x^2-x-2} (x+4) x-8 x+84}-1}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )}+\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{\frac {25}{-e^{2 x^2-2 x-4} x^2-x^2-2 e^{x^2-x-2} (x+4) x-8 x+84}-1}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -25 \int \frac {e^{2 x+4} x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}} \left (e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x-2 \left (e^{2 x+4} (x+4)+e^{2 x^2} x \left (2 x^2-x+1\right )+e^{x^2+x+2} \left (2 x^3+7 x^2-2 x+4\right )\right ) \log (x) x+e^{2 x+4} \left (x^2+8 x-84\right )\right )}{\left (e^{x+2} (x-6)+e^{x^2} x\right )^2 \left (e^{x^2} x+e^{x+2} (x+14)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -25 \int \left (\frac {e^{2 x+4} \left (2 x^3-13 x^2+6 x-6\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )^2}-\frac {e^{2 x+4} \left (2 x^3+27 x^2-14 x+14\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )^2}-\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )}+\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -25 \int \frac {e^{2 x+4} x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}} \left (e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x-2 \left (e^{2 x+4} (x+4)+e^{2 x^2} x \left (2 x^2-x+1\right )+e^{x^2+x+2} \left (2 x^3+7 x^2-2 x+4\right )\right ) \log (x) x+e^{2 x+4} \left (x^2+8 x-84\right )\right )}{\left (e^{x+2} (x-6)+e^{x^2} x\right )^2 \left (e^{x^2} x+e^{x+2} (x+14)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -25 \int \left (\frac {e^{2 x+4} \left (2 x^3-13 x^2+6 x-6\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )^2}-\frac {e^{2 x+4} \left (2 x^3+27 x^2-14 x+14\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )^2}-\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )}+\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -25 \int \frac {e^{2 x+4} x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}} \left (e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x-2 \left (e^{2 x+4} (x+4)+e^{2 x^2} x \left (2 x^2-x+1\right )+e^{x^2+x+2} \left (2 x^3+7 x^2-2 x+4\right )\right ) \log (x) x+e^{2 x+4} \left (x^2+8 x-84\right )\right )}{\left (e^{x+2} (x-6)+e^{x^2} x\right )^2 \left (e^{x^2} x+e^{x+2} (x+14)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -25 \int \left (\frac {e^{2 x+4} \left (2 x^3-13 x^2+6 x-6\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )^2}-\frac {e^{2 x+4} \left (2 x^3+27 x^2-14 x+14\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )^2}-\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )}+\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -25 \int \frac {e^{2 x+4} x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}} \left (e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x-2 \left (e^{2 x+4} (x+4)+e^{2 x^2} x \left (2 x^2-x+1\right )+e^{x^2+x+2} \left (2 x^3+7 x^2-2 x+4\right )\right ) \log (x) x+e^{2 x+4} \left (x^2+8 x-84\right )\right )}{\left (e^{x+2} (x-6)+e^{x^2} x\right )^2 \left (e^{x^2} x+e^{x+2} (x+14)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -25 \int \left (\frac {e^{2 x+4} \left (2 x^3-13 x^2+6 x-6\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )^2}-\frac {e^{2 x+4} \left (2 x^3+27 x^2-14 x+14\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )^2}-\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )}+\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -25 \int \frac {e^{2 x+4} x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}} \left (e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x-2 \left (e^{2 x+4} (x+4)+e^{2 x^2} x \left (2 x^2-x+1\right )+e^{x^2+x+2} \left (2 x^3+7 x^2-2 x+4\right )\right ) \log (x) x+e^{2 x+4} \left (x^2+8 x-84\right )\right )}{\left (e^{x+2} (x-6)+e^{x^2} x\right )^2 \left (e^{x^2} x+e^{x+2} (x+14)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -25 \int \left (\frac {e^{2 x+4} \left (2 x^3-13 x^2+6 x-6\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )^2}-\frac {e^{2 x+4} \left (2 x^3+27 x^2-14 x+14\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )^2}-\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )}+\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -25 \int \frac {e^{2 x+4} x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}} \left (e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x-2 \left (e^{2 x+4} (x+4)+e^{2 x^2} x \left (2 x^2-x+1\right )+e^{x^2+x+2} \left (2 x^3+7 x^2-2 x+4\right )\right ) \log (x) x+e^{2 x+4} \left (x^2+8 x-84\right )\right )}{\left (e^{x+2} (x-6)+e^{x^2} x\right )^2 \left (e^{x^2} x+e^{x+2} (x+14)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -25 \int \left (\frac {e^{2 x+4} \left (2 x^3-13 x^2+6 x-6\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )^2}-\frac {e^{2 x+4} \left (2 x^3+27 x^2-14 x+14\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )^2}-\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )}+\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -25 \int \frac {e^{2 x+4} x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}} \left (e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x-2 \left (e^{2 x+4} (x+4)+e^{2 x^2} x \left (2 x^2-x+1\right )+e^{x^2+x+2} \left (2 x^3+7 x^2-2 x+4\right )\right ) \log (x) x+e^{2 x+4} \left (x^2+8 x-84\right )\right )}{\left (e^{x+2} (x-6)+e^{x^2} x\right )^2 \left (e^{x^2} x+e^{x+2} (x+14)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -25 \int \left (\frac {e^{2 x+4} \left (2 x^3-13 x^2+6 x-6\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )^2}-\frac {e^{2 x+4} \left (2 x^3+27 x^2-14 x+14\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )^2}-\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )}+\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -25 \int \frac {e^{2 x+4} x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}} \left (e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x-2 \left (e^{2 x+4} (x+4)+e^{2 x^2} x \left (2 x^2-x+1\right )+e^{x^2+x+2} \left (2 x^3+7 x^2-2 x+4\right )\right ) \log (x) x+e^{2 x+4} \left (x^2+8 x-84\right )\right )}{\left (e^{x+2} (x-6)+e^{x^2} x\right )^2 \left (e^{x^2} x+e^{x+2} (x+14)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -25 \int \left (\frac {e^{2 x+4} \left (2 x^3-13 x^2+6 x-6\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )^2}-\frac {e^{2 x+4} \left (2 x^3+27 x^2-14 x+14\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )^2}-\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )}+\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -25 \int \frac {e^{2 x+4} x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}} \left (e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x-2 \left (e^{2 x+4} (x+4)+e^{2 x^2} x \left (2 x^2-x+1\right )+e^{x^2+x+2} \left (2 x^3+7 x^2-2 x+4\right )\right ) \log (x) x+e^{2 x+4} \left (x^2+8 x-84\right )\right )}{\left (e^{x+2} (x-6)+e^{x^2} x\right )^2 \left (e^{x^2} x+e^{x+2} (x+14)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -25 \int \left (\frac {e^{2 x+4} \left (2 x^3-13 x^2+6 x-6\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )^2}-\frac {e^{2 x+4} \left (2 x^3+27 x^2-14 x+14\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )^2}-\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )}+\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -25 \int \frac {e^{2 x+4} x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}} \left (e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x-2 \left (e^{2 x+4} (x+4)+e^{2 x^2} x \left (2 x^2-x+1\right )+e^{x^2+x+2} \left (2 x^3+7 x^2-2 x+4\right )\right ) \log (x) x+e^{2 x+4} \left (x^2+8 x-84\right )\right )}{\left (e^{x+2} (x-6)+e^{x^2} x\right )^2 \left (e^{x^2} x+e^{x+2} (x+14)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -25 \int \left (\frac {e^{2 x+4} \left (2 x^3-13 x^2+6 x-6\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )^2}-\frac {e^{2 x+4} \left (2 x^3+27 x^2-14 x+14\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )^2}-\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )}+\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -25 \int \frac {e^{2 x+4} x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}} \left (e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x-2 \left (e^{2 x+4} (x+4)+e^{2 x^2} x \left (2 x^2-x+1\right )+e^{x^2+x+2} \left (2 x^3+7 x^2-2 x+4\right )\right ) \log (x) x+e^{2 x+4} \left (x^2+8 x-84\right )\right )}{\left (e^{x+2} (x-6)+e^{x^2} x\right )^2 \left (e^{x^2} x+e^{x+2} (x+14)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -25 \int \left (\frac {e^{2 x+4} \left (2 x^3-13 x^2+6 x-6\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )^2}-\frac {e^{2 x+4} \left (2 x^3+27 x^2-14 x+14\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )^2}-\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )}+\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -25 \int \frac {e^{2 x+4} x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}} \left (e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x-2 \left (e^{2 x+4} (x+4)+e^{2 x^2} x \left (2 x^2-x+1\right )+e^{x^2+x+2} \left (2 x^3+7 x^2-2 x+4\right )\right ) \log (x) x+e^{2 x+4} \left (x^2+8 x-84\right )\right )}{\left (e^{x+2} (x-6)+e^{x^2} x\right )^2 \left (e^{x^2} x+e^{x+2} (x+14)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -25 \int \left (\frac {e^{2 x+4} \left (2 x^3-13 x^2+6 x-6\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )^2}-\frac {e^{2 x+4} \left (2 x^3+27 x^2-14 x+14\right ) \log (x) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )^2}-\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x-6 e^{x+2}\right )}+\frac {e^{x+2} \left (2 \log (x) x^2-\log (x) x+\log (x)-1\right ) x^{-1-\frac {25 e^{2 x+4}}{e^{2 x^2} x^2+2 e^{x^2+x+2} (x+4) x+e^{2 x+4} \left (x^2+8 x-84\right )}}}{20 \left (e^{x^2} x+e^{x+2} x+14 e^{x+2}\right )}\right )dx\) |
Input:
Int[(2100 - 200*x - 25*x^2 + (200*x + 50*x^2)*Log[x] + E^(-4 - 2*x + 2*x^2 )*x^2*(-25 + (50 - 50*x + 100*x^2)*Log[x]) + E^(-2 - x + x^2)*x*(-200 - 50 *x + (200 - 100*x + 350*x^2 + 100*x^3)*Log[x]))/(x^(25/(-84 + 8*x + x^2 + E^(-4 - 2*x + 2*x^2)*x^2 + E^(-2 - x + x^2)*x*(8 + 2*x)))*(7056*x - 1344*x ^2 - 104*x^3 + 16*x^4 + x^5 + E^(-8 - 4*x + 4*x^2)*x^5 + E^(-6 - 3*x + 3*x ^2)*x^3*(16*x + 4*x^2) + E^(-4 - 2*x + 2*x^2)*x^2*(-104*x + 48*x^2 + 6*x^3 ) + E^(-2 - x + x^2)*x*(-1344*x - 208*x^2 + 48*x^3 + 4*x^4))),x]
Output:
$Aborted
Time = 0.12 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.93
\[x^{-\frac {25}{x^{2} {\mathrm e}^{2 \left (1+x \right ) \left (-2+x \right )}+2 \,{\mathrm e}^{\left (1+x \right ) \left (-2+x \right )} x^{2}+8 x \,{\mathrm e}^{\left (1+x \right ) \left (-2+x \right )}+x^{2}+8 x -84}}\]
Input:
int((((100*x^2-50*x+50)*ln(x)-25)*exp(ln(x)+x^2-x-2)^2+((100*x^3+350*x^2-1 00*x+200)*ln(x)-50*x-200)*exp(ln(x)+x^2-x-2)+(50*x^2+200*x)*ln(x)-25*x^2-2 00*x+2100)*exp(-25*ln(x)/(exp(ln(x)+x^2-x-2)^2+(2*x+8)*exp(ln(x)+x^2-x-2)+ x^2+8*x-84))/(x*exp(ln(x)+x^2-x-2)^4+(4*x^2+16*x)*exp(ln(x)+x^2-x-2)^3+(6* x^3+48*x^2-104*x)*exp(ln(x)+x^2-x-2)^2+(4*x^4+48*x^3-208*x^2-1344*x)*exp(l n(x)+x^2-x-2)+x^5+16*x^4-104*x^3-1344*x^2+7056*x),x)
Output:
x^(-25/(x^2*exp(2*(1+x)*(-2+x))+2*exp((1+x)*(-2+x))*x^2+8*x*exp((1+x)*(-2+ x))+x^2+8*x-84))
Time = 0.10 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {x^{-\frac {25}{-84+8 x+x^2+e^{-4-2 x+2 x^2} x^2+e^{-2-x+x^2} x (8+2 x)}} \left (2100-200 x-25 x^2+\left (200 x+50 x^2\right ) \log (x)+e^{-4-2 x+2 x^2} x^2 \left (-25+\left (50-50 x+100 x^2\right ) \log (x)\right )+e^{-2-x+x^2} x \left (-200-50 x+\left (200-100 x+350 x^2+100 x^3\right ) \log (x)\right )\right )}{7056 x-1344 x^2-104 x^3+16 x^4+x^5+e^{-8-4 x+4 x^2} x^5+e^{-6-3 x+3 x^2} x^3 \left (16 x+4 x^2\right )+e^{-4-2 x+2 x^2} x^2 \left (-104 x+48 x^2+6 x^3\right )+e^{-2-x+x^2} x \left (-1344 x-208 x^2+48 x^3+4 x^4\right )} \, dx=\frac {1}{x^{\frac {25}{x^{2} + 2 \, {\left (x + 4\right )} e^{\left (x^{2} - x + \log \left (x\right ) - 2\right )} + 8 \, x + e^{\left (2 \, x^{2} - 2 \, x + 2 \, \log \left (x\right ) - 4\right )} - 84}}} \] Input:
integrate((((100*x^2-50*x+50)*log(x)-25)*exp(log(x)+x^2-x-2)^2+((100*x^3+3 50*x^2-100*x+200)*log(x)-50*x-200)*exp(log(x)+x^2-x-2)+(50*x^2+200*x)*log( x)-25*x^2-200*x+2100)*exp(-25*log(x)/(exp(log(x)+x^2-x-2)^2+(2*x+8)*exp(lo g(x)+x^2-x-2)+x^2+8*x-84))/(x*exp(log(x)+x^2-x-2)^4+(4*x^2+16*x)*exp(log(x )+x^2-x-2)^3+(6*x^3+48*x^2-104*x)*exp(log(x)+x^2-x-2)^2+(4*x^4+48*x^3-208* x^2-1344*x)*exp(log(x)+x^2-x-2)+x^5+16*x^4-104*x^3-1344*x^2+7056*x),x, alg orithm="fricas")
Output:
1/(x^(25/(x^2 + 2*(x + 4)*e^(x^2 - x + log(x) - 2) + 8*x + e^(2*x^2 - 2*x + 2*log(x) - 4) - 84)))
Time = 4.33 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {x^{-\frac {25}{-84+8 x+x^2+e^{-4-2 x+2 x^2} x^2+e^{-2-x+x^2} x (8+2 x)}} \left (2100-200 x-25 x^2+\left (200 x+50 x^2\right ) \log (x)+e^{-4-2 x+2 x^2} x^2 \left (-25+\left (50-50 x+100 x^2\right ) \log (x)\right )+e^{-2-x+x^2} x \left (-200-50 x+\left (200-100 x+350 x^2+100 x^3\right ) \log (x)\right )\right )}{7056 x-1344 x^2-104 x^3+16 x^4+x^5+e^{-8-4 x+4 x^2} x^5+e^{-6-3 x+3 x^2} x^3 \left (16 x+4 x^2\right )+e^{-4-2 x+2 x^2} x^2 \left (-104 x+48 x^2+6 x^3\right )+e^{-2-x+x^2} x \left (-1344 x-208 x^2+48 x^3+4 x^4\right )} \, dx=e^{- \frac {25 \log {\left (x \right )}}{x^{2} e^{2 x^{2} - 2 x - 4} + x^{2} + x \left (2 x + 8\right ) e^{x^{2} - x - 2} + 8 x - 84}} \] Input:
integrate((((100*x**2-50*x+50)*ln(x)-25)*exp(ln(x)+x**2-x-2)**2+((100*x**3 +350*x**2-100*x+200)*ln(x)-50*x-200)*exp(ln(x)+x**2-x-2)+(50*x**2+200*x)*l n(x)-25*x**2-200*x+2100)*exp(-25*ln(x)/(exp(ln(x)+x**2-x-2)**2+(2*x+8)*exp (ln(x)+x**2-x-2)+x**2+8*x-84))/(x*exp(ln(x)+x**2-x-2)**4+(4*x**2+16*x)*exp (ln(x)+x**2-x-2)**3+(6*x**3+48*x**2-104*x)*exp(ln(x)+x**2-x-2)**2+(4*x**4+ 48*x**3-208*x**2-1344*x)*exp(ln(x)+x**2-x-2)+x**5+16*x**4-104*x**3-1344*x* *2+7056*x),x)
Output:
exp(-25*log(x)/(x**2*exp(2*x**2 - 2*x - 4) + x**2 + x*(2*x + 8)*exp(x**2 - x - 2) + 8*x - 84))
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (26) = 52\).
Time = 0.43 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {x^{-\frac {25}{-84+8 x+x^2+e^{-4-2 x+2 x^2} x^2+e^{-2-x+x^2} x (8+2 x)}} \left (2100-200 x-25 x^2+\left (200 x+50 x^2\right ) \log (x)+e^{-4-2 x+2 x^2} x^2 \left (-25+\left (50-50 x+100 x^2\right ) \log (x)\right )+e^{-2-x+x^2} x \left (-200-50 x+\left (200-100 x+350 x^2+100 x^3\right ) \log (x)\right )\right )}{7056 x-1344 x^2-104 x^3+16 x^4+x^5+e^{-8-4 x+4 x^2} x^5+e^{-6-3 x+3 x^2} x^3 \left (16 x+4 x^2\right )+e^{-4-2 x+2 x^2} x^2 \left (-104 x+48 x^2+6 x^3\right )+e^{-2-x+x^2} x \left (-1344 x-208 x^2+48 x^3+4 x^4\right )} \, dx=e^{\left (\frac {5 \, e^{\left (x + 2\right )} \log \left (x\right )}{4 \, {\left (x e^{\left (x^{2}\right )} + {\left (x e^{2} + 14 \, e^{2}\right )} e^{x}\right )}} - \frac {5 \, e^{\left (x + 2\right )} \log \left (x\right )}{4 \, {\left (x e^{\left (x^{2}\right )} + {\left (x e^{2} - 6 \, e^{2}\right )} e^{x}\right )}}\right )} \] Input:
integrate((((100*x^2-50*x+50)*log(x)-25)*exp(log(x)+x^2-x-2)^2+((100*x^3+3 50*x^2-100*x+200)*log(x)-50*x-200)*exp(log(x)+x^2-x-2)+(50*x^2+200*x)*log( x)-25*x^2-200*x+2100)*exp(-25*log(x)/(exp(log(x)+x^2-x-2)^2+(2*x+8)*exp(lo g(x)+x^2-x-2)+x^2+8*x-84))/(x*exp(log(x)+x^2-x-2)^4+(4*x^2+16*x)*exp(log(x )+x^2-x-2)^3+(6*x^3+48*x^2-104*x)*exp(log(x)+x^2-x-2)^2+(4*x^4+48*x^3-208* x^2-1344*x)*exp(log(x)+x^2-x-2)+x^5+16*x^4-104*x^3-1344*x^2+7056*x),x, alg orithm="maxima")
Output:
e^(5/4*e^(x + 2)*log(x)/(x*e^(x^2) + (x*e^2 + 14*e^2)*e^x) - 5/4*e^(x + 2) *log(x)/(x*e^(x^2) + (x*e^2 - 6*e^2)*e^x))
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (26) = 52\).
Time = 8.53 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.11 \[ \int \frac {x^{-\frac {25}{-84+8 x+x^2+e^{-4-2 x+2 x^2} x^2+e^{-2-x+x^2} x (8+2 x)}} \left (2100-200 x-25 x^2+\left (200 x+50 x^2\right ) \log (x)+e^{-4-2 x+2 x^2} x^2 \left (-25+\left (50-50 x+100 x^2\right ) \log (x)\right )+e^{-2-x+x^2} x \left (-200-50 x+\left (200-100 x+350 x^2+100 x^3\right ) \log (x)\right )\right )}{7056 x-1344 x^2-104 x^3+16 x^4+x^5+e^{-8-4 x+4 x^2} x^5+e^{-6-3 x+3 x^2} x^3 \left (16 x+4 x^2\right )+e^{-4-2 x+2 x^2} x^2 \left (-104 x+48 x^2+6 x^3\right )+e^{-2-x+x^2} x \left (-1344 x-208 x^2+48 x^3+4 x^4\right )} \, dx=\frac {1}{x^{\frac {25}{x^{2} e^{\left (2 \, x^{2} - 2 \, x - 4\right )} + 2 \, x^{2} e^{\left (x^{2} - x - 2\right )} + x^{2} + 8 \, x e^{\left (x^{2} - x - 2\right )} + 8 \, x - 84}}} \] Input:
integrate((((100*x^2-50*x+50)*log(x)-25)*exp(log(x)+x^2-x-2)^2+((100*x^3+3 50*x^2-100*x+200)*log(x)-50*x-200)*exp(log(x)+x^2-x-2)+(50*x^2+200*x)*log( x)-25*x^2-200*x+2100)*exp(-25*log(x)/(exp(log(x)+x^2-x-2)^2+(2*x+8)*exp(lo g(x)+x^2-x-2)+x^2+8*x-84))/(x*exp(log(x)+x^2-x-2)^4+(4*x^2+16*x)*exp(log(x )+x^2-x-2)^3+(6*x^3+48*x^2-104*x)*exp(log(x)+x^2-x-2)^2+(4*x^4+48*x^3-208* x^2-1344*x)*exp(log(x)+x^2-x-2)+x^5+16*x^4-104*x^3-1344*x^2+7056*x),x, alg orithm="giac")
Output:
1/(x^(25/(x^2*e^(2*x^2 - 2*x - 4) + 2*x^2*e^(x^2 - x - 2) + x^2 + 8*x*e^(x ^2 - x - 2) + 8*x - 84)))
Time = 3.21 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {x^{-\frac {25}{-84+8 x+x^2+e^{-4-2 x+2 x^2} x^2+e^{-2-x+x^2} x (8+2 x)}} \left (2100-200 x-25 x^2+\left (200 x+50 x^2\right ) \log (x)+e^{-4-2 x+2 x^2} x^2 \left (-25+\left (50-50 x+100 x^2\right ) \log (x)\right )+e^{-2-x+x^2} x \left (-200-50 x+\left (200-100 x+350 x^2+100 x^3\right ) \log (x)\right )\right )}{7056 x-1344 x^2-104 x^3+16 x^4+x^5+e^{-8-4 x+4 x^2} x^5+e^{-6-3 x+3 x^2} x^3 \left (16 x+4 x^2\right )+e^{-4-2 x+2 x^2} x^2 \left (-104 x+48 x^2+6 x^3\right )+e^{-2-x+x^2} x \left (-1344 x-208 x^2+48 x^3+4 x^4\right )} \, dx=\frac {1}{x^{\frac {25}{8\,x+x^2+2\,x^2\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-2}+x^2\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{2\,x^2}+8\,x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-2}-84}}} \] Input:
int(-(exp(-(25*log(x))/(8*x + exp(2*log(x) - 2*x + 2*x^2 - 4) + x^2 + exp( log(x) - x + x^2 - 2)*(2*x + 8) - 84))*(200*x - exp(2*log(x) - 2*x + 2*x^2 - 4)*(log(x)*(100*x^2 - 50*x + 50) - 25) - log(x)*(200*x + 50*x^2) + exp( log(x) - x + x^2 - 2)*(50*x - log(x)*(350*x^2 - 100*x + 100*x^3 + 200) + 2 00) + 25*x^2 - 2100))/(7056*x + exp(2*log(x) - 2*x + 2*x^2 - 4)*(48*x^2 - 104*x + 6*x^3) - exp(log(x) - x + x^2 - 2)*(1344*x + 208*x^2 - 48*x^3 - 4* x^4) - 1344*x^2 - 104*x^3 + 16*x^4 + x^5 + exp(3*log(x) - 3*x + 3*x^2 - 6) *(16*x + 4*x^2) + x*exp(4*log(x) - 4*x + 4*x^2 - 8)),x)
Output:
1/x^(25/(8*x + x^2 + 2*x^2*exp(-x)*exp(x^2)*exp(-2) + x^2*exp(-2*x)*exp(-4 )*exp(2*x^2) + 8*x*exp(-x)*exp(x^2)*exp(-2) - 84))
\[ \int \frac {x^{-\frac {25}{-84+8 x+x^2+e^{-4-2 x+2 x^2} x^2+e^{-2-x+x^2} x (8+2 x)}} \left (2100-200 x-25 x^2+\left (200 x+50 x^2\right ) \log (x)+e^{-4-2 x+2 x^2} x^2 \left (-25+\left (50-50 x+100 x^2\right ) \log (x)\right )+e^{-2-x+x^2} x \left (-200-50 x+\left (200-100 x+350 x^2+100 x^3\right ) \log (x)\right )\right )}{7056 x-1344 x^2-104 x^3+16 x^4+x^5+e^{-8-4 x+4 x^2} x^5+e^{-6-3 x+3 x^2} x^3 \left (16 x+4 x^2\right )+e^{-4-2 x+2 x^2} x^2 \left (-104 x+48 x^2+6 x^3\right )+e^{-2-x+x^2} x \left (-1344 x-208 x^2+48 x^3+4 x^4\right )} \, dx=\int \frac {\left (\left (\left (100 x^{2}-50 x +50\right ) \mathrm {log}\left (x \right )-25\right ) \left ({\mathrm e}^{\mathrm {log}\left (x \right )+x^{2}-x -2}\right )^{2}+\left (\left (100 x^{3}+350 x^{2}-100 x +200\right ) \mathrm {log}\left (x \right )-50 x -200\right ) {\mathrm e}^{\mathrm {log}\left (x \right )+x^{2}-x -2}+\left (50 x^{2}+200 x \right ) \mathrm {log}\left (x \right )-25 x^{2}-200 x +2100\right ) {\mathrm e}^{-\frac {25 \,\mathrm {log}\left (x \right )}{\left ({\mathrm e}^{\mathrm {log}\left (x \right )+x^{2}-x -2}\right )^{2}+\left (2 x +8\right ) {\mathrm e}^{\mathrm {log}\left (x \right )+x^{2}-x -2}+x^{2}+8 x -84}}}{x \left ({\mathrm e}^{\mathrm {log}\left (x \right )+x^{2}-x -2}\right )^{4}+\left (4 x^{2}+16 x \right ) \left ({\mathrm e}^{\mathrm {log}\left (x \right )+x^{2}-x -2}\right )^{3}+\left (6 x^{3}+48 x^{2}-104 x \right ) \left ({\mathrm e}^{\mathrm {log}\left (x \right )+x^{2}-x -2}\right )^{2}+\left (4 x^{4}+48 x^{3}-208 x^{2}-1344 x \right ) {\mathrm e}^{\mathrm {log}\left (x \right )+x^{2}-x -2}+x^{5}+16 x^{4}-104 x^{3}-1344 x^{2}+7056 x}d x \] Input:
int((((100*x^2-50*x+50)*log(x)-25)*exp(log(x)+x^2-x-2)^2+((100*x^3+350*x^2 -100*x+200)*log(x)-50*x-200)*exp(log(x)+x^2-x-2)+(50*x^2+200*x)*log(x)-25* x^2-200*x+2100)*exp(-25*log(x)/(exp(log(x)+x^2-x-2)^2+(2*x+8)*exp(log(x)+x ^2-x-2)+x^2+8*x-84))/(x*exp(log(x)+x^2-x-2)^4+(4*x^2+16*x)*exp(log(x)+x^2- x-2)^3+(6*x^3+48*x^2-104*x)*exp(log(x)+x^2-x-2)^2+(4*x^4+48*x^3-208*x^2-13 44*x)*exp(log(x)+x^2-x-2)+x^5+16*x^4-104*x^3-1344*x^2+7056*x),x)
Output:
int((((100*x^2-50*x+50)*log(x)-25)*exp(log(x)+x^2-x-2)^2+((100*x^3+350*x^2 -100*x+200)*log(x)-50*x-200)*exp(log(x)+x^2-x-2)+(50*x^2+200*x)*log(x)-25* x^2-200*x+2100)*exp(-25*log(x)/(exp(log(x)+x^2-x-2)^2+(2*x+8)*exp(log(x)+x ^2-x-2)+x^2+8*x-84))/(x*exp(log(x)+x^2-x-2)^4+(4*x^2+16*x)*exp(log(x)+x^2- x-2)^3+(6*x^3+48*x^2-104*x)*exp(log(x)+x^2-x-2)^2+(4*x^4+48*x^3-208*x^2-13 44*x)*exp(log(x)+x^2-x-2)+x^5+16*x^4-104*x^3-1344*x^2+7056*x),x)