Integrand size = 246, antiderivative size = 36 \[ \int \frac {e^x \left (54+9 x-9 x^2\right )+e^{3 x} \left (-108 x^2-36 x^3+33 x^4+6 x^5-3 x^6\right )+e^{2 x} \left (108 x^2+36 x^3-33 x^4-6 x^5+3 x^6\right )+e^x \left (-54-72 x+18 x^2+9 x^3\right ) \log (x)}{e^{4 x} \left (36 x^2+12 x^3-11 x^4-2 x^5+x^6\right )+e^{3 x} \left (-72 x^3-24 x^4+22 x^5+4 x^6-2 x^7\right )+e^{2 x} \left (36 x^4+12 x^5-11 x^6-2 x^7+x^8\right )+\left (e^{2 x} \left (-36 x-6 x^2+6 x^3\right )+e^x \left (36 x^2+6 x^3-6 x^4\right )\right ) \log (x)+9 \log ^2(x)} \, dx=\frac {3}{e^x-x-\frac {e^{-x} \log (x)}{(2+x) \left (x-\frac {x^2}{3}\right )}} \]
Time = 0.13 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.14 \[ \int \frac {e^x \left (54+9 x-9 x^2\right )+e^{3 x} \left (-108 x^2-36 x^3+33 x^4+6 x^5-3 x^6\right )+e^{2 x} \left (108 x^2+36 x^3-33 x^4-6 x^5+3 x^6\right )+e^x \left (-54-72 x+18 x^2+9 x^3\right ) \log (x)}{e^{4 x} \left (36 x^2+12 x^3-11 x^4-2 x^5+x^6\right )+e^{3 x} \left (-72 x^3-24 x^4+22 x^5+4 x^6-2 x^7\right )+e^{2 x} \left (36 x^4+12 x^5-11 x^6-2 x^7+x^8\right )+\left (e^{2 x} \left (-36 x-6 x^2+6 x^3\right )+e^x \left (36 x^2+6 x^3-6 x^4\right )\right ) \log (x)+9 \log ^2(x)} \, dx=\frac {3 e^x x \left (-6-x+x^2\right )}{e^x \left (e^x-x\right ) x \left (-6-x+x^2\right )+3 \log (x)} \]
Integrate[(E^x*(54 + 9*x - 9*x^2) + E^(3*x)*(-108*x^2 - 36*x^3 + 33*x^4 + 6*x^5 - 3*x^6) + E^(2*x)*(108*x^2 + 36*x^3 - 33*x^4 - 6*x^5 + 3*x^6) + E^x *(-54 - 72*x + 18*x^2 + 9*x^3)*Log[x])/(E^(4*x)*(36*x^2 + 12*x^3 - 11*x^4 - 2*x^5 + x^6) + E^(3*x)*(-72*x^3 - 24*x^4 + 22*x^5 + 4*x^6 - 2*x^7) + E^( 2*x)*(36*x^4 + 12*x^5 - 11*x^6 - 2*x^7 + x^8) + (E^(2*x)*(-36*x - 6*x^2 + 6*x^3) + E^x*(36*x^2 + 6*x^3 - 6*x^4))*Log[x] + 9*Log[x]^2),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.
| \(\displaystyle \int \frac {e^x \left (-9 x^2+9 x+54\right )+e^x \left (9 x^3+18 x^2-72 x-54\right ) \log (x)+e^{3 x} \left (-3 x^6+6 x^5+33 x^4-36 x^3-108 x^2\right )+e^{2 x} \left (3 x^6-6 x^5-33 x^4+36 x^3+108 x^2\right )}{\left (e^{2 x} \left (6 x^3-6 x^2-36 x\right )+e^x \left (-6 x^4+6 x^3+36 x^2\right )\right ) \log (x)+e^{2 x} \left (x^8-2 x^7-11 x^6+12 x^5+36 x^4\right )+e^{3 x} \left (-2 x^7+4 x^6+22 x^5-24 x^4-72 x^3\right )+e^{4 x} \left (x^6-2 x^5-11 x^4+12 x^3+36 x^2\right )+9 \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {3 e^x \left (3 \left (x^3+2 x^2-8 x-6\right ) \log (x)-\left (x^2-x-6\right ) \left (-e^x \left (x^2-x-6\right ) x^2+e^{2 x} \left (x^2-x-6\right ) x^2+3\right )\right )}{\left (e^x \left (e^x-x\right ) x \left (x^2-x-6\right )+3 \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \int \frac {e^x \left (\left (-x^2+x+6\right ) \left (e^x \left (-x^2+x+6\right ) x^2-e^{2 x} \left (-x^2+x+6\right ) x^2+3\right )-3 \left (-x^3-2 x^2+8 x+6\right ) \log (x)\right )}{\left (e^x \left (e^x-x\right ) x \left (-x^2+x+6\right )-3 \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^x x \left (x^2-x-6\right )}{e^x x^4-e^x x^3-e^{2 x} x^3-6 e^x x^2+e^{2 x} x^2+6 e^{2 x} x-3 \log (x)}-\frac {e^x \left (e^x x^7-3 e^x x^6-9 e^x x^5+23 e^x x^4+24 e^x x^3-6 \log (x) x^3-36 e^x x^2-3 \log (x) x^2+3 x^2+42 \log (x) x-3 x+18 \log (x)-18\right )}{\left (-e^x x^4+e^x x^3+e^{2 x} x^3+6 e^x x^2-e^{2 x} x^2-6 e^{2 x} x+3 \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^x \left (3 \left (x^3+2 x^2-8 x-6\right ) \log (x)-\left (x^2-x-6\right ) \left (-e^x \left (x^2-x-6\right ) x^2+e^{2 x} \left (x^2-x-6\right ) x^2+3\right )\right )}{\left (e^x \left (e^x-x\right ) x \left (x^2-x-6\right )+3 \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^x x \left (x^2-x-6\right )}{e^x x^4-e^x x^3-e^{2 x} x^3-6 e^x x^2+e^{2 x} x^2+6 e^{2 x} x-3 \log (x)}-\frac {e^x \left (e^x x^7-3 e^x x^6-9 e^x x^5+23 e^x x^4+24 e^x x^3-6 \log (x) x^3-36 e^x x^2-3 \log (x) x^2+3 x^2+42 \log (x) x-3 x+18 \log (x)-18\right )}{\left (-e^x x^4+e^x x^3+e^{2 x} x^3+6 e^x x^2-e^{2 x} x^2-6 e^{2 x} x+3 \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^x \left (3 \left (x^3+2 x^2-8 x-6\right ) \log (x)-\left (x^2-x-6\right ) \left (-e^x \left (x^2-x-6\right ) x^2+e^{2 x} \left (x^2-x-6\right ) x^2+3\right )\right )}{\left (e^x \left (e^x-x\right ) x \left (x^2-x-6\right )+3 \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^x x \left (x^2-x-6\right )}{e^x x^4-e^x x^3-e^{2 x} x^3-6 e^x x^2+e^{2 x} x^2+6 e^{2 x} x-3 \log (x)}-\frac {e^x \left (e^x x^7-3 e^x x^6-9 e^x x^5+23 e^x x^4+24 e^x x^3-6 \log (x) x^3-36 e^x x^2-3 \log (x) x^2+3 x^2+42 \log (x) x-3 x+18 \log (x)-18\right )}{\left (-e^x x^4+e^x x^3+e^{2 x} x^3+6 e^x x^2-e^{2 x} x^2-6 e^{2 x} x+3 \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^x \left (3 \left (x^3+2 x^2-8 x-6\right ) \log (x)-\left (x^2-x-6\right ) \left (-e^x \left (x^2-x-6\right ) x^2+e^{2 x} \left (x^2-x-6\right ) x^2+3\right )\right )}{\left (e^x \left (e^x-x\right ) x \left (x^2-x-6\right )+3 \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^x x \left (x^2-x-6\right )}{e^x x^4-e^x x^3-e^{2 x} x^3-6 e^x x^2+e^{2 x} x^2+6 e^{2 x} x-3 \log (x)}-\frac {e^x \left (e^x x^7-3 e^x x^6-9 e^x x^5+23 e^x x^4+24 e^x x^3-6 \log (x) x^3-36 e^x x^2-3 \log (x) x^2+3 x^2+42 \log (x) x-3 x+18 \log (x)-18\right )}{\left (-e^x x^4+e^x x^3+e^{2 x} x^3+6 e^x x^2-e^{2 x} x^2-6 e^{2 x} x+3 \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^x \left (3 \left (x^3+2 x^2-8 x-6\right ) \log (x)-\left (x^2-x-6\right ) \left (-e^x \left (x^2-x-6\right ) x^2+e^{2 x} \left (x^2-x-6\right ) x^2+3\right )\right )}{\left (e^x \left (e^x-x\right ) x \left (x^2-x-6\right )+3 \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^x x \left (x^2-x-6\right )}{e^x x^4-e^x x^3-e^{2 x} x^3-6 e^x x^2+e^{2 x} x^2+6 e^{2 x} x-3 \log (x)}-\frac {e^x \left (e^x x^7-3 e^x x^6-9 e^x x^5+23 e^x x^4+24 e^x x^3-6 \log (x) x^3-36 e^x x^2-3 \log (x) x^2+3 x^2+42 \log (x) x-3 x+18 \log (x)-18\right )}{\left (-e^x x^4+e^x x^3+e^{2 x} x^3+6 e^x x^2-e^{2 x} x^2-6 e^{2 x} x+3 \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^x \left (3 \left (x^3+2 x^2-8 x-6\right ) \log (x)-\left (x^2-x-6\right ) \left (-e^x \left (x^2-x-6\right ) x^2+e^{2 x} \left (x^2-x-6\right ) x^2+3\right )\right )}{\left (e^x \left (e^x-x\right ) x \left (x^2-x-6\right )+3 \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^x x \left (x^2-x-6\right )}{e^x x^4-e^x x^3-e^{2 x} x^3-6 e^x x^2+e^{2 x} x^2+6 e^{2 x} x-3 \log (x)}-\frac {e^x \left (e^x x^7-3 e^x x^6-9 e^x x^5+23 e^x x^4+24 e^x x^3-6 \log (x) x^3-36 e^x x^2-3 \log (x) x^2+3 x^2+42 \log (x) x-3 x+18 \log (x)-18\right )}{\left (-e^x x^4+e^x x^3+e^{2 x} x^3+6 e^x x^2-e^{2 x} x^2-6 e^{2 x} x+3 \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^x \left (3 \left (x^3+2 x^2-8 x-6\right ) \log (x)-\left (x^2-x-6\right ) \left (-e^x \left (x^2-x-6\right ) x^2+e^{2 x} \left (x^2-x-6\right ) x^2+3\right )\right )}{\left (e^x \left (e^x-x\right ) x \left (x^2-x-6\right )+3 \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^x x \left (x^2-x-6\right )}{e^x x^4-e^x x^3-e^{2 x} x^3-6 e^x x^2+e^{2 x} x^2+6 e^{2 x} x-3 \log (x)}-\frac {e^x \left (e^x x^7-3 e^x x^6-9 e^x x^5+23 e^x x^4+24 e^x x^3-6 \log (x) x^3-36 e^x x^2-3 \log (x) x^2+3 x^2+42 \log (x) x-3 x+18 \log (x)-18\right )}{\left (-e^x x^4+e^x x^3+e^{2 x} x^3+6 e^x x^2-e^{2 x} x^2-6 e^{2 x} x+3 \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^x \left (3 \left (x^3+2 x^2-8 x-6\right ) \log (x)-\left (x^2-x-6\right ) \left (-e^x \left (x^2-x-6\right ) x^2+e^{2 x} \left (x^2-x-6\right ) x^2+3\right )\right )}{\left (e^x \left (e^x-x\right ) x \left (x^2-x-6\right )+3 \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^x x \left (x^2-x-6\right )}{e^x x^4-e^x x^3-e^{2 x} x^3-6 e^x x^2+e^{2 x} x^2+6 e^{2 x} x-3 \log (x)}-\frac {e^x \left (e^x x^7-3 e^x x^6-9 e^x x^5+23 e^x x^4+24 e^x x^3-6 \log (x) x^3-36 e^x x^2-3 \log (x) x^2+3 x^2+42 \log (x) x-3 x+18 \log (x)-18\right )}{\left (-e^x x^4+e^x x^3+e^{2 x} x^3+6 e^x x^2-e^{2 x} x^2-6 e^{2 x} x+3 \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^x \left (3 \left (x^3+2 x^2-8 x-6\right ) \log (x)-\left (x^2-x-6\right ) \left (-e^x \left (x^2-x-6\right ) x^2+e^{2 x} \left (x^2-x-6\right ) x^2+3\right )\right )}{\left (e^x \left (e^x-x\right ) x \left (x^2-x-6\right )+3 \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^x x \left (x^2-x-6\right )}{e^x x^4-e^x x^3-e^{2 x} x^3-6 e^x x^2+e^{2 x} x^2+6 e^{2 x} x-3 \log (x)}-\frac {e^x \left (e^x x^7-3 e^x x^6-9 e^x x^5+23 e^x x^4+24 e^x x^3-6 \log (x) x^3-36 e^x x^2-3 \log (x) x^2+3 x^2+42 \log (x) x-3 x+18 \log (x)-18\right )}{\left (-e^x x^4+e^x x^3+e^{2 x} x^3+6 e^x x^2-e^{2 x} x^2-6 e^{2 x} x+3 \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^x \left (3 \left (x^3+2 x^2-8 x-6\right ) \log (x)-\left (x^2-x-6\right ) \left (-e^x \left (x^2-x-6\right ) x^2+e^{2 x} \left (x^2-x-6\right ) x^2+3\right )\right )}{\left (e^x \left (e^x-x\right ) x \left (x^2-x-6\right )+3 \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^x x \left (x^2-x-6\right )}{e^x x^4-e^x x^3-e^{2 x} x^3-6 e^x x^2+e^{2 x} x^2+6 e^{2 x} x-3 \log (x)}-\frac {e^x \left (e^x x^7-3 e^x x^6-9 e^x x^5+23 e^x x^4+24 e^x x^3-6 \log (x) x^3-36 e^x x^2-3 \log (x) x^2+3 x^2+42 \log (x) x-3 x+18 \log (x)-18\right )}{\left (-e^x x^4+e^x x^3+e^{2 x} x^3+6 e^x x^2-e^{2 x} x^2-6 e^{2 x} x+3 \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^x \left (3 \left (x^3+2 x^2-8 x-6\right ) \log (x)-\left (x^2-x-6\right ) \left (-e^x \left (x^2-x-6\right ) x^2+e^{2 x} \left (x^2-x-6\right ) x^2+3\right )\right )}{\left (e^x \left (e^x-x\right ) x \left (x^2-x-6\right )+3 \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^x x \left (x^2-x-6\right )}{e^x x^4-e^x x^3-e^{2 x} x^3-6 e^x x^2+e^{2 x} x^2+6 e^{2 x} x-3 \log (x)}-\frac {e^x \left (e^x x^7-3 e^x x^6-9 e^x x^5+23 e^x x^4+24 e^x x^3-6 \log (x) x^3-36 e^x x^2-3 \log (x) x^2+3 x^2+42 \log (x) x-3 x+18 \log (x)-18\right )}{\left (-e^x x^4+e^x x^3+e^{2 x} x^3+6 e^x x^2-e^{2 x} x^2-6 e^{2 x} x+3 \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^x \left (3 \left (x^3+2 x^2-8 x-6\right ) \log (x)-\left (x^2-x-6\right ) \left (-e^x \left (x^2-x-6\right ) x^2+e^{2 x} \left (x^2-x-6\right ) x^2+3\right )\right )}{\left (e^x \left (e^x-x\right ) x \left (x^2-x-6\right )+3 \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^x x \left (x^2-x-6\right )}{e^x x^4-e^x x^3-e^{2 x} x^3-6 e^x x^2+e^{2 x} x^2+6 e^{2 x} x-3 \log (x)}-\frac {e^x \left (e^x x^7-3 e^x x^6-9 e^x x^5+23 e^x x^4+24 e^x x^3-6 \log (x) x^3-36 e^x x^2-3 \log (x) x^2+3 x^2+42 \log (x) x-3 x+18 \log (x)-18\right )}{\left (-e^x x^4+e^x x^3+e^{2 x} x^3+6 e^x x^2-e^{2 x} x^2-6 e^{2 x} x+3 \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^x \left (3 \left (x^3+2 x^2-8 x-6\right ) \log (x)-\left (x^2-x-6\right ) \left (-e^x \left (x^2-x-6\right ) x^2+e^{2 x} \left (x^2-x-6\right ) x^2+3\right )\right )}{\left (e^x \left (e^x-x\right ) x \left (x^2-x-6\right )+3 \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^x x \left (x^2-x-6\right )}{e^x x^4-e^x x^3-e^{2 x} x^3-6 e^x x^2+e^{2 x} x^2+6 e^{2 x} x-3 \log (x)}-\frac {e^x \left (e^x x^7-3 e^x x^6-9 e^x x^5+23 e^x x^4+24 e^x x^3-6 \log (x) x^3-36 e^x x^2-3 \log (x) x^2+3 x^2+42 \log (x) x-3 x+18 \log (x)-18\right )}{\left (-e^x x^4+e^x x^3+e^{2 x} x^3+6 e^x x^2-e^{2 x} x^2-6 e^{2 x} x+3 \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^x \left (3 \left (x^3+2 x^2-8 x-6\right ) \log (x)-\left (x^2-x-6\right ) \left (-e^x \left (x^2-x-6\right ) x^2+e^{2 x} \left (x^2-x-6\right ) x^2+3\right )\right )}{\left (e^x \left (e^x-x\right ) x \left (x^2-x-6\right )+3 \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^x x \left (x^2-x-6\right )}{e^x x^4-e^x x^3-e^{2 x} x^3-6 e^x x^2+e^{2 x} x^2+6 e^{2 x} x-3 \log (x)}-\frac {e^x \left (e^x x^7-3 e^x x^6-9 e^x x^5+23 e^x x^4+24 e^x x^3-6 \log (x) x^3-36 e^x x^2-3 \log (x) x^2+3 x^2+42 \log (x) x-3 x+18 \log (x)-18\right )}{\left (-e^x x^4+e^x x^3+e^{2 x} x^3+6 e^x x^2-e^{2 x} x^2-6 e^{2 x} x+3 \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^x \left (3 \left (x^3+2 x^2-8 x-6\right ) \log (x)-\left (x^2-x-6\right ) \left (-e^x \left (x^2-x-6\right ) x^2+e^{2 x} \left (x^2-x-6\right ) x^2+3\right )\right )}{\left (e^x \left (e^x-x\right ) x \left (x^2-x-6\right )+3 \log (x)\right )^2}dx\) |
Int[(E^x*(54 + 9*x - 9*x^2) + E^(3*x)*(-108*x^2 - 36*x^3 + 33*x^4 + 6*x^5 - 3*x^6) + E^(2*x)*(108*x^2 + 36*x^3 - 33*x^4 - 6*x^5 + 3*x^6) + E^x*(-54 - 72*x + 18*x^2 + 9*x^3)*Log[x])/(E^(4*x)*(36*x^2 + 12*x^3 - 11*x^4 - 2*x^ 5 + x^6) + E^(3*x)*(-72*x^3 - 24*x^4 + 22*x^5 + 4*x^6 - 2*x^7) + E^(2*x)*( 36*x^4 + 12*x^5 - 11*x^6 - 2*x^7 + x^8) + (E^(2*x)*(-36*x - 6*x^2 + 6*x^3) + E^x*(36*x^2 + 6*x^3 - 6*x^4))*Log[x] + 9*Log[x]^2),x]
3.11.36.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 2.07 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.81
| method | result | size |
| risch | \(-\frac {3 \left (x^{2}-x -6\right ) x \,{\mathrm e}^{x}}{{\mathrm e}^{x} x^{4}-{\mathrm e}^{2 x} x^{3}-{\mathrm e}^{x} x^{3}+{\mathrm e}^{2 x} x^{2}-6 \,{\mathrm e}^{x} x^{2}+6 x \,{\mathrm e}^{2 x}-3 \ln \left (x \right )}\) | \(65\) |
| parallelrisch | \(\frac {-9 \,{\mathrm e}^{x} x^{3}+9 \,{\mathrm e}^{x} x^{2}+54 \,{\mathrm e}^{x} x}{3 \,{\mathrm e}^{x} x^{4}-3 \,{\mathrm e}^{2 x} x^{3}-3 \,{\mathrm e}^{x} x^{3}+3 \,{\mathrm e}^{2 x} x^{2}-18 \,{\mathrm e}^{x} x^{2}+18 x \,{\mathrm e}^{2 x}-9 \ln \left (x \right )}\) | \(74\) |
int(((9*x^3+18*x^2-72*x-54)*exp(x)*ln(x)+(-3*x^6+6*x^5+33*x^4-36*x^3-108*x ^2)*exp(x)^3+(3*x^6-6*x^5-33*x^4+36*x^3+108*x^2)*exp(x)^2+(-9*x^2+9*x+54)* exp(x))/(9*ln(x)^2+((6*x^3-6*x^2-36*x)*exp(x)^2+(-6*x^4+6*x^3+36*x^2)*exp( x))*ln(x)+(x^6-2*x^5-11*x^4+12*x^3+36*x^2)*exp(x)^4+(-2*x^7+4*x^6+22*x^5-2 4*x^4-72*x^3)*exp(x)^3+(x^8-2*x^7-11*x^6+12*x^5+36*x^4)*exp(x)^2),x,method =_RETURNVERBOSE)
-3*(x^2-x-6)*x*exp(x)/(exp(x)*x^4-exp(x)^2*x^3-exp(x)*x^3+exp(x)^2*x^2-6*e xp(x)*x^2+6*x*exp(x)^2-3*ln(x))
Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.61 \[ \int \frac {e^x \left (54+9 x-9 x^2\right )+e^{3 x} \left (-108 x^2-36 x^3+33 x^4+6 x^5-3 x^6\right )+e^{2 x} \left (108 x^2+36 x^3-33 x^4-6 x^5+3 x^6\right )+e^x \left (-54-72 x+18 x^2+9 x^3\right ) \log (x)}{e^{4 x} \left (36 x^2+12 x^3-11 x^4-2 x^5+x^6\right )+e^{3 x} \left (-72 x^3-24 x^4+22 x^5+4 x^6-2 x^7\right )+e^{2 x} \left (36 x^4+12 x^5-11 x^6-2 x^7+x^8\right )+\left (e^{2 x} \left (-36 x-6 x^2+6 x^3\right )+e^x \left (36 x^2+6 x^3-6 x^4\right )\right ) \log (x)+9 \log ^2(x)} \, dx=\frac {3 \, {\left (x^{3} - x^{2} - 6 \, x\right )} e^{x}}{{\left (x^{3} - x^{2} - 6 \, x\right )} e^{\left (2 \, x\right )} - {\left (x^{4} - x^{3} - 6 \, x^{2}\right )} e^{x} + 3 \, \log \left (x\right )} \]
integrate(((9*x^3+18*x^2-72*x-54)*exp(x)*log(x)+(-3*x^6+6*x^5+33*x^4-36*x^ 3-108*x^2)*exp(x)^3+(3*x^6-6*x^5-33*x^4+36*x^3+108*x^2)*exp(x)^2+(-9*x^2+9 *x+54)*exp(x))/(9*log(x)^2+((6*x^3-6*x^2-36*x)*exp(x)^2+(-6*x^4+6*x^3+36*x ^2)*exp(x))*log(x)+(x^6-2*x^5-11*x^4+12*x^3+36*x^2)*exp(x)^4+(-2*x^7+4*x^6 +22*x^5-24*x^4-72*x^3)*exp(x)^3+(x^8-2*x^7-11*x^6+12*x^5+36*x^4)*exp(x)^2) ,x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (22) = 44\).
Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.42 \[ \int \frac {e^x \left (54+9 x-9 x^2\right )+e^{3 x} \left (-108 x^2-36 x^3+33 x^4+6 x^5-3 x^6\right )+e^{2 x} \left (108 x^2+36 x^3-33 x^4-6 x^5+3 x^6\right )+e^x \left (-54-72 x+18 x^2+9 x^3\right ) \log (x)}{e^{4 x} \left (36 x^2+12 x^3-11 x^4-2 x^5+x^6\right )+e^{3 x} \left (-72 x^3-24 x^4+22 x^5+4 x^6-2 x^7\right )+e^{2 x} \left (36 x^4+12 x^5-11 x^6-2 x^7+x^8\right )+\left (e^{2 x} \left (-36 x-6 x^2+6 x^3\right )+e^x \left (36 x^2+6 x^3-6 x^4\right )\right ) \log (x)+9 \log ^2(x)} \, dx=\frac {\left (3 x^{3} - 3 x^{2} - 18 x\right ) e^{x}}{\left (x^{3} - x^{2} - 6 x\right ) e^{2 x} + \left (- x^{4} + x^{3} + 6 x^{2}\right ) e^{x} + 3 \log {\left (x \right )}} \]
integrate(((9*x**3+18*x**2-72*x-54)*exp(x)*ln(x)+(-3*x**6+6*x**5+33*x**4-3 6*x**3-108*x**2)*exp(x)**3+(3*x**6-6*x**5-33*x**4+36*x**3+108*x**2)*exp(x) **2+(-9*x**2+9*x+54)*exp(x))/(9*ln(x)**2+((6*x**3-6*x**2-36*x)*exp(x)**2+( -6*x**4+6*x**3+36*x**2)*exp(x))*ln(x)+(x**6-2*x**5-11*x**4+12*x**3+36*x**2 )*exp(x)**4+(-2*x**7+4*x**6+22*x**5-24*x**4-72*x**3)*exp(x)**3+(x**8-2*x** 7-11*x**6+12*x**5+36*x**4)*exp(x)**2),x)
(3*x**3 - 3*x**2 - 18*x)*exp(x)/((x**3 - x**2 - 6*x)*exp(2*x) + (-x**4 + x **3 + 6*x**2)*exp(x) + 3*log(x))
Time = 0.30 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.61 \[ \int \frac {e^x \left (54+9 x-9 x^2\right )+e^{3 x} \left (-108 x^2-36 x^3+33 x^4+6 x^5-3 x^6\right )+e^{2 x} \left (108 x^2+36 x^3-33 x^4-6 x^5+3 x^6\right )+e^x \left (-54-72 x+18 x^2+9 x^3\right ) \log (x)}{e^{4 x} \left (36 x^2+12 x^3-11 x^4-2 x^5+x^6\right )+e^{3 x} \left (-72 x^3-24 x^4+22 x^5+4 x^6-2 x^7\right )+e^{2 x} \left (36 x^4+12 x^5-11 x^6-2 x^7+x^8\right )+\left (e^{2 x} \left (-36 x-6 x^2+6 x^3\right )+e^x \left (36 x^2+6 x^3-6 x^4\right )\right ) \log (x)+9 \log ^2(x)} \, dx=\frac {3 \, {\left (x^{3} - x^{2} - 6 \, x\right )} e^{x}}{{\left (x^{3} - x^{2} - 6 \, x\right )} e^{\left (2 \, x\right )} - {\left (x^{4} - x^{3} - 6 \, x^{2}\right )} e^{x} + 3 \, \log \left (x\right )} \]
integrate(((9*x^3+18*x^2-72*x-54)*exp(x)*log(x)+(-3*x^6+6*x^5+33*x^4-36*x^ 3-108*x^2)*exp(x)^3+(3*x^6-6*x^5-33*x^4+36*x^3+108*x^2)*exp(x)^2+(-9*x^2+9 *x+54)*exp(x))/(9*log(x)^2+((6*x^3-6*x^2-36*x)*exp(x)^2+(-6*x^4+6*x^3+36*x ^2)*exp(x))*log(x)+(x^6-2*x^5-11*x^4+12*x^3+36*x^2)*exp(x)^4+(-2*x^7+4*x^6 +22*x^5-24*x^4-72*x^3)*exp(x)^3+(x^8-2*x^7-11*x^6+12*x^5+36*x^4)*exp(x)^2) ,x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (32) = 64\).
Time = 0.34 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.00 \[ \int \frac {e^x \left (54+9 x-9 x^2\right )+e^{3 x} \left (-108 x^2-36 x^3+33 x^4+6 x^5-3 x^6\right )+e^{2 x} \left (108 x^2+36 x^3-33 x^4-6 x^5+3 x^6\right )+e^x \left (-54-72 x+18 x^2+9 x^3\right ) \log (x)}{e^{4 x} \left (36 x^2+12 x^3-11 x^4-2 x^5+x^6\right )+e^{3 x} \left (-72 x^3-24 x^4+22 x^5+4 x^6-2 x^7\right )+e^{2 x} \left (36 x^4+12 x^5-11 x^6-2 x^7+x^8\right )+\left (e^{2 x} \left (-36 x-6 x^2+6 x^3\right )+e^x \left (36 x^2+6 x^3-6 x^4\right )\right ) \log (x)+9 \log ^2(x)} \, dx=-\frac {3 \, {\left (x^{3} e^{x} - x^{2} e^{x} - 6 \, x e^{x}\right )}}{x^{4} e^{x} - x^{3} e^{\left (2 \, x\right )} - x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )} - 6 \, x^{2} e^{x} + 6 \, x e^{\left (2 \, x\right )} - 3 \, \log \left (x\right )} \]
integrate(((9*x^3+18*x^2-72*x-54)*exp(x)*log(x)+(-3*x^6+6*x^5+33*x^4-36*x^ 3-108*x^2)*exp(x)^3+(3*x^6-6*x^5-33*x^4+36*x^3+108*x^2)*exp(x)^2+(-9*x^2+9 *x+54)*exp(x))/(9*log(x)^2+((6*x^3-6*x^2-36*x)*exp(x)^2+(-6*x^4+6*x^3+36*x ^2)*exp(x))*log(x)+(x^6-2*x^5-11*x^4+12*x^3+36*x^2)*exp(x)^4+(-2*x^7+4*x^6 +22*x^5-24*x^4-72*x^3)*exp(x)^3+(x^8-2*x^7-11*x^6+12*x^5+36*x^4)*exp(x)^2) ,x, algorithm=\
-3*(x^3*e^x - x^2*e^x - 6*x*e^x)/(x^4*e^x - x^3*e^(2*x) - x^3*e^x + x^2*e^ (2*x) - 6*x^2*e^x + 6*x*e^(2*x) - 3*log(x))
Timed out. \[ \int \frac {e^x \left (54+9 x-9 x^2\right )+e^{3 x} \left (-108 x^2-36 x^3+33 x^4+6 x^5-3 x^6\right )+e^{2 x} \left (108 x^2+36 x^3-33 x^4-6 x^5+3 x^6\right )+e^x \left (-54-72 x+18 x^2+9 x^3\right ) \log (x)}{e^{4 x} \left (36 x^2+12 x^3-11 x^4-2 x^5+x^6\right )+e^{3 x} \left (-72 x^3-24 x^4+22 x^5+4 x^6-2 x^7\right )+e^{2 x} \left (36 x^4+12 x^5-11 x^6-2 x^7+x^8\right )+\left (e^{2 x} \left (-36 x-6 x^2+6 x^3\right )+e^x \left (36 x^2+6 x^3-6 x^4\right )\right ) \log (x)+9 \log ^2(x)} \, dx=\int \frac {{\mathrm {e}}^{2\,x}\,\left (3\,x^6-6\,x^5-33\,x^4+36\,x^3+108\,x^2\right )-{\mathrm {e}}^{3\,x}\,\left (3\,x^6-6\,x^5-33\,x^4+36\,x^3+108\,x^2\right )+{\mathrm {e}}^x\,\left (-9\,x^2+9\,x+54\right )-{\mathrm {e}}^x\,\ln \left (x\right )\,\left (-9\,x^3-18\,x^2+72\,x+54\right )}{9\,{\ln \left (x\right )}^2+\left ({\mathrm {e}}^x\,\left (-6\,x^4+6\,x^3+36\,x^2\right )-{\mathrm {e}}^{2\,x}\,\left (-6\,x^3+6\,x^2+36\,x\right )\right )\,\ln \left (x\right )-{\mathrm {e}}^{3\,x}\,\left (2\,x^7-4\,x^6-22\,x^5+24\,x^4+72\,x^3\right )+{\mathrm {e}}^{4\,x}\,\left (x^6-2\,x^5-11\,x^4+12\,x^3+36\,x^2\right )+{\mathrm {e}}^{2\,x}\,\left (x^8-2\,x^7-11\,x^6+12\,x^5+36\,x^4\right )} \,d x \]
int((exp(2*x)*(108*x^2 + 36*x^3 - 33*x^4 - 6*x^5 + 3*x^6) - exp(3*x)*(108* x^2 + 36*x^3 - 33*x^4 - 6*x^5 + 3*x^6) + exp(x)*(9*x - 9*x^2 + 54) - exp(x )*log(x)*(72*x - 18*x^2 - 9*x^3 + 54))/(9*log(x)^2 - log(x)*(exp(2*x)*(36* x + 6*x^2 - 6*x^3) - exp(x)*(36*x^2 + 6*x^3 - 6*x^4)) - exp(3*x)*(72*x^3 + 24*x^4 - 22*x^5 - 4*x^6 + 2*x^7) + exp(4*x)*(36*x^2 + 12*x^3 - 11*x^4 - 2 *x^5 + x^6) + exp(2*x)*(36*x^4 + 12*x^5 - 11*x^6 - 2*x^7 + x^8)),x)
int((exp(2*x)*(108*x^2 + 36*x^3 - 33*x^4 - 6*x^5 + 3*x^6) - exp(3*x)*(108* x^2 + 36*x^3 - 33*x^4 - 6*x^5 + 3*x^6) + exp(x)*(9*x - 9*x^2 + 54) - exp(x )*log(x)*(72*x - 18*x^2 - 9*x^3 + 54))/(9*log(x)^2 - log(x)*(exp(2*x)*(36* x + 6*x^2 - 6*x^3) - exp(x)*(36*x^2 + 6*x^3 - 6*x^4)) - exp(3*x)*(72*x^3 + 24*x^4 - 22*x^5 - 4*x^6 + 2*x^7) + exp(4*x)*(36*x^2 + 12*x^3 - 11*x^4 - 2 *x^5 + x^6) + exp(2*x)*(36*x^4 + 12*x^5 - 11*x^6 - 2*x^7 + x^8)), x)