Integrand size = 173, antiderivative size = 30 \[ \int \frac {e^x \left (-40 x+192 x^2-156 x^3-20 x^4+18 x^5\right )+e^{2 x} \left (-40 x^2+36 x^3+56 x^4-42 x^5-16 x^6+6 x^7\right )+\left (-40+192 x-156 x^2-20 x^3+18 x^4+e^x \left (-40 x+36 x^2+56 x^3-42 x^4-16 x^5+6 x^6\right )\right ) \log \left (\frac {10 x-24 x^2+17 x^3-3 x^4}{2+x}\right )}{-20 x+38 x^2-10 x^3-11 x^4+3 x^5} \, dx=\left (e^x x+\log \left (\left (x-x^2\right ) \left (5+3 x-\frac {25 x}{2+x}\right )\right )\right )^2 \] Output:
(ln((-x^2+x)*(3*x+5-x/(2/25+1/25*x)))+exp(x)*x)^2
Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {e^x \left (-40 x+192 x^2-156 x^3-20 x^4+18 x^5\right )+e^{2 x} \left (-40 x^2+36 x^3+56 x^4-42 x^5-16 x^6+6 x^7\right )+\left (-40+192 x-156 x^2-20 x^3+18 x^4+e^x \left (-40 x+36 x^2+56 x^3-42 x^4-16 x^5+6 x^6\right )\right ) \log \left (\frac {10 x-24 x^2+17 x^3-3 x^4}{2+x}\right )}{-20 x+38 x^2-10 x^3-11 x^4+3 x^5} \, dx=\left (e^x x+\log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )\right )^2 \] Input:
Integrate[(E^x*(-40*x + 192*x^2 - 156*x^3 - 20*x^4 + 18*x^5) + E^(2*x)*(-4 0*x^2 + 36*x^3 + 56*x^4 - 42*x^5 - 16*x^6 + 6*x^7) + (-40 + 192*x - 156*x^ 2 - 20*x^3 + 18*x^4 + E^x*(-40*x + 36*x^2 + 56*x^3 - 42*x^4 - 16*x^5 + 6*x ^6))*Log[(10*x - 24*x^2 + 17*x^3 - 3*x^4)/(2 + x)])/(-20*x + 38*x^2 - 10*x ^3 - 11*x^4 + 3*x^5),x]
Output:
(E^x*x + Log[(x*(10 - 24*x + 17*x^2 - 3*x^3))/(2 + x)])^2
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {e^x \left (18 x^5-20 x^4-156 x^3+192 x^2-40 x\right )+\left (18 x^4-20 x^3-156 x^2+e^x \left (6 x^6-16 x^5-42 x^4+56 x^3+36 x^2-40 x\right )+192 x-40\right ) \log \left (\frac {-3 x^4+17 x^3-24 x^2+10 x}{x+2}\right )+e^{2 x} \left (6 x^7-16 x^6-42 x^5+56 x^4+36 x^3-40 x^2\right )}{3 x^5-11 x^4-10 x^3+38 x^2-20 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {e^x \left (18 x^5-20 x^4-156 x^3+192 x^2-40 x\right )+\left (18 x^4-20 x^3-156 x^2+e^x \left (6 x^6-16 x^5-42 x^4+56 x^3+36 x^2-40 x\right )+192 x-40\right ) \log \left (\frac {-3 x^4+17 x^3-24 x^2+10 x}{x+2}\right )+e^{2 x} \left (6 x^7-16 x^6-42 x^5+56 x^4+36 x^3-40 x^2\right )}{x \left (3 x^4-11 x^3-10 x^2+38 x-20\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {e^x \left (18 x^5-20 x^4-156 x^3+192 x^2-40 x\right )+\left (18 x^4-20 x^3-156 x^2+e^x \left (6 x^6-16 x^5-42 x^4+56 x^3+36 x^2-40 x\right )+192 x-40\right ) \log \left (\frac {-3 x^4+17 x^3-24 x^2+10 x}{x+2}\right )+e^{2 x} \left (6 x^7-16 x^6-42 x^5+56 x^4+36 x^3-40 x^2\right )}{3 (x-1) x}-\frac {e^x \left (18 x^5-20 x^4-156 x^3+192 x^2-40 x\right )+\left (18 x^4-20 x^3-156 x^2+e^x \left (6 x^6-16 x^5-42 x^4+56 x^3+36 x^2-40 x\right )+192 x-40\right ) \log \left (\frac {-3 x^4+17 x^3-24 x^2+10 x}{x+2}\right )+e^{2 x} \left (6 x^7-16 x^6-42 x^5+56 x^4+36 x^3-40 x^2\right )}{150 x (x+2)}+\frac {(51 x-190) \left (e^x \left (18 x^5-20 x^4-156 x^3+192 x^2-40 x\right )+\left (18 x^4-20 x^3-156 x^2+e^x \left (6 x^6-16 x^5-42 x^4+56 x^3+36 x^2-40 x\right )+192 x-40\right ) \log \left (\frac {-3 x^4+17 x^3-24 x^2+10 x}{x+2}\right )+e^{2 x} \left (6 x^7-16 x^6-42 x^5+56 x^4+36 x^3-40 x^2\right )\right )}{50 x \left (3 x^2-14 x+10\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (-3 e^x x^6+8 e^x x^5-\left (9-21 e^x\right ) x^4-2 \left (14 e^x-5\right ) x^3-6 \left (3 e^x-13\right ) x^2-\left (96-20 e^x\right ) x+20\right ) \left (\log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )+e^x x\right )}{(1-x) x (x+2) \left (3 x^2-14 x+10\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {\left (-3 e^x x^6+8 e^x x^5-3 \left (3-7 e^x\right ) x^4+2 \left (5-14 e^x\right ) x^3+6 \left (13-3 e^x\right ) x^2-4 \left (24-5 e^x\right ) x+20\right ) \left (e^x x+\log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )\right )}{(1-x) x (x+2) \left (3 x^2-14 x+10\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle 2 \int \left (\frac {9 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {10 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+e^{2 x} (x+1) x-\frac {78 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {96 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {e^x \left (3 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^5-8 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^4+9 x^4-21 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3-10 x^3+28 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2-78 x^2+18 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x+96 x-20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )-20\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\left (-3 e^x x^6+8 e^x x^5-\left (9-21 e^x\right ) x^4-2 \left (-5+14 e^x\right ) x^3-6 \left (-13+3 e^x\right ) x^2-\left (96-20 e^x\right ) x+20\right ) \left (e^x x+\log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )\right )}{(1-x) x (x+2) \left (3 x^2-14 x+10\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle 2 \int \left (\frac {9 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {10 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+e^{2 x} (x+1) x-\frac {78 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {96 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {e^x \left (3 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^5-8 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^4+9 x^4-21 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3-10 x^3+28 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2-78 x^2+18 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x+96 x-20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )-20\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\left (-3 e^x x^6+8 e^x x^5-\left (9-21 e^x\right ) x^4-2 \left (-5+14 e^x\right ) x^3-6 \left (-13+3 e^x\right ) x^2-\left (96-20 e^x\right ) x+20\right ) \left (e^x x+\log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )\right )}{(1-x) x (x+2) \left (3 x^2-14 x+10\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle 2 \int \left (\frac {9 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {10 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+e^{2 x} (x+1) x-\frac {78 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {96 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {e^x \left (3 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^5-8 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^4+9 x^4-21 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3-10 x^3+28 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2-78 x^2+18 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x+96 x-20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )-20\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\left (-3 e^x x^6+8 e^x x^5-\left (9-21 e^x\right ) x^4-2 \left (-5+14 e^x\right ) x^3-6 \left (-13+3 e^x\right ) x^2-\left (96-20 e^x\right ) x+20\right ) \left (e^x x+\log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )\right )}{(1-x) x (x+2) \left (3 x^2-14 x+10\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle 2 \int \left (\frac {9 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {10 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+e^{2 x} (x+1) x-\frac {78 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {96 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {e^x \left (3 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^5-8 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^4+9 x^4-21 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3-10 x^3+28 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2-78 x^2+18 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x+96 x-20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )-20\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\left (-3 e^x x^6+8 e^x x^5-\left (9-21 e^x\right ) x^4-2 \left (-5+14 e^x\right ) x^3-6 \left (-13+3 e^x\right ) x^2-\left (96-20 e^x\right ) x+20\right ) \left (e^x x+\log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )\right )}{(1-x) x (x+2) \left (3 x^2-14 x+10\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle 2 \int \left (\frac {9 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {10 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+e^{2 x} (x+1) x-\frac {78 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {96 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {e^x \left (3 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^5-8 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^4+9 x^4-21 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3-10 x^3+28 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2-78 x^2+18 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x+96 x-20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )-20\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\left (-3 e^x x^6+8 e^x x^5-\left (9-21 e^x\right ) x^4-2 \left (-5+14 e^x\right ) x^3-6 \left (-13+3 e^x\right ) x^2-\left (96-20 e^x\right ) x+20\right ) \left (e^x x+\log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )\right )}{(1-x) x (x+2) \left (3 x^2-14 x+10\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle 2 \int \left (\frac {9 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {10 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+e^{2 x} (x+1) x-\frac {78 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {96 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {e^x \left (3 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^5-8 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^4+9 x^4-21 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3-10 x^3+28 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2-78 x^2+18 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x+96 x-20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )-20\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\left (-3 e^x x^6+8 e^x x^5-\left (9-21 e^x\right ) x^4-2 \left (-5+14 e^x\right ) x^3-6 \left (-13+3 e^x\right ) x^2-\left (96-20 e^x\right ) x+20\right ) \left (e^x x+\log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )\right )}{(1-x) x (x+2) \left (3 x^2-14 x+10\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle 2 \int \left (\frac {9 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {10 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+e^{2 x} (x+1) x-\frac {78 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {96 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {e^x \left (3 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^5-8 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^4+9 x^4-21 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3-10 x^3+28 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2-78 x^2+18 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x+96 x-20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )-20\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\left (-3 e^x x^6+8 e^x x^5-\left (9-21 e^x\right ) x^4-2 \left (-5+14 e^x\right ) x^3-6 \left (-13+3 e^x\right ) x^2-\left (96-20 e^x\right ) x+20\right ) \left (e^x x+\log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )\right )}{(1-x) x (x+2) \left (3 x^2-14 x+10\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle 2 \int \left (\frac {9 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {10 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+e^{2 x} (x+1) x-\frac {78 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {96 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {e^x \left (3 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^5-8 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^4+9 x^4-21 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3-10 x^3+28 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2-78 x^2+18 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x+96 x-20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )-20\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\left (-3 e^x x^6+8 e^x x^5-\left (9-21 e^x\right ) x^4-2 \left (-5+14 e^x\right ) x^3-6 \left (-13+3 e^x\right ) x^2-\left (96-20 e^x\right ) x+20\right ) \left (e^x x+\log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )\right )}{(1-x) x (x+2) \left (3 x^2-14 x+10\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle 2 \int \left (\frac {9 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {10 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+e^{2 x} (x+1) x-\frac {78 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {96 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {e^x \left (3 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^5-8 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^4+9 x^4-21 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3-10 x^3+28 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2-78 x^2+18 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x+96 x-20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )-20\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\left (-3 e^x x^6+8 e^x x^5-\left (9-21 e^x\right ) x^4-2 \left (-5+14 e^x\right ) x^3-6 \left (-13+3 e^x\right ) x^2-\left (96-20 e^x\right ) x+20\right ) \left (e^x x+\log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )\right )}{(1-x) x (x+2) \left (3 x^2-14 x+10\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle 2 \int \left (\frac {9 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {10 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+e^{2 x} (x+1) x-\frac {78 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {96 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {e^x \left (3 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^5-8 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^4+9 x^4-21 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3-10 x^3+28 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2-78 x^2+18 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x+96 x-20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )-20\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\left (-3 e^x x^6+8 e^x x^5-\left (9-21 e^x\right ) x^4-2 \left (-5+14 e^x\right ) x^3-6 \left (-13+3 e^x\right ) x^2-\left (96-20 e^x\right ) x+20\right ) \left (e^x x+\log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )\right )}{(1-x) x (x+2) \left (3 x^2-14 x+10\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle 2 \int \left (\frac {9 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {10 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+e^{2 x} (x+1) x-\frac {78 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {96 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {e^x \left (3 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^5-8 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^4+9 x^4-21 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3-10 x^3+28 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2-78 x^2+18 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x+96 x-20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )-20\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\left (-3 e^x x^6+8 e^x x^5-\left (9-21 e^x\right ) x^4-2 \left (-5+14 e^x\right ) x^3-6 \left (-13+3 e^x\right ) x^2-\left (96-20 e^x\right ) x+20\right ) \left (e^x x+\log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )\right )}{(1-x) x (x+2) \left (3 x^2-14 x+10\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle 2 \int \left (\frac {9 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {10 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+e^{2 x} (x+1) x-\frac {78 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {96 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {e^x \left (3 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^5-8 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^4+9 x^4-21 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3-10 x^3+28 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2-78 x^2+18 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x+96 x-20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )-20\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\left (-3 e^x x^6+8 e^x x^5-\left (9-21 e^x\right ) x^4-2 \left (-5+14 e^x\right ) x^3-6 \left (-13+3 e^x\right ) x^2-\left (96-20 e^x\right ) x+20\right ) \left (e^x x+\log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )\right )}{(1-x) x (x+2) \left (3 x^2-14 x+10\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle 2 \int \left (\frac {9 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {10 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+e^{2 x} (x+1) x-\frac {78 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {96 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}+\frac {e^x \left (3 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^5-8 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^4+9 x^4-21 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^3-10 x^3+28 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x^2-78 x^2+18 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right ) x+96 x-20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )-20\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right )}-\frac {20 \log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )}{(x-1) (x+2) \left (3 x^2-14 x+10\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\left (-3 e^x x^6+8 e^x x^5-\left (9-21 e^x\right ) x^4-2 \left (-5+14 e^x\right ) x^3-6 \left (-13+3 e^x\right ) x^2-\left (96-20 e^x\right ) x+20\right ) \left (e^x x+\log \left (\frac {x \left (-3 x^3+17 x^2-24 x+10\right )}{x+2}\right )\right )}{(1-x) x (x+2) \left (3 x^2-14 x+10\right )}dx\) |
Input:
Int[(E^x*(-40*x + 192*x^2 - 156*x^3 - 20*x^4 + 18*x^5) + E^(2*x)*(-40*x^2 + 36*x^3 + 56*x^4 - 42*x^5 - 16*x^6 + 6*x^7) + (-40 + 192*x - 156*x^2 - 20 *x^3 + 18*x^4 + E^x*(-40*x + 36*x^2 + 56*x^3 - 42*x^4 - 16*x^5 + 6*x^6))*L og[(10*x - 24*x^2 + 17*x^3 - 3*x^4)/(2 + x)])/(-20*x + 38*x^2 - 10*x^3 - 1 1*x^4 + 3*x^5),x]
Output:
$Aborted
\[\int \frac {\left (\left (6 x^{6}-16 x^{5}-42 x^{4}+56 x^{3}+36 x^{2}-40 x \right ) {\mathrm e}^{x}+18 x^{4}-20 x^{3}-156 x^{2}+192 x -40\right ) \ln \left (\frac {-3 x^{4}+17 x^{3}-24 x^{2}+10 x}{2+x}\right )+\left (6 x^{7}-16 x^{6}-42 x^{5}+56 x^{4}+36 x^{3}-40 x^{2}\right ) {\mathrm e}^{2 x}+\left (18 x^{5}-20 x^{4}-156 x^{3}+192 x^{2}-40 x \right ) {\mathrm e}^{x}}{3 x^{5}-11 x^{4}-10 x^{3}+38 x^{2}-20 x}d x\]
Input:
int((((6*x^6-16*x^5-42*x^4+56*x^3+36*x^2-40*x)*exp(x)+18*x^4-20*x^3-156*x^ 2+192*x-40)*ln((-3*x^4+17*x^3-24*x^2+10*x)/(2+x))+(6*x^7-16*x^6-42*x^5+56* x^4+36*x^3-40*x^2)*exp(x)^2+(18*x^5-20*x^4-156*x^3+192*x^2-40*x)*exp(x))/( 3*x^5-11*x^4-10*x^3+38*x^2-20*x),x)
Output:
int((((6*x^6-16*x^5-42*x^4+56*x^3+36*x^2-40*x)*exp(x)+18*x^4-20*x^3-156*x^ 2+192*x-40)*ln((-3*x^4+17*x^3-24*x^2+10*x)/(2+x))+(6*x^7-16*x^6-42*x^5+56* x^4+36*x^3-40*x^2)*exp(x)^2+(18*x^5-20*x^4-156*x^3+192*x^2-40*x)*exp(x))/( 3*x^5-11*x^4-10*x^3+38*x^2-20*x),x)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (30) = 60\).
Time = 0.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.33 \[ \int \frac {e^x \left (-40 x+192 x^2-156 x^3-20 x^4+18 x^5\right )+e^{2 x} \left (-40 x^2+36 x^3+56 x^4-42 x^5-16 x^6+6 x^7\right )+\left (-40+192 x-156 x^2-20 x^3+18 x^4+e^x \left (-40 x+36 x^2+56 x^3-42 x^4-16 x^5+6 x^6\right )\right ) \log \left (\frac {10 x-24 x^2+17 x^3-3 x^4}{2+x}\right )}{-20 x+38 x^2-10 x^3-11 x^4+3 x^5} \, dx=x^{2} e^{\left (2 \, x\right )} + 2 \, x e^{x} \log \left (-\frac {3 \, x^{4} - 17 \, x^{3} + 24 \, x^{2} - 10 \, x}{x + 2}\right ) + \log \left (-\frac {3 \, x^{4} - 17 \, x^{3} + 24 \, x^{2} - 10 \, x}{x + 2}\right )^{2} \] Input:
integrate((((6*x^6-16*x^5-42*x^4+56*x^3+36*x^2-40*x)*exp(x)+18*x^4-20*x^3- 156*x^2+192*x-40)*log((-3*x^4+17*x^3-24*x^2+10*x)/(2+x))+(6*x^7-16*x^6-42* x^5+56*x^4+36*x^3-40*x^2)*exp(x)^2+(18*x^5-20*x^4-156*x^3+192*x^2-40*x)*ex p(x))/(3*x^5-11*x^4-10*x^3+38*x^2-20*x),x, algorithm="fricas")
Output:
x^2*e^(2*x) + 2*x*e^x*log(-(3*x^4 - 17*x^3 + 24*x^2 - 10*x)/(x + 2)) + log (-(3*x^4 - 17*x^3 + 24*x^2 - 10*x)/(x + 2))^2
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (26) = 52\).
Time = 0.59 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.10 \[ \int \frac {e^x \left (-40 x+192 x^2-156 x^3-20 x^4+18 x^5\right )+e^{2 x} \left (-40 x^2+36 x^3+56 x^4-42 x^5-16 x^6+6 x^7\right )+\left (-40+192 x-156 x^2-20 x^3+18 x^4+e^x \left (-40 x+36 x^2+56 x^3-42 x^4-16 x^5+6 x^6\right )\right ) \log \left (\frac {10 x-24 x^2+17 x^3-3 x^4}{2+x}\right )}{-20 x+38 x^2-10 x^3-11 x^4+3 x^5} \, dx=x^{2} e^{2 x} + 2 x e^{x} \log {\left (\frac {- 3 x^{4} + 17 x^{3} - 24 x^{2} + 10 x}{x + 2} \right )} + \log {\left (\frac {- 3 x^{4} + 17 x^{3} - 24 x^{2} + 10 x}{x + 2} \right )}^{2} \] Input:
integrate((((6*x**6-16*x**5-42*x**4+56*x**3+36*x**2-40*x)*exp(x)+18*x**4-2 0*x**3-156*x**2+192*x-40)*ln((-3*x**4+17*x**3-24*x**2+10*x)/(2+x))+(6*x**7 -16*x**6-42*x**5+56*x**4+36*x**3-40*x**2)*exp(x)**2+(18*x**5-20*x**4-156*x **3+192*x**2-40*x)*exp(x))/(3*x**5-11*x**4-10*x**3+38*x**2-20*x),x)
Output:
x**2*exp(2*x) + 2*x*exp(x)*log((-3*x**4 + 17*x**3 - 24*x**2 + 10*x)/(x + 2 )) + log((-3*x**4 + 17*x**3 - 24*x**2 + 10*x)/(x + 2))**2
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (30) = 60\).
Time = 0.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.50 \[ \int \frac {e^x \left (-40 x+192 x^2-156 x^3-20 x^4+18 x^5\right )+e^{2 x} \left (-40 x^2+36 x^3+56 x^4-42 x^5-16 x^6+6 x^7\right )+\left (-40+192 x-156 x^2-20 x^3+18 x^4+e^x \left (-40 x+36 x^2+56 x^3-42 x^4-16 x^5+6 x^6\right )\right ) \log \left (\frac {10 x-24 x^2+17 x^3-3 x^4}{2+x}\right )}{-20 x+38 x^2-10 x^3-11 x^4+3 x^5} \, dx=x^{2} e^{\left (2 \, x\right )} + 2 \, x e^{x} \log \left (x\right ) + 2 \, {\left (x e^{x} - \log \left (x + 2\right ) + \log \left (x - 1\right ) + \log \left (x\right )\right )} \log \left (-3 \, x^{2} + 14 \, x - 10\right ) + \log \left (-3 \, x^{2} + 14 \, x - 10\right )^{2} - 2 \, {\left (x e^{x} + \log \left (x - 1\right ) + \log \left (x\right )\right )} \log \left (x + 2\right ) + \log \left (x + 2\right )^{2} + 2 \, {\left (x e^{x} + \log \left (x\right )\right )} \log \left (x - 1\right ) + \log \left (x - 1\right )^{2} + \log \left (x\right )^{2} \] Input:
integrate((((6*x^6-16*x^5-42*x^4+56*x^3+36*x^2-40*x)*exp(x)+18*x^4-20*x^3- 156*x^2+192*x-40)*log((-3*x^4+17*x^3-24*x^2+10*x)/(2+x))+(6*x^7-16*x^6-42* x^5+56*x^4+36*x^3-40*x^2)*exp(x)^2+(18*x^5-20*x^4-156*x^3+192*x^2-40*x)*ex p(x))/(3*x^5-11*x^4-10*x^3+38*x^2-20*x),x, algorithm="maxima")
Output:
x^2*e^(2*x) + 2*x*e^x*log(x) + 2*(x*e^x - log(x + 2) + log(x - 1) + log(x) )*log(-3*x^2 + 14*x - 10) + log(-3*x^2 + 14*x - 10)^2 - 2*(x*e^x + log(x - 1) + log(x))*log(x + 2) + log(x + 2)^2 + 2*(x*e^x + log(x))*log(x - 1) + log(x - 1)^2 + log(x)^2
\[ \int \frac {e^x \left (-40 x+192 x^2-156 x^3-20 x^4+18 x^5\right )+e^{2 x} \left (-40 x^2+36 x^3+56 x^4-42 x^5-16 x^6+6 x^7\right )+\left (-40+192 x-156 x^2-20 x^3+18 x^4+e^x \left (-40 x+36 x^2+56 x^3-42 x^4-16 x^5+6 x^6\right )\right ) \log \left (\frac {10 x-24 x^2+17 x^3-3 x^4}{2+x}\right )}{-20 x+38 x^2-10 x^3-11 x^4+3 x^5} \, dx=\int { \frac {2 \, {\left ({\left (3 \, x^{7} - 8 \, x^{6} - 21 \, x^{5} + 28 \, x^{4} + 18 \, x^{3} - 20 \, x^{2}\right )} e^{\left (2 \, x\right )} + {\left (9 \, x^{5} - 10 \, x^{4} - 78 \, x^{3} + 96 \, x^{2} - 20 \, x\right )} e^{x} + {\left (9 \, x^{4} - 10 \, x^{3} - 78 \, x^{2} + {\left (3 \, x^{6} - 8 \, x^{5} - 21 \, x^{4} + 28 \, x^{3} + 18 \, x^{2} - 20 \, x\right )} e^{x} + 96 \, x - 20\right )} \log \left (-\frac {3 \, x^{4} - 17 \, x^{3} + 24 \, x^{2} - 10 \, x}{x + 2}\right )\right )}}{3 \, x^{5} - 11 \, x^{4} - 10 \, x^{3} + 38 \, x^{2} - 20 \, x} \,d x } \] Input:
integrate((((6*x^6-16*x^5-42*x^4+56*x^3+36*x^2-40*x)*exp(x)+18*x^4-20*x^3- 156*x^2+192*x-40)*log((-3*x^4+17*x^3-24*x^2+10*x)/(2+x))+(6*x^7-16*x^6-42* x^5+56*x^4+36*x^3-40*x^2)*exp(x)^2+(18*x^5-20*x^4-156*x^3+192*x^2-40*x)*ex p(x))/(3*x^5-11*x^4-10*x^3+38*x^2-20*x),x, algorithm="giac")
Output:
integrate(2*((3*x^7 - 8*x^6 - 21*x^5 + 28*x^4 + 18*x^3 - 20*x^2)*e^(2*x) + (9*x^5 - 10*x^4 - 78*x^3 + 96*x^2 - 20*x)*e^x + (9*x^4 - 10*x^3 - 78*x^2 + (3*x^6 - 8*x^5 - 21*x^4 + 28*x^3 + 18*x^2 - 20*x)*e^x + 96*x - 20)*log(- (3*x^4 - 17*x^3 + 24*x^2 - 10*x)/(x + 2)))/(3*x^5 - 11*x^4 - 10*x^3 + 38*x ^2 - 20*x), x)
Time = 3.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {e^x \left (-40 x+192 x^2-156 x^3-20 x^4+18 x^5\right )+e^{2 x} \left (-40 x^2+36 x^3+56 x^4-42 x^5-16 x^6+6 x^7\right )+\left (-40+192 x-156 x^2-20 x^3+18 x^4+e^x \left (-40 x+36 x^2+56 x^3-42 x^4-16 x^5+6 x^6\right )\right ) \log \left (\frac {10 x-24 x^2+17 x^3-3 x^4}{2+x}\right )}{-20 x+38 x^2-10 x^3-11 x^4+3 x^5} \, dx={\left (\ln \left (\frac {-3\,x^4+17\,x^3-24\,x^2+10\,x}{x+2}\right )+x\,{\mathrm {e}}^x\right )}^2 \] Input:
int((exp(2*x)*(40*x^2 - 36*x^3 - 56*x^4 + 42*x^5 + 16*x^6 - 6*x^7) + exp(x )*(40*x - 192*x^2 + 156*x^3 + 20*x^4 - 18*x^5) + log((10*x - 24*x^2 + 17*x ^3 - 3*x^4)/(x + 2))*(exp(x)*(40*x - 36*x^2 - 56*x^3 + 42*x^4 + 16*x^5 - 6 *x^6) - 192*x + 156*x^2 + 20*x^3 - 18*x^4 + 40))/(20*x - 38*x^2 + 10*x^3 + 11*x^4 - 3*x^5),x)
Output:
(log((10*x - 24*x^2 + 17*x^3 - 3*x^4)/(x + 2)) + x*exp(x))^2
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.33 \[ \int \frac {e^x \left (-40 x+192 x^2-156 x^3-20 x^4+18 x^5\right )+e^{2 x} \left (-40 x^2+36 x^3+56 x^4-42 x^5-16 x^6+6 x^7\right )+\left (-40+192 x-156 x^2-20 x^3+18 x^4+e^x \left (-40 x+36 x^2+56 x^3-42 x^4-16 x^5+6 x^6\right )\right ) \log \left (\frac {10 x-24 x^2+17 x^3-3 x^4}{2+x}\right )}{-20 x+38 x^2-10 x^3-11 x^4+3 x^5} \, dx=e^{2 x} x^{2}+2 e^{x} \mathrm {log}\left (\frac {-3 x^{4}+17 x^{3}-24 x^{2}+10 x}{x +2}\right ) x +\mathrm {log}\left (\frac {-3 x^{4}+17 x^{3}-24 x^{2}+10 x}{x +2}\right )^{2} \] Input:
int((((6*x^6-16*x^5-42*x^4+56*x^3+36*x^2-40*x)*exp(x)+18*x^4-20*x^3-156*x^ 2+192*x-40)*log((-3*x^4+17*x^3-24*x^2+10*x)/(2+x))+(6*x^7-16*x^6-42*x^5+56 *x^4+36*x^3-40*x^2)*exp(x)^2+(18*x^5-20*x^4-156*x^3+192*x^2-40*x)*exp(x))/ (3*x^5-11*x^4-10*x^3+38*x^2-20*x),x)
Output:
e**(2*x)*x**2 + 2*e**x*log(( - 3*x**4 + 17*x**3 - 24*x**2 + 10*x)/(x + 2)) *x + log(( - 3*x**4 + 17*x**3 - 24*x**2 + 10*x)/(x + 2))**2