Integrand size = 185, antiderivative size = 28 \[ \int \frac {(-128+320 x) \log ^2(9)+e^{x^2} \left (-20 x \log (9)+\left (64 x^3-160 x^4\right ) \log ^2(9)\right )+e^{x^2} \left (-8+20 x+16 x^2-40 x^3\right ) \log (9) \log (2-5 x)}{(-512+1280 x) \log ^2(9)+e^{x^2} \left (-256 x+640 x^2\right ) \log ^2(9)+e^{2 x^2} \left (-32 x^2+80 x^3\right ) \log ^2(9)+\left (e^{x^2} (-64+160 x) \log (9)+e^{2 x^2} \left (-16 x+40 x^2\right ) \log (9)\right ) \log (2-5 x)+e^{2 x^2} (-2+5 x) \log ^2(2-5 x)} \, dx=\frac {x}{4+e^{x^2} \left (x+\frac {\log (2-5 x)}{4 \log (9)}\right )} \]
Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {(-128+320 x) \log ^2(9)+e^{x^2} \left (-20 x \log (9)+\left (64 x^3-160 x^4\right ) \log ^2(9)\right )+e^{x^2} \left (-8+20 x+16 x^2-40 x^3\right ) \log (9) \log (2-5 x)}{(-512+1280 x) \log ^2(9)+e^{x^2} \left (-256 x+640 x^2\right ) \log ^2(9)+e^{2 x^2} \left (-32 x^2+80 x^3\right ) \log ^2(9)+\left (e^{x^2} (-64+160 x) \log (9)+e^{2 x^2} \left (-16 x+40 x^2\right ) \log (9)\right ) \log (2-5 x)+e^{2 x^2} (-2+5 x) \log ^2(2-5 x)} \, dx=\frac {4 x \log (9)}{16 \log (9)+4 e^{x^2} x \log (9)+e^{x^2} \log (2-5 x)} \]
Integrate[((-128 + 320*x)*Log[9]^2 + E^x^2*(-20*x*Log[9] + (64*x^3 - 160*x ^4)*Log[9]^2) + E^x^2*(-8 + 20*x + 16*x^2 - 40*x^3)*Log[9]*Log[2 - 5*x])/( (-512 + 1280*x)*Log[9]^2 + E^x^2*(-256*x + 640*x^2)*Log[9]^2 + E^(2*x^2)*( -32*x^2 + 80*x^3)*Log[9]^2 + (E^x^2*(-64 + 160*x)*Log[9] + E^(2*x^2)*(-16* x + 40*x^2)*Log[9])*Log[2 - 5*x] + E^(2*x^2)*(-2 + 5*x)*Log[2 - 5*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.
| \(\displaystyle \int \frac {e^{x^2} \left (-40 x^3+16 x^2+20 x-8\right ) \log (9) \log (2-5 x)+e^{x^2} \left (\left (64 x^3-160 x^4\right ) \log ^2(9)-20 x \log (9)\right )+(320 x-128) \log ^2(9)}{e^{2 x^2} (5 x-2) \log ^2(2-5 x)+e^{x^2} \left (640 x^2-256 x\right ) \log ^2(9)+\left (e^{x^2} (160 x-64) \log (9)+e^{2 x^2} \left (40 x^2-16 x\right ) \log (9)\right ) \log (2-5 x)+e^{2 x^2} \left (80 x^3-32 x^2\right ) \log ^2(9)+(1280 x-512) \log ^2(9)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {4 \log (9) \left (e^{x^2} x \left (40 x^3 \log (9)-16 x^2 \log (9)+5\right )+e^{x^2} \left (10 x^3-4 x^2-5 x+2\right ) \log (2-5 x)+16 (2-5 x) \log (9)\right )}{(2-5 x) \left (4 \left (e^{x^2} x+4\right ) \log (9)+e^{x^2} \log (2-5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \log (9) \int \frac {16 \log (9) (2-5 x)+e^{x^2} x \left (40 \log (9) x^3-16 \log (9) x^2+5\right )+e^{x^2} \left (10 x^3-4 x^2-5 x+2\right ) \log (2-5 x)}{(2-5 x) \left (4 \log (9) \left (e^{x^2} x+4\right )+e^{x^2} \log (2-5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \log (9) \int \left (\frac {16 x \log (9) \left (-40 \log (9) x^3-10 \log (2-5 x) x^2+16 \log (9) x^2+4 \log (2-5 x) x-20 \log (9) x-5 \left (1-\frac {8 \log (9)}{5}\right )\right )}{(2-5 x) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )^2}-\frac {40 \log (9) x^4+10 \log (2-5 x) x^3-16 \log (9) x^3-4 \log (2-5 x) x^2-5 \log (2-5 x) x+5 x+2 \log (2-5 x)}{(5 x-2) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \log (9) \int \frac {16 \log (9) (2-5 x)+e^{x^2} x \left (40 \log (9) x^3-16 \log (9) x^2+5\right )+e^{x^2} \left (10 x^3-4 x^2-5 x+2\right ) \log (2-5 x)}{(2-5 x) \left (4 \log (9) \left (e^{x^2} x+4\right )+e^{x^2} \log (2-5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \log (9) \int \left (\frac {16 x \log (9) \left (-40 \log (9) x^3-10 \log (2-5 x) x^2+16 \log (9) x^2+4 \log (2-5 x) x-20 \log (9) x-5 \left (1-\frac {8 \log (9)}{5}\right )\right )}{(2-5 x) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )^2}-\frac {40 \log (9) x^4+10 \log (2-5 x) x^3-16 \log (9) x^3-4 \log (2-5 x) x^2-5 \log (2-5 x) x+5 x+2 \log (2-5 x)}{(5 x-2) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \log (9) \int \frac {16 \log (9) (2-5 x)+e^{x^2} x \left (40 \log (9) x^3-16 \log (9) x^2+5\right )+e^{x^2} \left (10 x^3-4 x^2-5 x+2\right ) \log (2-5 x)}{(2-5 x) \left (4 \log (9) \left (e^{x^2} x+4\right )+e^{x^2} \log (2-5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \log (9) \int \left (\frac {16 x \log (9) \left (-40 \log (9) x^3-10 \log (2-5 x) x^2+16 \log (9) x^2+4 \log (2-5 x) x-20 \log (9) x-5 \left (1-\frac {8 \log (9)}{5}\right )\right )}{(2-5 x) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )^2}-\frac {40 \log (9) x^4+10 \log (2-5 x) x^3-16 \log (9) x^3-4 \log (2-5 x) x^2-5 \log (2-5 x) x+5 x+2 \log (2-5 x)}{(5 x-2) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \log (9) \int \frac {16 \log (9) (2-5 x)+e^{x^2} x \left (40 \log (9) x^3-16 \log (9) x^2+5\right )+e^{x^2} \left (10 x^3-4 x^2-5 x+2\right ) \log (2-5 x)}{(2-5 x) \left (4 \log (9) \left (e^{x^2} x+4\right )+e^{x^2} \log (2-5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \log (9) \int \left (\frac {16 x \log (9) \left (-40 \log (9) x^3-10 \log (2-5 x) x^2+16 \log (9) x^2+4 \log (2-5 x) x-20 \log (9) x-5 \left (1-\frac {8 \log (9)}{5}\right )\right )}{(2-5 x) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )^2}-\frac {40 \log (9) x^4+10 \log (2-5 x) x^3-16 \log (9) x^3-4 \log (2-5 x) x^2-5 \log (2-5 x) x+5 x+2 \log (2-5 x)}{(5 x-2) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \log (9) \int \frac {16 \log (9) (2-5 x)+e^{x^2} x \left (40 \log (9) x^3-16 \log (9) x^2+5\right )+e^{x^2} \left (10 x^3-4 x^2-5 x+2\right ) \log (2-5 x)}{(2-5 x) \left (4 \log (9) \left (e^{x^2} x+4\right )+e^{x^2} \log (2-5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \log (9) \int \left (\frac {16 x \log (9) \left (-40 \log (9) x^3-10 \log (2-5 x) x^2+16 \log (9) x^2+4 \log (2-5 x) x-20 \log (9) x-5 \left (1-\frac {8 \log (9)}{5}\right )\right )}{(2-5 x) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )^2}-\frac {40 \log (9) x^4+10 \log (2-5 x) x^3-16 \log (9) x^3-4 \log (2-5 x) x^2-5 \log (2-5 x) x+5 x+2 \log (2-5 x)}{(5 x-2) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \log (9) \int \frac {16 \log (9) (2-5 x)+e^{x^2} x \left (40 \log (9) x^3-16 \log (9) x^2+5\right )+e^{x^2} \left (10 x^3-4 x^2-5 x+2\right ) \log (2-5 x)}{(2-5 x) \left (4 \log (9) \left (e^{x^2} x+4\right )+e^{x^2} \log (2-5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \log (9) \int \left (\frac {16 x \log (9) \left (-40 \log (9) x^3-10 \log (2-5 x) x^2+16 \log (9) x^2+4 \log (2-5 x) x-20 \log (9) x-5 \left (1-\frac {8 \log (9)}{5}\right )\right )}{(2-5 x) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )^2}-\frac {40 \log (9) x^4+10 \log (2-5 x) x^3-16 \log (9) x^3-4 \log (2-5 x) x^2-5 \log (2-5 x) x+5 x+2 \log (2-5 x)}{(5 x-2) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \log (9) \int \frac {16 \log (9) (2-5 x)+e^{x^2} x \left (40 \log (9) x^3-16 \log (9) x^2+5\right )+e^{x^2} \left (10 x^3-4 x^2-5 x+2\right ) \log (2-5 x)}{(2-5 x) \left (4 \log (9) \left (e^{x^2} x+4\right )+e^{x^2} \log (2-5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \log (9) \int \left (\frac {16 x \log (9) \left (-40 \log (9) x^3-10 \log (2-5 x) x^2+16 \log (9) x^2+4 \log (2-5 x) x-20 \log (9) x-5 \left (1-\frac {8 \log (9)}{5}\right )\right )}{(2-5 x) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )^2}-\frac {40 \log (9) x^4+10 \log (2-5 x) x^3-16 \log (9) x^3-4 \log (2-5 x) x^2-5 \log (2-5 x) x+5 x+2 \log (2-5 x)}{(5 x-2) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \log (9) \int \frac {16 \log (9) (2-5 x)+e^{x^2} x \left (40 \log (9) x^3-16 \log (9) x^2+5\right )+e^{x^2} \left (10 x^3-4 x^2-5 x+2\right ) \log (2-5 x)}{(2-5 x) \left (4 \log (9) \left (e^{x^2} x+4\right )+e^{x^2} \log (2-5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \log (9) \int \left (\frac {16 x \log (9) \left (-40 \log (9) x^3-10 \log (2-5 x) x^2+16 \log (9) x^2+4 \log (2-5 x) x-20 \log (9) x-5 \left (1-\frac {8 \log (9)}{5}\right )\right )}{(2-5 x) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )^2}-\frac {40 \log (9) x^4+10 \log (2-5 x) x^3-16 \log (9) x^3-4 \log (2-5 x) x^2-5 \log (2-5 x) x+5 x+2 \log (2-5 x)}{(5 x-2) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \log (9) \int \frac {16 \log (9) (2-5 x)+e^{x^2} x \left (40 \log (9) x^3-16 \log (9) x^2+5\right )+e^{x^2} \left (10 x^3-4 x^2-5 x+2\right ) \log (2-5 x)}{(2-5 x) \left (4 \log (9) \left (e^{x^2} x+4\right )+e^{x^2} \log (2-5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \log (9) \int \left (\frac {16 x \log (9) \left (-40 \log (9) x^3-10 \log (2-5 x) x^2+16 \log (9) x^2+4 \log (2-5 x) x-20 \log (9) x-5 \left (1-\frac {8 \log (9)}{5}\right )\right )}{(2-5 x) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )^2}-\frac {40 \log (9) x^4+10 \log (2-5 x) x^3-16 \log (9) x^3-4 \log (2-5 x) x^2-5 \log (2-5 x) x+5 x+2 \log (2-5 x)}{(5 x-2) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \log (9) \int \frac {16 \log (9) (2-5 x)+e^{x^2} x \left (40 \log (9) x^3-16 \log (9) x^2+5\right )+e^{x^2} \left (10 x^3-4 x^2-5 x+2\right ) \log (2-5 x)}{(2-5 x) \left (4 \log (9) \left (e^{x^2} x+4\right )+e^{x^2} \log (2-5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \log (9) \int \left (\frac {16 x \log (9) \left (-40 \log (9) x^3-10 \log (2-5 x) x^2+16 \log (9) x^2+4 \log (2-5 x) x-20 \log (9) x-5 \left (1-\frac {8 \log (9)}{5}\right )\right )}{(2-5 x) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )^2}-\frac {40 \log (9) x^4+10 \log (2-5 x) x^3-16 \log (9) x^3-4 \log (2-5 x) x^2-5 \log (2-5 x) x+5 x+2 \log (2-5 x)}{(5 x-2) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \log (9) \int \frac {16 \log (9) (2-5 x)+e^{x^2} x \left (40 \log (9) x^3-16 \log (9) x^2+5\right )+e^{x^2} \left (10 x^3-4 x^2-5 x+2\right ) \log (2-5 x)}{(2-5 x) \left (4 \log (9) \left (e^{x^2} x+4\right )+e^{x^2} \log (2-5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \log (9) \int \left (\frac {16 x \log (9) \left (-40 \log (9) x^3-10 \log (2-5 x) x^2+16 \log (9) x^2+4 \log (2-5 x) x-20 \log (9) x-5 \left (1-\frac {8 \log (9)}{5}\right )\right )}{(2-5 x) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )^2}-\frac {40 \log (9) x^4+10 \log (2-5 x) x^3-16 \log (9) x^3-4 \log (2-5 x) x^2-5 \log (2-5 x) x+5 x+2 \log (2-5 x)}{(5 x-2) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \log (9) \int \frac {16 \log (9) (2-5 x)+e^{x^2} x \left (40 \log (9) x^3-16 \log (9) x^2+5\right )+e^{x^2} \left (10 x^3-4 x^2-5 x+2\right ) \log (2-5 x)}{(2-5 x) \left (4 \log (9) \left (e^{x^2} x+4\right )+e^{x^2} \log (2-5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \log (9) \int \left (\frac {16 x \log (9) \left (-40 \log (9) x^3-10 \log (2-5 x) x^2+16 \log (9) x^2+4 \log (2-5 x) x-20 \log (9) x-5 \left (1-\frac {8 \log (9)}{5}\right )\right )}{(2-5 x) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )^2}-\frac {40 \log (9) x^4+10 \log (2-5 x) x^3-16 \log (9) x^3-4 \log (2-5 x) x^2-5 \log (2-5 x) x+5 x+2 \log (2-5 x)}{(5 x-2) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \log (9) \int \frac {16 \log (9) (2-5 x)+e^{x^2} x \left (40 \log (9) x^3-16 \log (9) x^2+5\right )+e^{x^2} \left (10 x^3-4 x^2-5 x+2\right ) \log (2-5 x)}{(2-5 x) \left (4 \log (9) \left (e^{x^2} x+4\right )+e^{x^2} \log (2-5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \log (9) \int \left (\frac {16 x \log (9) \left (-40 \log (9) x^3-10 \log (2-5 x) x^2+16 \log (9) x^2+4 \log (2-5 x) x-20 \log (9) x-5 \left (1-\frac {8 \log (9)}{5}\right )\right )}{(2-5 x) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )^2}-\frac {40 \log (9) x^4+10 \log (2-5 x) x^3-16 \log (9) x^3-4 \log (2-5 x) x^2-5 \log (2-5 x) x+5 x+2 \log (2-5 x)}{(5 x-2) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \log (9) \int \frac {16 \log (9) (2-5 x)+e^{x^2} x \left (40 \log (9) x^3-16 \log (9) x^2+5\right )+e^{x^2} \left (10 x^3-4 x^2-5 x+2\right ) \log (2-5 x)}{(2-5 x) \left (4 \log (9) \left (e^{x^2} x+4\right )+e^{x^2} \log (2-5 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \log (9) \int \left (\frac {16 x \log (9) \left (-40 \log (9) x^3-10 \log (2-5 x) x^2+16 \log (9) x^2+4 \log (2-5 x) x-20 \log (9) x-5 \left (1-\frac {8 \log (9)}{5}\right )\right )}{(2-5 x) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )^2}-\frac {40 \log (9) x^4+10 \log (2-5 x) x^3-16 \log (9) x^3-4 \log (2-5 x) x^2-5 \log (2-5 x) x+5 x+2 \log (2-5 x)}{(5 x-2) (4 \log (9) x+\log (2-5 x)) \left (4 e^{x^2} \log (9) x+e^{x^2} \log (2-5 x)+16 \log (9)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \log (9) \int \frac {16 \log (9) (2-5 x)+e^{x^2} x \left (40 \log (9) x^3-16 \log (9) x^2+5\right )+e^{x^2} \left (10 x^3-4 x^2-5 x+2\right ) \log (2-5 x)}{(2-5 x) \left (4 \log (9) \left (e^{x^2} x+4\right )+e^{x^2} \log (2-5 x)\right )^2}dx\) |
Int[((-128 + 320*x)*Log[9]^2 + E^x^2*(-20*x*Log[9] + (64*x^3 - 160*x^4)*Lo g[9]^2) + E^x^2*(-8 + 20*x + 16*x^2 - 40*x^3)*Log[9]*Log[2 - 5*x])/((-512 + 1280*x)*Log[9]^2 + E^x^2*(-256*x + 640*x^2)*Log[9]^2 + E^(2*x^2)*(-32*x^ 2 + 80*x^3)*Log[9]^2 + (E^x^2*(-64 + 160*x)*Log[9] + E^(2*x^2)*(-16*x + 40 *x^2)*Log[9])*Log[2 - 5*x] + E^(2*x^2)*(-2 + 5*x)*Log[2 - 5*x]^2),x]
3.18.68.3.1 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 = 0.53 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18
| method | result | size |
| risch | \(\frac {8 \ln \left (3\right ) x}{8 \ln \left (3\right ) {\mathrm e}^{x^{2}} x +{\mathrm e}^{x^{2}} \ln \left (-5 x +2\right )+32 \ln \left (3\right )}\) | \(33\) |
| parallelrisch | \(\frac {8 \ln \left (3\right ) x}{8 \ln \left (3\right ) {\mathrm e}^{x^{2}} x +{\mathrm e}^{x^{2}} \ln \left (-5 x +2\right )+32 \ln \left (3\right )}\) | \(33\) |
int((2*(-40*x^3+16*x^2+20*x-8)*ln(3)*exp(x^2)*ln(-5*x+2)+(4*(-160*x^4+64*x ^3)*ln(3)^2-40*x*ln(3))*exp(x^2)+4*(320*x-128)*ln(3)^2)/((5*x-2)*exp(x^2)^ 2*ln(-5*x+2)^2+(2*(40*x^2-16*x)*ln(3)*exp(x^2)^2+2*(160*x-64)*ln(3)*exp(x^ 2))*ln(-5*x+2)+4*(80*x^3-32*x^2)*ln(3)^2*exp(x^2)^2+4*(640*x^2-256*x)*ln(3 )^2*exp(x^2)+4*(1280*x-512)*ln(3)^2),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {(-128+320 x) \log ^2(9)+e^{x^2} \left (-20 x \log (9)+\left (64 x^3-160 x^4\right ) \log ^2(9)\right )+e^{x^2} \left (-8+20 x+16 x^2-40 x^3\right ) \log (9) \log (2-5 x)}{(-512+1280 x) \log ^2(9)+e^{x^2} \left (-256 x+640 x^2\right ) \log ^2(9)+e^{2 x^2} \left (-32 x^2+80 x^3\right ) \log ^2(9)+\left (e^{x^2} (-64+160 x) \log (9)+e^{2 x^2} \left (-16 x+40 x^2\right ) \log (9)\right ) \log (2-5 x)+e^{2 x^2} (-2+5 x) \log ^2(2-5 x)} \, dx=\frac {8 \, x \log \left (3\right )}{8 \, x e^{\left (x^{2}\right )} \log \left (3\right ) + e^{\left (x^{2}\right )} \log \left (-5 \, x + 2\right ) + 32 \, \log \left (3\right )} \]
integrate((2*(-40*x^3+16*x^2+20*x-8)*log(3)*exp(x^2)*log(-5*x+2)+(4*(-160* x^4+64*x^3)*log(3)^2-40*x*log(3))*exp(x^2)+4*(320*x-128)*log(3)^2)/((5*x-2 )*exp(x^2)^2*log(-5*x+2)^2+(2*(40*x^2-16*x)*log(3)*exp(x^2)^2+2*(160*x-64) *log(3)*exp(x^2))*log(-5*x+2)+4*(80*x^3-32*x^2)*log(3)^2*exp(x^2)^2+4*(640 *x^2-256*x)*log(3)^2*exp(x^2)+4*(1280*x-512)*log(3)^2),x, algorithm=\
Time = 0.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {(-128+320 x) \log ^2(9)+e^{x^2} \left (-20 x \log (9)+\left (64 x^3-160 x^4\right ) \log ^2(9)\right )+e^{x^2} \left (-8+20 x+16 x^2-40 x^3\right ) \log (9) \log (2-5 x)}{(-512+1280 x) \log ^2(9)+e^{x^2} \left (-256 x+640 x^2\right ) \log ^2(9)+e^{2 x^2} \left (-32 x^2+80 x^3\right ) \log ^2(9)+\left (e^{x^2} (-64+160 x) \log (9)+e^{2 x^2} \left (-16 x+40 x^2\right ) \log (9)\right ) \log (2-5 x)+e^{2 x^2} (-2+5 x) \log ^2(2-5 x)} \, dx=\frac {8 x \log {\left (3 \right )}}{\left (8 x \log {\left (3 \right )} + \log {\left (2 - 5 x \right )}\right ) e^{x^{2}} + 32 \log {\left (3 \right )}} \]
integrate((2*(-40*x**3+16*x**2+20*x-8)*ln(3)*exp(x**2)*ln(-5*x+2)+(4*(-160 *x**4+64*x**3)*ln(3)**2-40*x*ln(3))*exp(x**2)+4*(320*x-128)*ln(3)**2)/((5* x-2)*exp(x**2)**2*ln(-5*x+2)**2+(2*(40*x**2-16*x)*ln(3)*exp(x**2)**2+2*(16 0*x-64)*ln(3)*exp(x**2))*ln(-5*x+2)+4*(80*x**3-32*x**2)*ln(3)**2*exp(x**2) **2+4*(640*x**2-256*x)*ln(3)**2*exp(x**2)+4*(1280*x-512)*ln(3)**2),x)
Time = 0.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {(-128+320 x) \log ^2(9)+e^{x^2} \left (-20 x \log (9)+\left (64 x^3-160 x^4\right ) \log ^2(9)\right )+e^{x^2} \left (-8+20 x+16 x^2-40 x^3\right ) \log (9) \log (2-5 x)}{(-512+1280 x) \log ^2(9)+e^{x^2} \left (-256 x+640 x^2\right ) \log ^2(9)+e^{2 x^2} \left (-32 x^2+80 x^3\right ) \log ^2(9)+\left (e^{x^2} (-64+160 x) \log (9)+e^{2 x^2} \left (-16 x+40 x^2\right ) \log (9)\right ) \log (2-5 x)+e^{2 x^2} (-2+5 x) \log ^2(2-5 x)} \, dx=\frac {8 \, x \log \left (3\right )}{8 \, x e^{\left (x^{2}\right )} \log \left (3\right ) + e^{\left (x^{2}\right )} \log \left (-5 \, x + 2\right ) + 32 \, \log \left (3\right )} \]
integrate((2*(-40*x^3+16*x^2+20*x-8)*log(3)*exp(x^2)*log(-5*x+2)+(4*(-160* x^4+64*x^3)*log(3)^2-40*x*log(3))*exp(x^2)+4*(320*x-128)*log(3)^2)/((5*x-2 )*exp(x^2)^2*log(-5*x+2)^2+(2*(40*x^2-16*x)*log(3)*exp(x^2)^2+2*(160*x-64) *log(3)*exp(x^2))*log(-5*x+2)+4*(80*x^3-32*x^2)*log(3)^2*exp(x^2)^2+4*(640 *x^2-256*x)*log(3)^2*exp(x^2)+4*(1280*x-512)*log(3)^2),x, algorithm=\
Time = 0.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {(-128+320 x) \log ^2(9)+e^{x^2} \left (-20 x \log (9)+\left (64 x^3-160 x^4\right ) \log ^2(9)\right )+e^{x^2} \left (-8+20 x+16 x^2-40 x^3\right ) \log (9) \log (2-5 x)}{(-512+1280 x) \log ^2(9)+e^{x^2} \left (-256 x+640 x^2\right ) \log ^2(9)+e^{2 x^2} \left (-32 x^2+80 x^3\right ) \log ^2(9)+\left (e^{x^2} (-64+160 x) \log (9)+e^{2 x^2} \left (-16 x+40 x^2\right ) \log (9)\right ) \log (2-5 x)+e^{2 x^2} (-2+5 x) \log ^2(2-5 x)} \, dx=\frac {8 \, x \log \left (3\right )}{8 \, x e^{\left (x^{2}\right )} \log \left (3\right ) + e^{\left (x^{2}\right )} \log \left (-5 \, x + 2\right ) + 32 \, \log \left (3\right )} \]
integrate((2*(-40*x^3+16*x^2+20*x-8)*log(3)*exp(x^2)*log(-5*x+2)+(4*(-160* x^4+64*x^3)*log(3)^2-40*x*log(3))*exp(x^2)+4*(320*x-128)*log(3)^2)/((5*x-2 )*exp(x^2)^2*log(-5*x+2)^2+(2*(40*x^2-16*x)*log(3)*exp(x^2)^2+2*(160*x-64) *log(3)*exp(x^2))*log(-5*x+2)+4*(80*x^3-32*x^2)*log(3)^2*exp(x^2)^2+4*(640 *x^2-256*x)*log(3)^2*exp(x^2)+4*(1280*x-512)*log(3)^2),x, algorithm=\
Timed out. \[ \int \frac {(-128+320 x) \log ^2(9)+e^{x^2} \left (-20 x \log (9)+\left (64 x^3-160 x^4\right ) \log ^2(9)\right )+e^{x^2} \left (-8+20 x+16 x^2-40 x^3\right ) \log (9) \log (2-5 x)}{(-512+1280 x) \log ^2(9)+e^{x^2} \left (-256 x+640 x^2\right ) \log ^2(9)+e^{2 x^2} \left (-32 x^2+80 x^3\right ) \log ^2(9)+\left (e^{x^2} (-64+160 x) \log (9)+e^{2 x^2} \left (-16 x+40 x^2\right ) \log (9)\right ) \log (2-5 x)+e^{2 x^2} (-2+5 x) \log ^2(2-5 x)} \, dx=\int \frac {4\,{\ln \left (3\right )}^2\,\left (320\,x-128\right )-{\mathrm {e}}^{x^2}\,\left (40\,x\,\ln \left (3\right )-4\,{\ln \left (3\right )}^2\,\left (64\,x^3-160\,x^4\right )\right )+2\,{\mathrm {e}}^{x^2}\,\ln \left (3\right )\,\ln \left (2-5\,x\right )\,\left (-40\,x^3+16\,x^2+20\,x-8\right )}{{\mathrm {e}}^{2\,x^2}\,\left (5\,x-2\right )\,{\ln \left (2-5\,x\right )}^2+\left (2\,{\mathrm {e}}^{x^2}\,\ln \left (3\right )\,\left (160\,x-64\right )-2\,{\mathrm {e}}^{2\,x^2}\,\ln \left (3\right )\,\left (16\,x-40\,x^2\right )\right )\,\ln \left (2-5\,x\right )+4\,{\ln \left (3\right )}^2\,\left (1280\,x-512\right )-4\,{\mathrm {e}}^{2\,x^2}\,{\ln \left (3\right )}^2\,\left (32\,x^2-80\,x^3\right )-4\,{\mathrm {e}}^{x^2}\,{\ln \left (3\right )}^2\,\left (256\,x-640\,x^2\right )} \,d x \]
int((4*log(3)^2*(320*x - 128) - exp(x^2)*(40*x*log(3) - 4*log(3)^2*(64*x^3 - 160*x^4)) + 2*exp(x^2)*log(3)*log(2 - 5*x)*(20*x + 16*x^2 - 40*x^3 - 8) )/(4*log(3)^2*(1280*x - 512) + log(2 - 5*x)*(2*exp(x^2)*log(3)*(160*x - 64 ) - 2*exp(2*x^2)*log(3)*(16*x - 40*x^2)) + exp(2*x^2)*log(2 - 5*x)^2*(5*x - 2) - 4*exp(2*x^2)*log(3)^2*(32*x^2 - 80*x^3) - 4*exp(x^2)*log(3)^2*(256* x - 640*x^2)),x)
int((4*log(3)^2*(320*x - 128) - exp(x^2)*(40*x*log(3) - 4*log(3)^2*(64*x^3 - 160*x^4)) + 2*exp(x^2)*log(3)*log(2 - 5*x)*(20*x + 16*x^2 - 40*x^3 - 8) )/(4*log(3)^2*(1280*x - 512) + log(2 - 5*x)*(2*exp(x^2)*log(3)*(160*x - 64 ) - 2*exp(2*x^2)*log(3)*(16*x - 40*x^2)) + exp(2*x^2)*log(2 - 5*x)^2*(5*x - 2) - 4*exp(2*x^2)*log(3)^2*(32*x^2 - 80*x^3) - 4*exp(x^2)*log(3)^2*(256* x - 640*x^2)), x)