Integrand size = 227, antiderivative size = 25 \[ \int \frac {\left (960-960 x+300 x^2-30 x^3+e^x \left (-960+960 x-300 x^2+30 x^3\right )\right ) \log (-2+x)+\left (120-60 x+e^x (-120+60 x)\right ) \log ^2(-2+x)+\left (-480 x+240 x^2-30 x^3+e^x \left (480 x-240 x^2+30 x^3\right )+\left (480 x-360 x^2+60 x^3+e^x \left (-1440 x+1320 x^2-360 x^3+30 x^4\right )\right ) \log (-2+x)+e^x \left (-120 x+60 x^2\right ) \log ^2(-2+x)\right ) \log (x)}{-512 x+768 x^2-448 x^3+128 x^4-18 x^5+x^6+\left (-128 x+128 x^2-40 x^3+4 x^4\right ) \log (-2+x)+\left (-8 x+4 x^2\right ) \log ^2(-2+x)} \, dx=\frac {30 \left (-1+e^x\right ) \log (x)}{2+\frac {(-4+x)^2}{\log (-2+x)}} \]
Time = 0.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\left (960-960 x+300 x^2-30 x^3+e^x \left (-960+960 x-300 x^2+30 x^3\right )\right ) \log (-2+x)+\left (120-60 x+e^x (-120+60 x)\right ) \log ^2(-2+x)+\left (-480 x+240 x^2-30 x^3+e^x \left (480 x-240 x^2+30 x^3\right )+\left (480 x-360 x^2+60 x^3+e^x \left (-1440 x+1320 x^2-360 x^3+30 x^4\right )\right ) \log (-2+x)+e^x \left (-120 x+60 x^2\right ) \log ^2(-2+x)\right ) \log (x)}{-512 x+768 x^2-448 x^3+128 x^4-18 x^5+x^6+\left (-128 x+128 x^2-40 x^3+4 x^4\right ) \log (-2+x)+\left (-8 x+4 x^2\right ) \log ^2(-2+x)} \, dx=\frac {30 \left (-1+e^x\right ) \log (-2+x) \log (x)}{(-4+x)^2+2 \log (-2+x)} \]
Integrate[((960 - 960*x + 300*x^2 - 30*x^3 + E^x*(-960 + 960*x - 300*x^2 + 30*x^3))*Log[-2 + x] + (120 - 60*x + E^x*(-120 + 60*x))*Log[-2 + x]^2 + ( -480*x + 240*x^2 - 30*x^3 + E^x*(480*x - 240*x^2 + 30*x^3) + (480*x - 360* x^2 + 60*x^3 + E^x*(-1440*x + 1320*x^2 - 360*x^3 + 30*x^4))*Log[-2 + x] + E^x*(-120*x + 60*x^2)*Log[-2 + x]^2)*Log[x])/(-512*x + 768*x^2 - 448*x^3 + 128*x^4 - 18*x^5 + x^6 + (-128*x + 128*x^2 - 40*x^3 + 4*x^4)*Log[-2 + x] + (-8*x + 4*x^2)*Log[-2 + 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.
| \(\displaystyle \int \frac {\left (-30 x^3+300 x^2+e^x \left (30 x^3-300 x^2+960 x-960\right )-960 x+960\right ) \log (x-2)+\left (-30 x^3+240 x^2+e^x \left (60 x^2-120 x\right ) \log ^2(x-2)+e^x \left (30 x^3-240 x^2+480 x\right )+\left (60 x^3-360 x^2+e^x \left (30 x^4-360 x^3+1320 x^2-1440 x\right )+480 x\right ) \log (x-2)-480 x\right ) \log (x)+\left (-60 x+e^x (60 x-120)+120\right ) \log ^2(x-2)}{x^6-18 x^5+128 x^4-448 x^3+768 x^2+\left (4 x^2-8 x\right ) \log ^2(x-2)+\left (4 x^4-40 x^3+128 x^2-128 x\right ) \log (x-2)-512 x} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {30 \left (-\left (x^2-6 x+8\right ) \log (x-2) \left (\left (e^x-1\right ) (x-4)+\left (e^x (x-6)+2\right ) x \log (x)\right )-2 (x-2) \log ^2(x-2) \left (e^x+e^x x \log (x)-1\right )-\left (\left (e^x-1\right ) x (x-4)^2 \log (x)\right )\right )}{(2-x) x \left ((x-4)^2+2 \log (x-2)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 30 \int \frac {\left (1-e^x\right ) x \log (x) (4-x)^2-2 (2-x) \log ^2(x-2) \left (-e^x x \log (x)-e^x+1\right )-\left (x^2-6 x+8\right ) \log (x-2) \left (\left (1-e^x\right ) (4-x)+\left (2-e^x (6-x)\right ) x \log (x)\right )}{(2-x) x \left ((x-4)^2+2 \log (x-2)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 30 \int \left (-\frac {\log (x) (x-4)^2}{(x-2) \left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {2 \log (x-2) \log (x) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {4 \log (x-2) (x-4)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {\log (x-2) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {e^x \left (\log (x-2) \log (x) x^4+\log (x-2) x^3-12 \log (x-2) \log (x) x^3+\log (x) x^3-10 \log (x-2) x^2+2 \log ^2(x-2) \log (x) x^2+44 \log (x-2) \log (x) x^2-8 \log (x) x^2+2 \log ^2(x-2) x+32 \log (x-2) x-4 \log ^2(x-2) \log (x) x-48 \log (x-2) \log (x) x+16 \log (x) x-4 \log ^2(x-2)-32 \log (x-2)\right )}{(x-2) x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {2 \log ^2(x-2)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 30 \int \frac {-\left (\left (-1+e^x\right ) x \log (x) (x-4)^2\right )-2 (x-2) \log ^2(x-2) \left (e^x x \log (x)+e^x-1\right )-\left (x^2-6 x+8\right ) \log (x-2) \left (\left (-1+e^x\right ) (x-4)+\left (e^x (x-6)+2\right ) x \log (x)\right )}{(2-x) x \left ((x-4)^2+2 \log (x-2)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 30 \int \left (-\frac {\log (x) (x-4)^2}{(x-2) \left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {2 \log (x-2) \log (x) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {4 \log (x-2) (x-4)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {\log (x-2) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {e^x \left (\log (x-2) \log (x) x^4+\log (x-2) x^3-12 \log (x-2) \log (x) x^3+\log (x) x^3-10 \log (x-2) x^2+2 \log ^2(x-2) \log (x) x^2+44 \log (x-2) \log (x) x^2-8 \log (x) x^2+2 \log ^2(x-2) x+32 \log (x-2) x-4 \log ^2(x-2) \log (x) x-48 \log (x-2) \log (x) x+16 \log (x) x-4 \log ^2(x-2)-32 \log (x-2)\right )}{(x-2) x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {2 \log ^2(x-2)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 30 \int \frac {-\left (\left (-1+e^x\right ) x \log (x) (x-4)^2\right )-2 (x-2) \log ^2(x-2) \left (e^x x \log (x)+e^x-1\right )-\left (x^2-6 x+8\right ) \log (x-2) \left (\left (-1+e^x\right ) (x-4)+\left (e^x (x-6)+2\right ) x \log (x)\right )}{(2-x) x \left ((x-4)^2+2 \log (x-2)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 30 \int \left (-\frac {\log (x) (x-4)^2}{(x-2) \left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {2 \log (x-2) \log (x) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {4 \log (x-2) (x-4)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {\log (x-2) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {e^x \left (\log (x-2) \log (x) x^4+\log (x-2) x^3-12 \log (x-2) \log (x) x^3+\log (x) x^3-10 \log (x-2) x^2+2 \log ^2(x-2) \log (x) x^2+44 \log (x-2) \log (x) x^2-8 \log (x) x^2+2 \log ^2(x-2) x+32 \log (x-2) x-4 \log ^2(x-2) \log (x) x-48 \log (x-2) \log (x) x+16 \log (x) x-4 \log ^2(x-2)-32 \log (x-2)\right )}{(x-2) x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {2 \log ^2(x-2)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 30 \int \frac {-\left (\left (-1+e^x\right ) x \log (x) (x-4)^2\right )-2 (x-2) \log ^2(x-2) \left (e^x x \log (x)+e^x-1\right )-\left (x^2-6 x+8\right ) \log (x-2) \left (\left (-1+e^x\right ) (x-4)+\left (e^x (x-6)+2\right ) x \log (x)\right )}{(2-x) x \left ((x-4)^2+2 \log (x-2)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 30 \int \left (-\frac {\log (x) (x-4)^2}{(x-2) \left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {2 \log (x-2) \log (x) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {4 \log (x-2) (x-4)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {\log (x-2) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {e^x \left (\log (x-2) \log (x) x^4+\log (x-2) x^3-12 \log (x-2) \log (x) x^3+\log (x) x^3-10 \log (x-2) x^2+2 \log ^2(x-2) \log (x) x^2+44 \log (x-2) \log (x) x^2-8 \log (x) x^2+2 \log ^2(x-2) x+32 \log (x-2) x-4 \log ^2(x-2) \log (x) x-48 \log (x-2) \log (x) x+16 \log (x) x-4 \log ^2(x-2)-32 \log (x-2)\right )}{(x-2) x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {2 \log ^2(x-2)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 30 \int \frac {-\left (\left (-1+e^x\right ) x \log (x) (x-4)^2\right )-2 (x-2) \log ^2(x-2) \left (e^x x \log (x)+e^x-1\right )-\left (x^2-6 x+8\right ) \log (x-2) \left (\left (-1+e^x\right ) (x-4)+\left (e^x (x-6)+2\right ) x \log (x)\right )}{(2-x) x \left ((x-4)^2+2 \log (x-2)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 30 \int \left (-\frac {\log (x) (x-4)^2}{(x-2) \left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {2 \log (x-2) \log (x) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {4 \log (x-2) (x-4)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {\log (x-2) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {e^x \left (\log (x-2) \log (x) x^4+\log (x-2) x^3-12 \log (x-2) \log (x) x^3+\log (x) x^3-10 \log (x-2) x^2+2 \log ^2(x-2) \log (x) x^2+44 \log (x-2) \log (x) x^2-8 \log (x) x^2+2 \log ^2(x-2) x+32 \log (x-2) x-4 \log ^2(x-2) \log (x) x-48 \log (x-2) \log (x) x+16 \log (x) x-4 \log ^2(x-2)-32 \log (x-2)\right )}{(x-2) x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {2 \log ^2(x-2)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 30 \int \frac {-\left (\left (-1+e^x\right ) x \log (x) (x-4)^2\right )-2 (x-2) \log ^2(x-2) \left (e^x x \log (x)+e^x-1\right )-\left (x^2-6 x+8\right ) \log (x-2) \left (\left (-1+e^x\right ) (x-4)+\left (e^x (x-6)+2\right ) x \log (x)\right )}{(2-x) x \left ((x-4)^2+2 \log (x-2)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 30 \int \left (-\frac {\log (x) (x-4)^2}{(x-2) \left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {2 \log (x-2) \log (x) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {4 \log (x-2) (x-4)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {\log (x-2) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {e^x \left (\log (x-2) \log (x) x^4+\log (x-2) x^3-12 \log (x-2) \log (x) x^3+\log (x) x^3-10 \log (x-2) x^2+2 \log ^2(x-2) \log (x) x^2+44 \log (x-2) \log (x) x^2-8 \log (x) x^2+2 \log ^2(x-2) x+32 \log (x-2) x-4 \log ^2(x-2) \log (x) x-48 \log (x-2) \log (x) x+16 \log (x) x-4 \log ^2(x-2)-32 \log (x-2)\right )}{(x-2) x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {2 \log ^2(x-2)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 30 \int \frac {-\left (\left (-1+e^x\right ) x \log (x) (x-4)^2\right )-2 (x-2) \log ^2(x-2) \left (e^x x \log (x)+e^x-1\right )-\left (x^2-6 x+8\right ) \log (x-2) \left (\left (-1+e^x\right ) (x-4)+\left (e^x (x-6)+2\right ) x \log (x)\right )}{(2-x) x \left ((x-4)^2+2 \log (x-2)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 30 \int \left (-\frac {\log (x) (x-4)^2}{(x-2) \left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {2 \log (x-2) \log (x) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {4 \log (x-2) (x-4)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {\log (x-2) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {e^x \left (\log (x-2) \log (x) x^4+\log (x-2) x^3-12 \log (x-2) \log (x) x^3+\log (x) x^3-10 \log (x-2) x^2+2 \log ^2(x-2) \log (x) x^2+44 \log (x-2) \log (x) x^2-8 \log (x) x^2+2 \log ^2(x-2) x+32 \log (x-2) x-4 \log ^2(x-2) \log (x) x-48 \log (x-2) \log (x) x+16 \log (x) x-4 \log ^2(x-2)-32 \log (x-2)\right )}{(x-2) x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {2 \log ^2(x-2)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 30 \int \frac {-\left (\left (-1+e^x\right ) x \log (x) (x-4)^2\right )-2 (x-2) \log ^2(x-2) \left (e^x x \log (x)+e^x-1\right )-\left (x^2-6 x+8\right ) \log (x-2) \left (\left (-1+e^x\right ) (x-4)+\left (e^x (x-6)+2\right ) x \log (x)\right )}{(2-x) x \left ((x-4)^2+2 \log (x-2)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 30 \int \left (-\frac {\log (x) (x-4)^2}{(x-2) \left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {2 \log (x-2) \log (x) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {4 \log (x-2) (x-4)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {\log (x-2) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {e^x \left (\log (x-2) \log (x) x^4+\log (x-2) x^3-12 \log (x-2) \log (x) x^3+\log (x) x^3-10 \log (x-2) x^2+2 \log ^2(x-2) \log (x) x^2+44 \log (x-2) \log (x) x^2-8 \log (x) x^2+2 \log ^2(x-2) x+32 \log (x-2) x-4 \log ^2(x-2) \log (x) x-48 \log (x-2) \log (x) x+16 \log (x) x-4 \log ^2(x-2)-32 \log (x-2)\right )}{(x-2) x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {2 \log ^2(x-2)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 30 \int \frac {-\left (\left (-1+e^x\right ) x \log (x) (x-4)^2\right )-2 (x-2) \log ^2(x-2) \left (e^x x \log (x)+e^x-1\right )-\left (x^2-6 x+8\right ) \log (x-2) \left (\left (-1+e^x\right ) (x-4)+\left (e^x (x-6)+2\right ) x \log (x)\right )}{(2-x) x \left ((x-4)^2+2 \log (x-2)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 30 \int \left (-\frac {\log (x) (x-4)^2}{(x-2) \left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {2 \log (x-2) \log (x) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {4 \log (x-2) (x-4)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {\log (x-2) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {e^x \left (\log (x-2) \log (x) x^4+\log (x-2) x^3-12 \log (x-2) \log (x) x^3+\log (x) x^3-10 \log (x-2) x^2+2 \log ^2(x-2) \log (x) x^2+44 \log (x-2) \log (x) x^2-8 \log (x) x^2+2 \log ^2(x-2) x+32 \log (x-2) x-4 \log ^2(x-2) \log (x) x-48 \log (x-2) \log (x) x+16 \log (x) x-4 \log ^2(x-2)-32 \log (x-2)\right )}{(x-2) x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {2 \log ^2(x-2)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 30 \int \frac {-\left (\left (-1+e^x\right ) x \log (x) (x-4)^2\right )-2 (x-2) \log ^2(x-2) \left (e^x x \log (x)+e^x-1\right )-\left (x^2-6 x+8\right ) \log (x-2) \left (\left (-1+e^x\right ) (x-4)+\left (e^x (x-6)+2\right ) x \log (x)\right )}{(2-x) x \left ((x-4)^2+2 \log (x-2)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 30 \int \left (-\frac {\log (x) (x-4)^2}{(x-2) \left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {2 \log (x-2) \log (x) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {4 \log (x-2) (x-4)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {\log (x-2) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {e^x \left (\log (x-2) \log (x) x^4+\log (x-2) x^3-12 \log (x-2) \log (x) x^3+\log (x) x^3-10 \log (x-2) x^2+2 \log ^2(x-2) \log (x) x^2+44 \log (x-2) \log (x) x^2-8 \log (x) x^2+2 \log ^2(x-2) x+32 \log (x-2) x-4 \log ^2(x-2) \log (x) x-48 \log (x-2) \log (x) x+16 \log (x) x-4 \log ^2(x-2)-32 \log (x-2)\right )}{(x-2) x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {2 \log ^2(x-2)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 30 \int \frac {-\left (\left (-1+e^x\right ) x \log (x) (x-4)^2\right )-2 (x-2) \log ^2(x-2) \left (e^x x \log (x)+e^x-1\right )-\left (x^2-6 x+8\right ) \log (x-2) \left (\left (-1+e^x\right ) (x-4)+\left (e^x (x-6)+2\right ) x \log (x)\right )}{(2-x) x \left ((x-4)^2+2 \log (x-2)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 30 \int \left (-\frac {\log (x) (x-4)^2}{(x-2) \left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {2 \log (x-2) \log (x) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {4 \log (x-2) (x-4)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {\log (x-2) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {e^x \left (\log (x-2) \log (x) x^4+\log (x-2) x^3-12 \log (x-2) \log (x) x^3+\log (x) x^3-10 \log (x-2) x^2+2 \log ^2(x-2) \log (x) x^2+44 \log (x-2) \log (x) x^2-8 \log (x) x^2+2 \log ^2(x-2) x+32 \log (x-2) x-4 \log ^2(x-2) \log (x) x-48 \log (x-2) \log (x) x+16 \log (x) x-4 \log ^2(x-2)-32 \log (x-2)\right )}{(x-2) x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {2 \log ^2(x-2)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 30 \int \frac {-\left (\left (-1+e^x\right ) x \log (x) (x-4)^2\right )-2 (x-2) \log ^2(x-2) \left (e^x x \log (x)+e^x-1\right )-\left (x^2-6 x+8\right ) \log (x-2) \left (\left (-1+e^x\right ) (x-4)+\left (e^x (x-6)+2\right ) x \log (x)\right )}{(2-x) x \left ((x-4)^2+2 \log (x-2)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 30 \int \left (-\frac {\log (x) (x-4)^2}{(x-2) \left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {2 \log (x-2) \log (x) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {4 \log (x-2) (x-4)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {\log (x-2) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {e^x \left (\log (x-2) \log (x) x^4+\log (x-2) x^3-12 \log (x-2) \log (x) x^3+\log (x) x^3-10 \log (x-2) x^2+2 \log ^2(x-2) \log (x) x^2+44 \log (x-2) \log (x) x^2-8 \log (x) x^2+2 \log ^2(x-2) x+32 \log (x-2) x-4 \log ^2(x-2) \log (x) x-48 \log (x-2) \log (x) x+16 \log (x) x-4 \log ^2(x-2)-32 \log (x-2)\right )}{(x-2) x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {2 \log ^2(x-2)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 30 \int \frac {-\left (\left (-1+e^x\right ) x \log (x) (x-4)^2\right )-2 (x-2) \log ^2(x-2) \left (e^x x \log (x)+e^x-1\right )-\left (x^2-6 x+8\right ) \log (x-2) \left (\left (-1+e^x\right ) (x-4)+\left (e^x (x-6)+2\right ) x \log (x)\right )}{(2-x) x \left ((x-4)^2+2 \log (x-2)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 30 \int \left (-\frac {\log (x) (x-4)^2}{(x-2) \left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {2 \log (x-2) \log (x) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {4 \log (x-2) (x-4)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {\log (x-2) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {e^x \left (\log (x-2) \log (x) x^4+\log (x-2) x^3-12 \log (x-2) \log (x) x^3+\log (x) x^3-10 \log (x-2) x^2+2 \log ^2(x-2) \log (x) x^2+44 \log (x-2) \log (x) x^2-8 \log (x) x^2+2 \log ^2(x-2) x+32 \log (x-2) x-4 \log ^2(x-2) \log (x) x-48 \log (x-2) \log (x) x+16 \log (x) x-4 \log ^2(x-2)-32 \log (x-2)\right )}{(x-2) x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {2 \log ^2(x-2)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 30 \int \frac {-\left (\left (-1+e^x\right ) x \log (x) (x-4)^2\right )-2 (x-2) \log ^2(x-2) \left (e^x x \log (x)+e^x-1\right )-\left (x^2-6 x+8\right ) \log (x-2) \left (\left (-1+e^x\right ) (x-4)+\left (e^x (x-6)+2\right ) x \log (x)\right )}{(2-x) x \left ((x-4)^2+2 \log (x-2)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 30 \int \left (-\frac {\log (x) (x-4)^2}{(x-2) \left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {2 \log (x-2) \log (x) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {4 \log (x-2) (x-4)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {\log (x-2) (x-4)}{\left (x^2-8 x+2 \log (x-2)+16\right )^2}+\frac {e^x \left (\log (x-2) \log (x) x^4+\log (x-2) x^3-12 \log (x-2) \log (x) x^3+\log (x) x^3-10 \log (x-2) x^2+2 \log ^2(x-2) \log (x) x^2+44 \log (x-2) \log (x) x^2-8 \log (x) x^2+2 \log ^2(x-2) x+32 \log (x-2) x-4 \log ^2(x-2) \log (x) x-48 \log (x-2) \log (x) x+16 \log (x) x-4 \log ^2(x-2)-32 \log (x-2)\right )}{(x-2) x \left (x^2-8 x+2 \log (x-2)+16\right )^2}-\frac {2 \log ^2(x-2)}{x \left (x^2-8 x+2 \log (x-2)+16\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 30 \int \frac {-\left (\left (-1+e^x\right ) x \log (x) (x-4)^2\right )-2 (x-2) \log ^2(x-2) \left (e^x x \log (x)+e^x-1\right )-\left (x^2-6 x+8\right ) \log (x-2) \left (\left (-1+e^x\right ) (x-4)+\left (e^x (x-6)+2\right ) x \log (x)\right )}{(2-x) x \left ((x-4)^2+2 \log (x-2)\right )^2}dx\) |
Int[((960 - 960*x + 300*x^2 - 30*x^3 + E^x*(-960 + 960*x - 300*x^2 + 30*x^ 3))*Log[-2 + x] + (120 - 60*x + E^x*(-120 + 60*x))*Log[-2 + x]^2 + (-480*x + 240*x^2 - 30*x^3 + E^x*(480*x - 240*x^2 + 30*x^3) + (480*x - 360*x^2 + 60*x^3 + E^x*(-1440*x + 1320*x^2 - 360*x^3 + 30*x^4))*Log[-2 + x] + E^x*(- 120*x + 60*x^2)*Log[-2 + x]^2)*Log[x])/(-512*x + 768*x^2 - 448*x^3 + 128*x ^4 - 18*x^5 + x^6 + (-128*x + 128*x^2 - 40*x^3 + 4*x^4)*Log[-2 + x] + (-8* x + 4*x^2)*Log[-2 + x]^2),x]
3.2.84.3.1 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 = 23.53 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52
| method | result | size |
| parallelrisch | \(\frac {240 \ln \left (-2+x \right ) \ln \left (x \right ) {\mathrm e}^{x}-240 \ln \left (x \right ) \ln \left (-2+x \right )}{8 x^{2}+16 \ln \left (-2+x \right )-64 x +128}\) | \(38\) |
| risch | \(15 \,{\mathrm e}^{x} \ln \left (x \right )-15 \ln \left (x \right )-\frac {15 \left ({\mathrm e}^{x} x^{2}-x^{2}-8 \,{\mathrm e}^{x} x +8 x +16 \,{\mathrm e}^{x}-16\right ) \ln \left (x \right )}{x^{2}+2 \ln \left (-2+x \right )-8 x +16}\) | \(57\) |
int((((60*x^2-120*x)*exp(x)*ln(-2+x)^2+((30*x^4-360*x^3+1320*x^2-1440*x)*e xp(x)+60*x^3-360*x^2+480*x)*ln(-2+x)+(30*x^3-240*x^2+480*x)*exp(x)-30*x^3+ 240*x^2-480*x)*ln(x)+((60*x-120)*exp(x)-60*x+120)*ln(-2+x)^2+((30*x^3-300* x^2+960*x-960)*exp(x)-30*x^3+300*x^2-960*x+960)*ln(-2+x))/((4*x^2-8*x)*ln( -2+x)^2+(4*x^4-40*x^3+128*x^2-128*x)*ln(-2+x)+x^6-18*x^5+128*x^4-448*x^3+7 68*x^2-512*x),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {\left (960-960 x+300 x^2-30 x^3+e^x \left (-960+960 x-300 x^2+30 x^3\right )\right ) \log (-2+x)+\left (120-60 x+e^x (-120+60 x)\right ) \log ^2(-2+x)+\left (-480 x+240 x^2-30 x^3+e^x \left (480 x-240 x^2+30 x^3\right )+\left (480 x-360 x^2+60 x^3+e^x \left (-1440 x+1320 x^2-360 x^3+30 x^4\right )\right ) \log (-2+x)+e^x \left (-120 x+60 x^2\right ) \log ^2(-2+x)\right ) \log (x)}{-512 x+768 x^2-448 x^3+128 x^4-18 x^5+x^6+\left (-128 x+128 x^2-40 x^3+4 x^4\right ) \log (-2+x)+\left (-8 x+4 x^2\right ) \log ^2(-2+x)} \, dx=\frac {30 \, {\left (e^{x} - 1\right )} \log \left (x - 2\right ) \log \left (x\right )}{x^{2} - 8 \, x + 2 \, \log \left (x - 2\right ) + 16} \]
integrate((((60*x^2-120*x)*exp(x)*log(-2+x)^2+((30*x^4-360*x^3+1320*x^2-14 40*x)*exp(x)+60*x^3-360*x^2+480*x)*log(-2+x)+(30*x^3-240*x^2+480*x)*exp(x) -30*x^3+240*x^2-480*x)*log(x)+((60*x-120)*exp(x)-60*x+120)*log(-2+x)^2+((3 0*x^3-300*x^2+960*x-960)*exp(x)-30*x^3+300*x^2-960*x+960)*log(-2+x))/((4*x ^2-8*x)*log(-2+x)^2+(4*x^4-40*x^3+128*x^2-128*x)*log(-2+x)+x^6-18*x^5+128* x^4-448*x^3+768*x^2-512*x),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (20) = 40\).
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.72 \[ \int \frac {\left (960-960 x+300 x^2-30 x^3+e^x \left (-960+960 x-300 x^2+30 x^3\right )\right ) \log (-2+x)+\left (120-60 x+e^x (-120+60 x)\right ) \log ^2(-2+x)+\left (-480 x+240 x^2-30 x^3+e^x \left (480 x-240 x^2+30 x^3\right )+\left (480 x-360 x^2+60 x^3+e^x \left (-1440 x+1320 x^2-360 x^3+30 x^4\right )\right ) \log (-2+x)+e^x \left (-120 x+60 x^2\right ) \log ^2(-2+x)\right ) \log (x)}{-512 x+768 x^2-448 x^3+128 x^4-18 x^5+x^6+\left (-128 x+128 x^2-40 x^3+4 x^4\right ) \log (-2+x)+\left (-8 x+4 x^2\right ) \log ^2(-2+x)} \, dx=\frac {15 x^{2} \log {\left (x \right )} - 120 x \log {\left (x \right )} + 240 \log {\left (x \right )}}{x^{2} - 8 x + 2 \log {\left (x - 2 \right )} + 16} - 15 \log {\left (x \right )} + \frac {30 e^{x} \log {\left (x \right )} \log {\left (x - 2 \right )}}{x^{2} - 8 x + 2 \log {\left (x - 2 \right )} + 16} \]
integrate((((60*x**2-120*x)*exp(x)*ln(-2+x)**2+((30*x**4-360*x**3+1320*x** 2-1440*x)*exp(x)+60*x**3-360*x**2+480*x)*ln(-2+x)+(30*x**3-240*x**2+480*x) *exp(x)-30*x**3+240*x**2-480*x)*ln(x)+((60*x-120)*exp(x)-60*x+120)*ln(-2+x )**2+((30*x**3-300*x**2+960*x-960)*exp(x)-30*x**3+300*x**2-960*x+960)*ln(- 2+x))/((4*x**2-8*x)*ln(-2+x)**2+(4*x**4-40*x**3+128*x**2-128*x)*ln(-2+x)+x **6-18*x**5+128*x**4-448*x**3+768*x**2-512*x),x)
(15*x**2*log(x) - 120*x*log(x) + 240*log(x))/(x**2 - 8*x + 2*log(x - 2) + 16) - 15*log(x) + 30*exp(x)*log(x)*log(x - 2)/(x**2 - 8*x + 2*log(x - 2) + 16)
Time = 0.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80 \[ \int \frac {\left (960-960 x+300 x^2-30 x^3+e^x \left (-960+960 x-300 x^2+30 x^3\right )\right ) \log (-2+x)+\left (120-60 x+e^x (-120+60 x)\right ) \log ^2(-2+x)+\left (-480 x+240 x^2-30 x^3+e^x \left (480 x-240 x^2+30 x^3\right )+\left (480 x-360 x^2+60 x^3+e^x \left (-1440 x+1320 x^2-360 x^3+30 x^4\right )\right ) \log (-2+x)+e^x \left (-120 x+60 x^2\right ) \log ^2(-2+x)\right ) \log (x)}{-512 x+768 x^2-448 x^3+128 x^4-18 x^5+x^6+\left (-128 x+128 x^2-40 x^3+4 x^4\right ) \log (-2+x)+\left (-8 x+4 x^2\right ) \log ^2(-2+x)} \, dx=\frac {15 \, {\left (2 \, e^{x} \log \left (x - 2\right ) \log \left (x\right ) + {\left (x^{2} - 8 \, x + 16\right )} \log \left (x\right )\right )}}{x^{2} - 8 \, x + 2 \, \log \left (x - 2\right ) + 16} - 15 \, \log \left (x\right ) \]
integrate((((60*x^2-120*x)*exp(x)*log(-2+x)^2+((30*x^4-360*x^3+1320*x^2-14 40*x)*exp(x)+60*x^3-360*x^2+480*x)*log(-2+x)+(30*x^3-240*x^2+480*x)*exp(x) -30*x^3+240*x^2-480*x)*log(x)+((60*x-120)*exp(x)-60*x+120)*log(-2+x)^2+((3 0*x^3-300*x^2+960*x-960)*exp(x)-30*x^3+300*x^2-960*x+960)*log(-2+x))/((4*x ^2-8*x)*log(-2+x)^2+(4*x^4-40*x^3+128*x^2-128*x)*log(-2+x)+x^6-18*x^5+128* x^4-448*x^3+768*x^2-512*x),x, algorithm=\
15*(2*e^x*log(x - 2)*log(x) + (x^2 - 8*x + 16)*log(x))/(x^2 - 8*x + 2*log( x - 2) + 16) - 15*log(x)
Time = 0.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {\left (960-960 x+300 x^2-30 x^3+e^x \left (-960+960 x-300 x^2+30 x^3\right )\right ) \log (-2+x)+\left (120-60 x+e^x (-120+60 x)\right ) \log ^2(-2+x)+\left (-480 x+240 x^2-30 x^3+e^x \left (480 x-240 x^2+30 x^3\right )+\left (480 x-360 x^2+60 x^3+e^x \left (-1440 x+1320 x^2-360 x^3+30 x^4\right )\right ) \log (-2+x)+e^x \left (-120 x+60 x^2\right ) \log ^2(-2+x)\right ) \log (x)}{-512 x+768 x^2-448 x^3+128 x^4-18 x^5+x^6+\left (-128 x+128 x^2-40 x^3+4 x^4\right ) \log (-2+x)+\left (-8 x+4 x^2\right ) \log ^2(-2+x)} \, dx=\frac {30 \, {\left (e^{x} \log \left (x - 2\right ) \log \left (x\right ) - \log \left (x - 2\right ) \log \left (x\right )\right )}}{x^{2} - 8 \, x + 2 \, \log \left (x - 2\right ) + 16} \]
integrate((((60*x^2-120*x)*exp(x)*log(-2+x)^2+((30*x^4-360*x^3+1320*x^2-14 40*x)*exp(x)+60*x^3-360*x^2+480*x)*log(-2+x)+(30*x^3-240*x^2+480*x)*exp(x) -30*x^3+240*x^2-480*x)*log(x)+((60*x-120)*exp(x)-60*x+120)*log(-2+x)^2+((3 0*x^3-300*x^2+960*x-960)*exp(x)-30*x^3+300*x^2-960*x+960)*log(-2+x))/((4*x ^2-8*x)*log(-2+x)^2+(4*x^4-40*x^3+128*x^2-128*x)*log(-2+x)+x^6-18*x^5+128* x^4-448*x^3+768*x^2-512*x),x, algorithm=\
Timed out. \[ \int \frac {\left (960-960 x+300 x^2-30 x^3+e^x \left (-960+960 x-300 x^2+30 x^3\right )\right ) \log (-2+x)+\left (120-60 x+e^x (-120+60 x)\right ) \log ^2(-2+x)+\left (-480 x+240 x^2-30 x^3+e^x \left (480 x-240 x^2+30 x^3\right )+\left (480 x-360 x^2+60 x^3+e^x \left (-1440 x+1320 x^2-360 x^3+30 x^4\right )\right ) \log (-2+x)+e^x \left (-120 x+60 x^2\right ) \log ^2(-2+x)\right ) \log (x)}{-512 x+768 x^2-448 x^3+128 x^4-18 x^5+x^6+\left (-128 x+128 x^2-40 x^3+4 x^4\right ) \log (-2+x)+\left (-8 x+4 x^2\right ) \log ^2(-2+x)} \, dx=\int -\frac {\ln \left (x-2\right )\,\left (300\,x^2-960\,x-30\,x^3+{\mathrm {e}}^x\,\left (30\,x^3-300\,x^2+960\,x-960\right )+960\right )-\ln \left (x\right )\,\left (480\,x-\ln \left (x-2\right )\,\left (480\,x-{\mathrm {e}}^x\,\left (-30\,x^4+360\,x^3-1320\,x^2+1440\,x\right )-360\,x^2+60\,x^3\right )-240\,x^2+30\,x^3-{\mathrm {e}}^x\,\left (30\,x^3-240\,x^2+480\,x\right )+{\ln \left (x-2\right )}^2\,{\mathrm {e}}^x\,\left (120\,x-60\,x^2\right )\right )+{\ln \left (x-2\right )}^2\,\left ({\mathrm {e}}^x\,\left (60\,x-120\right )-60\,x+120\right )}{512\,x+{\ln \left (x-2\right )}^2\,\left (8\,x-4\,x^2\right )+\ln \left (x-2\right )\,\left (-4\,x^4+40\,x^3-128\,x^2+128\,x\right )-768\,x^2+448\,x^3-128\,x^4+18\,x^5-x^6} \,d x \]
int(-(log(x - 2)*(300*x^2 - 960*x - 30*x^3 + exp(x)*(960*x - 300*x^2 + 30* x^3 - 960) + 960) - log(x)*(480*x - log(x - 2)*(480*x - exp(x)*(1440*x - 1 320*x^2 + 360*x^3 - 30*x^4) - 360*x^2 + 60*x^3) - 240*x^2 + 30*x^3 - exp(x )*(480*x - 240*x^2 + 30*x^3) + log(x - 2)^2*exp(x)*(120*x - 60*x^2)) + log (x - 2)^2*(exp(x)*(60*x - 120) - 60*x + 120))/(512*x + log(x - 2)^2*(8*x - 4*x^2) + log(x - 2)*(128*x - 128*x^2 + 40*x^3 - 4*x^4) - 768*x^2 + 448*x^ 3 - 128*x^4 + 18*x^5 - x^6),x)
int(-(log(x - 2)*(300*x^2 - 960*x - 30*x^3 + exp(x)*(960*x - 300*x^2 + 30* x^3 - 960) + 960) - log(x)*(480*x - log(x - 2)*(480*x - exp(x)*(1440*x - 1 320*x^2 + 360*x^3 - 30*x^4) - 360*x^2 + 60*x^3) - 240*x^2 + 30*x^3 - exp(x )*(480*x - 240*x^2 + 30*x^3) + log(x - 2)^2*exp(x)*(120*x - 60*x^2)) + log (x - 2)^2*(exp(x)*(60*x - 120) - 60*x + 120))/(512*x + log(x - 2)^2*(8*x - 4*x^2) + log(x - 2)*(128*x - 128*x^2 + 40*x^3 - 4*x^4) - 768*x^2 + 448*x^ 3 - 128*x^4 + 18*x^5 - x^6), x)