Integrand size = 237, antiderivative size = 32 \[ \int \frac {\left (4-4 x^2\right ) \log (5)+\left (-1+x^2\right ) \log ^2(5) \log (-1+x)+\left (\left (-4+4 x^2+e^7 \left (-4-4 x+8 x^2\right )\right ) \log (5)+e^7 \left (-x-x^2\right ) \log ^2(5)+\left (1-x^2+e^7 \left (1+x-2 x^2\right )\right ) \log ^2(5) \log (-1+x)\right ) \log (x)+\left (\left (-4-4 x+8 x^2\right ) \log (5)+\left (-x-x^2\right ) \log ^2(5)+\left (1+x-2 x^2\right ) \log ^2(5) \log (-1+x)\right ) \log (x) \log \left (\frac {\log (x)}{x}\right )}{\left (-16 x^2-16 x^3+16 x^4+16 x^5+\left (8 x^2+8 x^3-8 x^4-8 x^5\right ) \log (5) \log (-1+x)+\left (-x^2-x^3+x^4+x^5\right ) \log ^2(5) \log ^2(-1+x)\right ) \log (x)} \, dx=\frac {e^7+\log \left (\frac {\log (x)}{x}\right )}{\left (x+x^2\right ) \left (-\frac {4}{\log (5)}+\log (-1+x)\right )} \]
Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {\left (4-4 x^2\right ) \log (5)+\left (-1+x^2\right ) \log ^2(5) \log (-1+x)+\left (\left (-4+4 x^2+e^7 \left (-4-4 x+8 x^2\right )\right ) \log (5)+e^7 \left (-x-x^2\right ) \log ^2(5)+\left (1-x^2+e^7 \left (1+x-2 x^2\right )\right ) \log ^2(5) \log (-1+x)\right ) \log (x)+\left (\left (-4-4 x+8 x^2\right ) \log (5)+\left (-x-x^2\right ) \log ^2(5)+\left (1+x-2 x^2\right ) \log ^2(5) \log (-1+x)\right ) \log (x) \log \left (\frac {\log (x)}{x}\right )}{\left (-16 x^2-16 x^3+16 x^4+16 x^5+\left (8 x^2+8 x^3-8 x^4-8 x^5\right ) \log (5) \log (-1+x)+\left (-x^2-x^3+x^4+x^5\right ) \log ^2(5) \log ^2(-1+x)\right ) \log (x)} \, dx=\frac {\log (5) \left (e^7+\log \left (\frac {\log (x)}{x}\right )\right )}{x (1+x) (-4+\log (5) \log (-1+x))} \]
Integrate[((4 - 4*x^2)*Log[5] + (-1 + x^2)*Log[5]^2*Log[-1 + x] + ((-4 + 4 *x^2 + E^7*(-4 - 4*x + 8*x^2))*Log[5] + E^7*(-x - x^2)*Log[5]^2 + (1 - x^2 + E^7*(1 + x - 2*x^2))*Log[5]^2*Log[-1 + x])*Log[x] + ((-4 - 4*x + 8*x^2) *Log[5] + (-x - x^2)*Log[5]^2 + (1 + x - 2*x^2)*Log[5]^2*Log[-1 + x])*Log[ x]*Log[Log[x]/x])/((-16*x^2 - 16*x^3 + 16*x^4 + 16*x^5 + (8*x^2 + 8*x^3 - 8*x^4 - 8*x^5)*Log[5]*Log[-1 + x] + (-x^2 - x^3 + x^4 + x^5)*Log[5]^2*Log[ -1 + x]^2)*Log[x]),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.
| \(\displaystyle \int \frac {\left (x^2-1\right ) \log ^2(5) \log (x-1)+\left (e^7 \left (-x^2-x\right ) \log ^2(5)+\left (-x^2+e^7 \left (-2 x^2+x+1\right )+1\right ) \log ^2(5) \log (x-1)+\left (4 x^2+e^7 \left (8 x^2-4 x-4\right )-4\right ) \log (5)\right ) \log (x)+\left (\left (-x^2-x\right ) \log ^2(5)+\left (-2 x^2+x+1\right ) \log ^2(5) \log (x-1)+\left (8 x^2-4 x-4\right ) \log (5)\right ) \log (x) \log \left (\frac {\log (x)}{x}\right )+\left (4-4 x^2\right ) \log (5)}{\left (16 x^5+16 x^4-16 x^3-16 x^2+\left (x^5+x^4-x^3-x^2\right ) \log ^2(5) \log ^2(x-1)+\left (-8 x^5-8 x^4+8 x^3+8 x^2\right ) \log (5) \log (x-1)\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log (5) \left (4 \left (x^2-1\right )+\log (x) \left (-4 x^2+e^7 \left (x^2 (\log (5)-8)+x (4+\log (5))+4\right )+\left (x^2 (\log (5)-8)+x (4+\log (5))+4\right ) \log \left (\frac {\log (x)}{x}\right )+4\right )+(x-1) \log (5) \log (x-1) \left (-x+\log (x) \left (2 e^7 x+x+(2 x+1) \log \left (\frac {\log (x)}{x}\right )+e^7+1\right )-1\right )\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2 \log (x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \log (5) \int -\frac {4 \left (1-x^2\right )-\log (x) \left (-4 x^2+e^7 \left (-\left ((8-\log (5)) x^2\right )+(4+\log (5)) x+4\right )+\left (-\left ((8-\log (5)) x^2\right )+(4+\log (5)) x+4\right ) \log \left (\frac {\log (x)}{x}\right )+4\right )-(1-x) \log (5) \log (x-1) \left (x-\log (x) \left (2 e^7 x+x+(2 x+1) \log \left (\frac {\log (x)}{x}\right )+e^7+1\right )+1\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2 \log (x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\log (5) \int \frac {4 \left (1-x^2\right )-\log (x) \left (-4 x^2+e^7 \left (-\left ((8-\log (5)) x^2\right )+(4+\log (5)) x+4\right )+\left (-\left ((8-\log (5)) x^2\right )+(4+\log (5)) x+4\right ) \log \left (\frac {\log (x)}{x}\right )+4\right )-(1-x) \log (5) \log (x-1) \left (x-\log (x) \left (2 e^7 x+x+(2 x+1) \log \left (\frac {\log (x)}{x}\right )+e^7+1\right )+1\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2 \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {e^7 \left ((8-\log (5)) x^2-(4+\log (5)) x-4\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}+\frac {\left (-2 \log (5) \log (x-1) x^2+8 \left (1-\frac {\log (5)}{8}\right ) x^2+\log (5) \log (x-1) x-4 \left (1+\frac {\log (5)}{4}\right ) x+\log (5) \log (x-1)-4\right ) \log \left (\frac {\log (x)}{x}\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}-\frac {\log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}-\frac {\log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {4}{x^2 (x+1) (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {\left (1+2 e^7\right ) \log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {\left (1+e^7\right ) \log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}-\frac {4}{(x-1) (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {4}{(x-1) x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {-4 \left (x^2-1\right )-\log (x) \left (-4 x^2+e^7 \left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right )+\left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right ) \log \left (\frac {\log (x)}{x}\right )+4\right )-(x-1) \log (5) \log (x-1) \left (-x+\log (x) \left (2 e^7 x+x+(2 x+1) \log \left (\frac {\log (x)}{x}\right )+e^7+1\right )-1\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2 \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {e^7 \left ((8-\log (5)) x^2-(4+\log (5)) x-4\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}+\frac {\left (-2 \log (5) \log (x-1) x^2+8 \left (1-\frac {\log (5)}{8}\right ) x^2+\log (5) \log (x-1) x-4 \left (1+\frac {\log (5)}{4}\right ) x+\log (5) \log (x-1)-4\right ) \log \left (\frac {\log (x)}{x}\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}-\frac {\log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}-\frac {\log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {4}{x^2 (x+1) (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {\left (1+2 e^7\right ) \log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {\left (1+e^7\right ) \log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}-\frac {4}{(x-1) (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {4}{(x-1) x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {-4 \left (x^2-1\right )-\log (x) \left (-4 x^2+e^7 \left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right )+\left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right ) \log \left (\frac {\log (x)}{x}\right )+4\right )-(x-1) \log (5) \log (x-1) \left (-x+\log (x) \left (2 e^7 x+x+(2 x+1) \log \left (\frac {\log (x)}{x}\right )+e^7+1\right )-1\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2 \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {e^7 \left ((8-\log (5)) x^2-(4+\log (5)) x-4\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}+\frac {\left (-2 \log (5) \log (x-1) x^2+8 \left (1-\frac {\log (5)}{8}\right ) x^2+\log (5) \log (x-1) x-4 \left (1+\frac {\log (5)}{4}\right ) x+\log (5) \log (x-1)-4\right ) \log \left (\frac {\log (x)}{x}\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}-\frac {\log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}-\frac {\log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {4}{x^2 (x+1) (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {\left (1+2 e^7\right ) \log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {\left (1+e^7\right ) \log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}-\frac {4}{(x-1) (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {4}{(x-1) x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {-4 \left (x^2-1\right )-\log (x) \left (-4 x^2+e^7 \left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right )+\left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right ) \log \left (\frac {\log (x)}{x}\right )+4\right )-(x-1) \log (5) \log (x-1) \left (-x+\log (x) \left (2 e^7 x+x+(2 x+1) \log \left (\frac {\log (x)}{x}\right )+e^7+1\right )-1\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2 \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {e^7 \left ((8-\log (5)) x^2-(4+\log (5)) x-4\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}+\frac {\left (-2 \log (5) \log (x-1) x^2+8 \left (1-\frac {\log (5)}{8}\right ) x^2+\log (5) \log (x-1) x-4 \left (1+\frac {\log (5)}{4}\right ) x+\log (5) \log (x-1)-4\right ) \log \left (\frac {\log (x)}{x}\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}-\frac {\log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}-\frac {\log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {4}{x^2 (x+1) (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {\left (1+2 e^7\right ) \log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {\left (1+e^7\right ) \log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}-\frac {4}{(x-1) (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {4}{(x-1) x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {-4 \left (x^2-1\right )-\log (x) \left (-4 x^2+e^7 \left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right )+\left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right ) \log \left (\frac {\log (x)}{x}\right )+4\right )-(x-1) \log (5) \log (x-1) \left (-x+\log (x) \left (2 e^7 x+x+(2 x+1) \log \left (\frac {\log (x)}{x}\right )+e^7+1\right )-1\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2 \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {e^7 \left ((8-\log (5)) x^2-(4+\log (5)) x-4\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}+\frac {\left (-2 \log (5) \log (x-1) x^2+8 \left (1-\frac {\log (5)}{8}\right ) x^2+\log (5) \log (x-1) x-4 \left (1+\frac {\log (5)}{4}\right ) x+\log (5) \log (x-1)-4\right ) \log \left (\frac {\log (x)}{x}\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}-\frac {\log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}-\frac {\log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {4}{x^2 (x+1) (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {\left (1+2 e^7\right ) \log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {\left (1+e^7\right ) \log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}-\frac {4}{(x-1) (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {4}{(x-1) x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {-4 \left (x^2-1\right )-\log (x) \left (-4 x^2+e^7 \left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right )+\left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right ) \log \left (\frac {\log (x)}{x}\right )+4\right )-(x-1) \log (5) \log (x-1) \left (-x+\log (x) \left (2 e^7 x+x+(2 x+1) \log \left (\frac {\log (x)}{x}\right )+e^7+1\right )-1\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2 \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {e^7 \left ((8-\log (5)) x^2-(4+\log (5)) x-4\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}+\frac {\left (-2 \log (5) \log (x-1) x^2+8 \left (1-\frac {\log (5)}{8}\right ) x^2+\log (5) \log (x-1) x-4 \left (1+\frac {\log (5)}{4}\right ) x+\log (5) \log (x-1)-4\right ) \log \left (\frac {\log (x)}{x}\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}-\frac {\log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}-\frac {\log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {4}{x^2 (x+1) (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {\left (1+2 e^7\right ) \log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {\left (1+e^7\right ) \log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}-\frac {4}{(x-1) (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {4}{(x-1) x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {-4 \left (x^2-1\right )-\log (x) \left (-4 x^2+e^7 \left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right )+\left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right ) \log \left (\frac {\log (x)}{x}\right )+4\right )-(x-1) \log (5) \log (x-1) \left (-x+\log (x) \left (2 e^7 x+x+(2 x+1) \log \left (\frac {\log (x)}{x}\right )+e^7+1\right )-1\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2 \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {e^7 \left ((8-\log (5)) x^2-(4+\log (5)) x-4\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}+\frac {\left (-2 \log (5) \log (x-1) x^2+8 \left (1-\frac {\log (5)}{8}\right ) x^2+\log (5) \log (x-1) x-4 \left (1+\frac {\log (5)}{4}\right ) x+\log (5) \log (x-1)-4\right ) \log \left (\frac {\log (x)}{x}\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}-\frac {\log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}-\frac {\log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {4}{x^2 (x+1) (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {\left (1+2 e^7\right ) \log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {\left (1+e^7\right ) \log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}-\frac {4}{(x-1) (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {4}{(x-1) x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {-4 \left (x^2-1\right )-\log (x) \left (-4 x^2+e^7 \left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right )+\left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right ) \log \left (\frac {\log (x)}{x}\right )+4\right )-(x-1) \log (5) \log (x-1) \left (-x+\log (x) \left (2 e^7 x+x+(2 x+1) \log \left (\frac {\log (x)}{x}\right )+e^7+1\right )-1\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2 \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {e^7 \left ((8-\log (5)) x^2-(4+\log (5)) x-4\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}+\frac {\left (-2 \log (5) \log (x-1) x^2+8 \left (1-\frac {\log (5)}{8}\right ) x^2+\log (5) \log (x-1) x-4 \left (1+\frac {\log (5)}{4}\right ) x+\log (5) \log (x-1)-4\right ) \log \left (\frac {\log (x)}{x}\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}-\frac {\log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}-\frac {\log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {4}{x^2 (x+1) (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {\left (1+2 e^7\right ) \log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {\left (1+e^7\right ) \log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}-\frac {4}{(x-1) (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {4}{(x-1) x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {-4 \left (x^2-1\right )-\log (x) \left (-4 x^2+e^7 \left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right )+\left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right ) \log \left (\frac {\log (x)}{x}\right )+4\right )-(x-1) \log (5) \log (x-1) \left (-x+\log (x) \left (2 e^7 x+x+(2 x+1) \log \left (\frac {\log (x)}{x}\right )+e^7+1\right )-1\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2 \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {e^7 \left ((8-\log (5)) x^2-(4+\log (5)) x-4\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}+\frac {\left (-2 \log (5) \log (x-1) x^2+8 \left (1-\frac {\log (5)}{8}\right ) x^2+\log (5) \log (x-1) x-4 \left (1+\frac {\log (5)}{4}\right ) x+\log (5) \log (x-1)-4\right ) \log \left (\frac {\log (x)}{x}\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}-\frac {\log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}-\frac {\log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {4}{x^2 (x+1) (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {\left (1+2 e^7\right ) \log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {\left (1+e^7\right ) \log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}-\frac {4}{(x-1) (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {4}{(x-1) x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {-4 \left (x^2-1\right )-\log (x) \left (-4 x^2+e^7 \left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right )+\left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right ) \log \left (\frac {\log (x)}{x}\right )+4\right )-(x-1) \log (5) \log (x-1) \left (-x+\log (x) \left (2 e^7 x+x+(2 x+1) \log \left (\frac {\log (x)}{x}\right )+e^7+1\right )-1\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2 \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {e^7 \left ((8-\log (5)) x^2-(4+\log (5)) x-4\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}+\frac {\left (-2 \log (5) \log (x-1) x^2+8 \left (1-\frac {\log (5)}{8}\right ) x^2+\log (5) \log (x-1) x-4 \left (1+\frac {\log (5)}{4}\right ) x+\log (5) \log (x-1)-4\right ) \log \left (\frac {\log (x)}{x}\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}-\frac {\log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}-\frac {\log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {4}{x^2 (x+1) (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {\left (1+2 e^7\right ) \log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {\left (1+e^7\right ) \log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}-\frac {4}{(x-1) (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {4}{(x-1) x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {-4 \left (x^2-1\right )-\log (x) \left (-4 x^2+e^7 \left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right )+\left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right ) \log \left (\frac {\log (x)}{x}\right )+4\right )-(x-1) \log (5) \log (x-1) \left (-x+\log (x) \left (2 e^7 x+x+(2 x+1) \log \left (\frac {\log (x)}{x}\right )+e^7+1\right )-1\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2 \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {e^7 \left ((8-\log (5)) x^2-(4+\log (5)) x-4\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}+\frac {\left (-2 \log (5) \log (x-1) x^2+8 \left (1-\frac {\log (5)}{8}\right ) x^2+\log (5) \log (x-1) x-4 \left (1+\frac {\log (5)}{4}\right ) x+\log (5) \log (x-1)-4\right ) \log \left (\frac {\log (x)}{x}\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}-\frac {\log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}-\frac {\log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {4}{x^2 (x+1) (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {\left (1+2 e^7\right ) \log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {\left (1+e^7\right ) \log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}-\frac {4}{(x-1) (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {4}{(x-1) x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {-4 \left (x^2-1\right )-\log (x) \left (-4 x^2+e^7 \left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right )+\left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right ) \log \left (\frac {\log (x)}{x}\right )+4\right )-(x-1) \log (5) \log (x-1) \left (-x+\log (x) \left (2 e^7 x+x+(2 x+1) \log \left (\frac {\log (x)}{x}\right )+e^7+1\right )-1\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2 \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {e^7 \left ((8-\log (5)) x^2-(4+\log (5)) x-4\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}+\frac {\left (-2 \log (5) \log (x-1) x^2+8 \left (1-\frac {\log (5)}{8}\right ) x^2+\log (5) \log (x-1) x-4 \left (1+\frac {\log (5)}{4}\right ) x+\log (5) \log (x-1)-4\right ) \log \left (\frac {\log (x)}{x}\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}-\frac {\log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}-\frac {\log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {4}{x^2 (x+1) (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {\left (1+2 e^7\right ) \log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {\left (1+e^7\right ) \log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}-\frac {4}{(x-1) (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {4}{(x-1) x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {-4 \left (x^2-1\right )-\log (x) \left (-4 x^2+e^7 \left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right )+\left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right ) \log \left (\frac {\log (x)}{x}\right )+4\right )-(x-1) \log (5) \log (x-1) \left (-x+\log (x) \left (2 e^7 x+x+(2 x+1) \log \left (\frac {\log (x)}{x}\right )+e^7+1\right )-1\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2 \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {e^7 \left ((8-\log (5)) x^2-(4+\log (5)) x-4\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}+\frac {\left (-2 \log (5) \log (x-1) x^2+8 \left (1-\frac {\log (5)}{8}\right ) x^2+\log (5) \log (x-1) x-4 \left (1+\frac {\log (5)}{4}\right ) x+\log (5) \log (x-1)-4\right ) \log \left (\frac {\log (x)}{x}\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}-\frac {\log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}-\frac {\log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {4}{x^2 (x+1) (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {\left (1+2 e^7\right ) \log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {\left (1+e^7\right ) \log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}-\frac {4}{(x-1) (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {4}{(x-1) x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {-4 \left (x^2-1\right )-\log (x) \left (-4 x^2+e^7 \left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right )+\left ((-8+\log (5)) x^2+(4+\log (5)) x+4\right ) \log \left (\frac {\log (x)}{x}\right )+4\right )-(x-1) \log (5) \log (x-1) \left (-x+\log (x) \left (2 e^7 x+x+(2 x+1) \log \left (\frac {\log (x)}{x}\right )+e^7+1\right )-1\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2 \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {e^7 \left ((8-\log (5)) x^2-(4+\log (5)) x-4\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}+\frac {\left (-2 \log (5) \log (x-1) x^2+8 \left (1-\frac {\log (5)}{8}\right ) x^2+\log (5) \log (x-1) x-4 \left (1+\frac {\log (5)}{4}\right ) x+\log (5) \log (x-1)-4\right ) \log \left (\frac {\log (x)}{x}\right )}{(1-x) x^2 (x+1)^2 (4-\log (5) \log (x-1))^2}-\frac {\log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}-\frac {\log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {4}{x^2 (x+1) (\log (5) \log (x-1)-4)^2 \log (x)}+\frac {\left (1+2 e^7\right ) \log (5) \log (x-1)}{x (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {\left (1+e^7\right ) \log (5) \log (x-1)}{x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}-\frac {4}{(x-1) (x+1)^2 (\log (5) \log (x-1)-4)^2}+\frac {4}{(x-1) x^2 (x+1)^2 (\log (5) \log (x-1)-4)^2}\right )dx\) |
Int[((4 - 4*x^2)*Log[5] + (-1 + x^2)*Log[5]^2*Log[-1 + x] + ((-4 + 4*x^2 + E^7*(-4 - 4*x + 8*x^2))*Log[5] + E^7*(-x - x^2)*Log[5]^2 + (1 - x^2 + E^7 *(1 + x - 2*x^2))*Log[5]^2*Log[-1 + x])*Log[x] + ((-4 - 4*x + 8*x^2)*Log[5 ] + (-x - x^2)*Log[5]^2 + (1 + x - 2*x^2)*Log[5]^2*Log[-1 + x])*Log[x]*Log [Log[x]/x])/((-16*x^2 - 16*x^3 + 16*x^4 + 16*x^5 + (8*x^2 + 8*x^3 - 8*x^4 - 8*x^5)*Log[5]*Log[-1 + x] + (-x^2 - x^3 + x^4 + x^5)*Log[5]^2*Log[-1 + x ]^2)*Log[x]),x]
3.7.75.3.1 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 = 105.99 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25
| method | result | size |
| parallelrisch | \(\frac {4 \ln \left (5\right ) \ln \left (\frac {\ln \left (x \right )}{x}\right )+4 \ln \left (5\right ) {\mathrm e}^{7}}{4 x \left (1+x \right ) \left (\ln \left (5\right ) \ln \left (-1+x \right )-4\right )}\) | \(40\) |
| risch | \(\frac {\ln \left (5\right ) \ln \left (\ln \left (x \right )\right )}{\left (1+x \right ) x \left (\ln \left (5\right ) \ln \left (-1+x \right )-4\right )}+\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right )^{2}+i \pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right )^{3}+2 \,{\mathrm e}^{7}-2 \ln \left (x \right )\right ) \ln \left (5\right )}{2 \left (1+x \right ) x \left (\ln \left (5\right ) \ln \left (-1+x \right )-4\right )}\) | \(143\) |
int((((-2*x^2+x+1)*ln(5)^2*ln(-1+x)+(-x^2-x)*ln(5)^2+(8*x^2-4*x-4)*ln(5))* ln(x)*ln(ln(x)/x)+(((-2*x^2+x+1)*exp(7)-x^2+1)*ln(5)^2*ln(-1+x)+(-x^2-x)*e xp(7)*ln(5)^2+((8*x^2-4*x-4)*exp(7)+4*x^2-4)*ln(5))*ln(x)+(x^2-1)*ln(5)^2* ln(-1+x)+(-4*x^2+4)*ln(5))/((x^5+x^4-x^3-x^2)*ln(5)^2*ln(-1+x)^2+(-8*x^5-8 *x^4+8*x^3+8*x^2)*ln(5)*ln(-1+x)+16*x^5+16*x^4-16*x^3-16*x^2)/ln(x),x,meth od=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int \frac {\left (4-4 x^2\right ) \log (5)+\left (-1+x^2\right ) \log ^2(5) \log (-1+x)+\left (\left (-4+4 x^2+e^7 \left (-4-4 x+8 x^2\right )\right ) \log (5)+e^7 \left (-x-x^2\right ) \log ^2(5)+\left (1-x^2+e^7 \left (1+x-2 x^2\right )\right ) \log ^2(5) \log (-1+x)\right ) \log (x)+\left (\left (-4-4 x+8 x^2\right ) \log (5)+\left (-x-x^2\right ) \log ^2(5)+\left (1+x-2 x^2\right ) \log ^2(5) \log (-1+x)\right ) \log (x) \log \left (\frac {\log (x)}{x}\right )}{\left (-16 x^2-16 x^3+16 x^4+16 x^5+\left (8 x^2+8 x^3-8 x^4-8 x^5\right ) \log (5) \log (-1+x)+\left (-x^2-x^3+x^4+x^5\right ) \log ^2(5) \log ^2(-1+x)\right ) \log (x)} \, dx=\frac {e^{7} \log \left (5\right ) + \log \left (5\right ) \log \left (\frac {\log \left (x\right )}{x}\right )}{{\left (x^{2} + x\right )} \log \left (5\right ) \log \left (x - 1\right ) - 4 \, x^{2} - 4 \, x} \]
integrate((((-2*x^2+x+1)*log(5)^2*log(-1+x)+(-x^2-x)*log(5)^2+(8*x^2-4*x-4 )*log(5))*log(x)*log(log(x)/x)+(((-2*x^2+x+1)*exp(7)-x^2+1)*log(5)^2*log(- 1+x)+(-x^2-x)*exp(7)*log(5)^2+((8*x^2-4*x-4)*exp(7)+4*x^2-4)*log(5))*log(x )+(x^2-1)*log(5)^2*log(-1+x)+(-4*x^2+4)*log(5))/((x^5+x^4-x^3-x^2)*log(5)^ 2*log(-1+x)^2+(-8*x^5-8*x^4+8*x^3+8*x^2)*log(5)*log(-1+x)+16*x^5+16*x^4-16 *x^3-16*x^2)/log(x),x, algorithm=\
Exception generated. \[ \int \frac {\left (4-4 x^2\right ) \log (5)+\left (-1+x^2\right ) \log ^2(5) \log (-1+x)+\left (\left (-4+4 x^2+e^7 \left (-4-4 x+8 x^2\right )\right ) \log (5)+e^7 \left (-x-x^2\right ) \log ^2(5)+\left (1-x^2+e^7 \left (1+x-2 x^2\right )\right ) \log ^2(5) \log (-1+x)\right ) \log (x)+\left (\left (-4-4 x+8 x^2\right ) \log (5)+\left (-x-x^2\right ) \log ^2(5)+\left (1+x-2 x^2\right ) \log ^2(5) \log (-1+x)\right ) \log (x) \log \left (\frac {\log (x)}{x}\right )}{\left (-16 x^2-16 x^3+16 x^4+16 x^5+\left (8 x^2+8 x^3-8 x^4-8 x^5\right ) \log (5) \log (-1+x)+\left (-x^2-x^3+x^4+x^5\right ) \log ^2(5) \log ^2(-1+x)\right ) \log (x)} \, dx=\text {Exception raised: TypeError} \]
integrate((((-2*x**2+x+1)*ln(5)**2*ln(-1+x)+(-x**2-x)*ln(5)**2+(8*x**2-4*x -4)*ln(5))*ln(x)*ln(ln(x)/x)+(((-2*x**2+x+1)*exp(7)-x**2+1)*ln(5)**2*ln(-1 +x)+(-x**2-x)*exp(7)*ln(5)**2+((8*x**2-4*x-4)*exp(7)+4*x**2-4)*ln(5))*ln(x )+(x**2-1)*ln(5)**2*ln(-1+x)+(-4*x**2+4)*ln(5))/((x**5+x**4-x**3-x**2)*ln( 5)**2*ln(-1+x)**2+(-8*x**5-8*x**4+8*x**3+8*x**2)*ln(5)*ln(-1+x)+16*x**5+16 *x**4-16*x**3-16*x**2)/ln(x),x)
Time = 0.34 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \frac {\left (4-4 x^2\right ) \log (5)+\left (-1+x^2\right ) \log ^2(5) \log (-1+x)+\left (\left (-4+4 x^2+e^7 \left (-4-4 x+8 x^2\right )\right ) \log (5)+e^7 \left (-x-x^2\right ) \log ^2(5)+\left (1-x^2+e^7 \left (1+x-2 x^2\right )\right ) \log ^2(5) \log (-1+x)\right ) \log (x)+\left (\left (-4-4 x+8 x^2\right ) \log (5)+\left (-x-x^2\right ) \log ^2(5)+\left (1+x-2 x^2\right ) \log ^2(5) \log (-1+x)\right ) \log (x) \log \left (\frac {\log (x)}{x}\right )}{\left (-16 x^2-16 x^3+16 x^4+16 x^5+\left (8 x^2+8 x^3-8 x^4-8 x^5\right ) \log (5) \log (-1+x)+\left (-x^2-x^3+x^4+x^5\right ) \log ^2(5) \log ^2(-1+x)\right ) \log (x)} \, dx=-\frac {e^{7} \log \left (5\right ) - \log \left (5\right ) \log \left (x\right ) + \log \left (5\right ) \log \left (\log \left (x\right )\right )}{4 \, x^{2} - {\left (x^{2} \log \left (5\right ) + x \log \left (5\right )\right )} \log \left (x - 1\right ) + 4 \, x} \]
integrate((((-2*x^2+x+1)*log(5)^2*log(-1+x)+(-x^2-x)*log(5)^2+(8*x^2-4*x-4 )*log(5))*log(x)*log(log(x)/x)+(((-2*x^2+x+1)*exp(7)-x^2+1)*log(5)^2*log(- 1+x)+(-x^2-x)*exp(7)*log(5)^2+((8*x^2-4*x-4)*exp(7)+4*x^2-4)*log(5))*log(x )+(x^2-1)*log(5)^2*log(-1+x)+(-4*x^2+4)*log(5))/((x^5+x^4-x^3-x^2)*log(5)^ 2*log(-1+x)^2+(-8*x^5-8*x^4+8*x^3+8*x^2)*log(5)*log(-1+x)+16*x^5+16*x^4-16 *x^3-16*x^2)/log(x),x, algorithm=\
-(e^7*log(5) - log(5)*log(x) + log(5)*log(log(x)))/(4*x^2 - (x^2*log(5) + x*log(5))*log(x - 1) + 4*x)
Time = 0.35 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \frac {\left (4-4 x^2\right ) \log (5)+\left (-1+x^2\right ) \log ^2(5) \log (-1+x)+\left (\left (-4+4 x^2+e^7 \left (-4-4 x+8 x^2\right )\right ) \log (5)+e^7 \left (-x-x^2\right ) \log ^2(5)+\left (1-x^2+e^7 \left (1+x-2 x^2\right )\right ) \log ^2(5) \log (-1+x)\right ) \log (x)+\left (\left (-4-4 x+8 x^2\right ) \log (5)+\left (-x-x^2\right ) \log ^2(5)+\left (1+x-2 x^2\right ) \log ^2(5) \log (-1+x)\right ) \log (x) \log \left (\frac {\log (x)}{x}\right )}{\left (-16 x^2-16 x^3+16 x^4+16 x^5+\left (8 x^2+8 x^3-8 x^4-8 x^5\right ) \log (5) \log (-1+x)+\left (-x^2-x^3+x^4+x^5\right ) \log ^2(5) \log ^2(-1+x)\right ) \log (x)} \, dx=\frac {e^{7} \log \left (5\right ) - \log \left (5\right ) \log \left (x\right ) + \log \left (5\right ) \log \left (\log \left (x\right )\right )}{x^{2} \log \left (5\right ) \log \left (x - 1\right ) + x \log \left (5\right ) \log \left (x - 1\right ) - 4 \, x^{2} - 4 \, x} \]
integrate((((-2*x^2+x+1)*log(5)^2*log(-1+x)+(-x^2-x)*log(5)^2+(8*x^2-4*x-4 )*log(5))*log(x)*log(log(x)/x)+(((-2*x^2+x+1)*exp(7)-x^2+1)*log(5)^2*log(- 1+x)+(-x^2-x)*exp(7)*log(5)^2+((8*x^2-4*x-4)*exp(7)+4*x^2-4)*log(5))*log(x )+(x^2-1)*log(5)^2*log(-1+x)+(-4*x^2+4)*log(5))/((x^5+x^4-x^3-x^2)*log(5)^ 2*log(-1+x)^2+(-8*x^5-8*x^4+8*x^3+8*x^2)*log(5)*log(-1+x)+16*x^5+16*x^4-16 *x^3-16*x^2)/log(x),x, algorithm=\
(e^7*log(5) - log(5)*log(x) + log(5)*log(log(x)))/(x^2*log(5)*log(x - 1) + x*log(5)*log(x - 1) - 4*x^2 - 4*x)
Time = 9.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {\left (4-4 x^2\right ) \log (5)+\left (-1+x^2\right ) \log ^2(5) \log (-1+x)+\left (\left (-4+4 x^2+e^7 \left (-4-4 x+8 x^2\right )\right ) \log (5)+e^7 \left (-x-x^2\right ) \log ^2(5)+\left (1-x^2+e^7 \left (1+x-2 x^2\right )\right ) \log ^2(5) \log (-1+x)\right ) \log (x)+\left (\left (-4-4 x+8 x^2\right ) \log (5)+\left (-x-x^2\right ) \log ^2(5)+\left (1+x-2 x^2\right ) \log ^2(5) \log (-1+x)\right ) \log (x) \log \left (\frac {\log (x)}{x}\right )}{\left (-16 x^2-16 x^3+16 x^4+16 x^5+\left (8 x^2+8 x^3-8 x^4-8 x^5\right ) \log (5) \log (-1+x)+\left (-x^2-x^3+x^4+x^5\right ) \log ^2(5) \log ^2(-1+x)\right ) \log (x)} \, dx=\frac {\ln \left (5\right )\,\left ({\mathrm {e}}^7+\ln \left (\frac {\ln \left (x\right )}{x}\right )\right )}{x\,\left (\ln \left (x-1\right )\,\ln \left (5\right )-4\right )\,\left (x+1\right )} \]
int((log(x)*(log(5)*(exp(7)*(4*x - 8*x^2 + 4) - 4*x^2 + 4) + exp(7)*log(5) ^2*(x + x^2) - log(x - 1)*log(5)^2*(exp(7)*(x - 2*x^2 + 1) - x^2 + 1)) + l og(5)*(4*x^2 - 4) - log(x - 1)*log(5)^2*(x^2 - 1) + log(log(x)/x)*log(x)*( log(5)*(4*x - 8*x^2 + 4) + log(5)^2*(x + x^2) - log(x - 1)*log(5)^2*(x - 2 *x^2 + 1)))/(log(x)*(16*x^2 + 16*x^3 - 16*x^4 - 16*x^5 - log(x - 1)*log(5) *(8*x^2 + 8*x^3 - 8*x^4 - 8*x^5) + log(x - 1)^2*log(5)^2*(x^2 + x^3 - x^4 - x^5))),x)