Integrand size = 277, antiderivative size = 30 \[ \int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx=\left (1-\frac {x}{-\frac {\log (3)}{x}+\frac {\log (-e+2 x)}{\log (x)}}\right )^2 \] Output:
(1-x/(ln(-exp(1)+2*x)/ln(x)-ln(3)/x))^2
\[ \int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx=\int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx \] Input:
Integrate[(((4*E*x^3 - 8*x^4)*Log[3] + (4*E*x - 8*x^2)*Log[3]^2)*Log[x]^3 + (2*E*x^3 - 4*x^4)*Log[-E + 2*x]^2 + Log[x]^2*(-4*x^5 - 4*x^3*Log[3] + (- 2*E*x^4 + 4*x^5 + (-6*E*x^2 + 12*x^3)*Log[3])*Log[-E + 2*x]) + Log[x]*((4* x^4 - 2*E*x^4 + 4*x^5 + (-2*E*x^2 + 4*x^3)*Log[3])*Log[-E + 2*x] + (2*E*x^ 3 - 4*x^4)*Log[-E + 2*x]^2))/((E - 2*x)*Log[3]^3*Log[x]^3 + (-3*E*x + 6*x^ 2)*Log[3]^2*Log[x]^2*Log[-E + 2*x] + (3*E*x^2 - 6*x^3)*Log[3]*Log[x]*Log[- E + 2*x]^2 + (-(E*x^3) + 2*x^4)*Log[-E + 2*x]^3),x]
Output:
Integrate[(((4*E*x^3 - 8*x^4)*Log[3] + (4*E*x - 8*x^2)*Log[3]^2)*Log[x]^3 + (2*E*x^3 - 4*x^4)*Log[-E + 2*x]^2 + Log[x]^2*(-4*x^5 - 4*x^3*Log[3] + (- 2*E*x^4 + 4*x^5 + (-6*E*x^2 + 12*x^3)*Log[3])*Log[-E + 2*x]) + Log[x]*((4* x^4 - 2*E*x^4 + 4*x^5 + (-2*E*x^2 + 4*x^3)*Log[3])*Log[-E + 2*x] + (2*E*x^ 3 - 4*x^4)*Log[-E + 2*x]^2))/((E - 2*x)*Log[3]^3*Log[x]^3 + (-3*E*x + 6*x^ 2)*Log[3]^2*Log[x]^2*Log[-E + 2*x] + (3*E*x^2 - 6*x^3)*Log[3]*Log[x]*Log[- E + 2*x]^2 + (-(E*x^3) + 2*x^4)*Log[-E + 2*x]^3), 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.
| \(\displaystyle \int \frac {\left (2 e x^3-4 x^4\right ) \log ^2(2 x-e)+\left (\left (4 e x-8 x^2\right ) \log ^2(3)+\left (4 e x^3-8 x^4\right ) \log (3)\right ) \log ^3(x)+\left (-4 x^5-4 x^3 \log (3)+\left (4 x^5-2 e x^4+\left (12 x^3-6 e x^2\right ) \log (3)\right ) \log (2 x-e)\right ) \log ^2(x)+\left (\left (2 e x^3-4 x^4\right ) \log ^2(2 x-e)+\left (4 x^5-2 e x^4+4 x^4+\left (4 x^3-2 e x^2\right ) \log (3)\right ) \log (2 x-e)\right ) \log (x)}{\left (6 x^2-3 e x\right ) \log ^2(3) \log (2 x-e) \log ^2(x)+\left (2 x^4-e x^3\right ) \log ^3(2 x-e)+\left (3 e x^2-6 x^3\right ) \log (3) \log ^2(2 x-e) \log (x)+(e-2 x) \log ^3(3) \log ^3(x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 x \left (\left (x^2+\log (3)\right ) \log (x)-x \log (2 x-e)\right ) \left (-\left ((x \log (81)-e \log (9)) \log ^2(x)\right )-x (2 x+(e-2 x) \log (2 x-e)) \log (x)-(e-2 x) x \log (2 x-e)\right )}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {x \left (\left (x^2+\log (3)\right ) \log (x)-x \log (2 x-e)\right ) \left ((e \log (9)-x \log (81)) \log ^2(x)-x (2 x+(e-2 x) \log (2 x-e)) \log (x)-(e-2 x) x \log (2 x-e)\right )}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (-\frac {\log ^2(x) \left (-2 x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x^3}{(e-2 x) (x \log (2 x-e)-\log (3) \log (x))^3}-\frac {(\log (x)+1) x}{x \log (2 x-e)-\log (3) \log (x)}+\frac {\log (x) \left (-2 \log (x) x^3-2 x^3+e \log (x) x^2-2 \left (1-\frac {e}{2}\right ) x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x \left (\left (x^2+\log (3)\right ) \log (x)-x \log (2 x-e)\right ) \left ((e-2 x) \log (9) \log ^2(x)-x (2 x+(e-2 x) \log (2 x-e)) \log (x)-(e-2 x) x \log (2 x-e)\right )}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (-\frac {\log ^2(x) \left (-2 x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x^3}{(e-2 x) (x \log (2 x-e)-\log (3) \log (x))^3}-\frac {(\log (x)+1) x}{x \log (2 x-e)-\log (3) \log (x)}+\frac {\log (x) \left (-2 \log (x) x^3-2 x^3+e \log (x) x^2-2 \left (1-\frac {e}{2}\right ) x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x \left (\left (x^2+\log (3)\right ) \log (x)-x \log (2 x-e)\right ) \left ((e-2 x) \log (9) \log ^2(x)-x (2 x+(e-2 x) \log (2 x-e)) \log (x)-(e-2 x) x \log (2 x-e)\right )}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (-\frac {\log ^2(x) \left (-2 x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x^3}{(e-2 x) (x \log (2 x-e)-\log (3) \log (x))^3}-\frac {(\log (x)+1) x}{x \log (2 x-e)-\log (3) \log (x)}+\frac {\log (x) \left (-2 \log (x) x^3-2 x^3+e \log (x) x^2-2 \left (1-\frac {e}{2}\right ) x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x \left (\left (x^2+\log (3)\right ) \log (x)-x \log (2 x-e)\right ) \left ((e-2 x) \log (9) \log ^2(x)-x (2 x+(e-2 x) \log (2 x-e)) \log (x)-(e-2 x) x \log (2 x-e)\right )}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (-\frac {\log ^2(x) \left (-2 x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x^3}{(e-2 x) (x \log (2 x-e)-\log (3) \log (x))^3}-\frac {(\log (x)+1) x}{x \log (2 x-e)-\log (3) \log (x)}+\frac {\log (x) \left (-2 \log (x) x^3-2 x^3+e \log (x) x^2-2 \left (1-\frac {e}{2}\right ) x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x \left (\left (x^2+\log (3)\right ) \log (x)-x \log (2 x-e)\right ) \left ((e-2 x) \log (9) \log ^2(x)-x (2 x+(e-2 x) \log (2 x-e)) \log (x)-(e-2 x) x \log (2 x-e)\right )}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (-\frac {\log ^2(x) \left (-2 x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x^3}{(e-2 x) (x \log (2 x-e)-\log (3) \log (x))^3}-\frac {(\log (x)+1) x}{x \log (2 x-e)-\log (3) \log (x)}+\frac {\log (x) \left (-2 \log (x) x^3-2 x^3+e \log (x) x^2-2 \left (1-\frac {e}{2}\right ) x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x \left (\left (x^2+\log (3)\right ) \log (x)-x \log (2 x-e)\right ) \left ((e-2 x) \log (9) \log ^2(x)-x (2 x+(e-2 x) \log (2 x-e)) \log (x)-(e-2 x) x \log (2 x-e)\right )}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (-\frac {\log ^2(x) \left (-2 x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x^3}{(e-2 x) (x \log (2 x-e)-\log (3) \log (x))^3}-\frac {(\log (x)+1) x}{x \log (2 x-e)-\log (3) \log (x)}+\frac {\log (x) \left (-2 \log (x) x^3-2 x^3+e \log (x) x^2-2 \left (1-\frac {e}{2}\right ) x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x \left (\left (x^2+\log (3)\right ) \log (x)-x \log (2 x-e)\right ) \left ((e-2 x) \log (9) \log ^2(x)-x (2 x+(e-2 x) \log (2 x-e)) \log (x)-(e-2 x) x \log (2 x-e)\right )}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (-\frac {\log ^2(x) \left (-2 x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x^3}{(e-2 x) (x \log (2 x-e)-\log (3) \log (x))^3}-\frac {(\log (x)+1) x}{x \log (2 x-e)-\log (3) \log (x)}+\frac {\log (x) \left (-2 \log (x) x^3-2 x^3+e \log (x) x^2-2 \left (1-\frac {e}{2}\right ) x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x \left (\left (x^2+\log (3)\right ) \log (x)-x \log (2 x-e)\right ) \left ((e-2 x) \log (9) \log ^2(x)-x (2 x+(e-2 x) \log (2 x-e)) \log (x)-(e-2 x) x \log (2 x-e)\right )}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (-\frac {\log ^2(x) \left (-2 x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x^3}{(e-2 x) (x \log (2 x-e)-\log (3) \log (x))^3}-\frac {(\log (x)+1) x}{x \log (2 x-e)-\log (3) \log (x)}+\frac {\log (x) \left (-2 \log (x) x^3-2 x^3+e \log (x) x^2-2 \left (1-\frac {e}{2}\right ) x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x \left (\left (x^2+\log (3)\right ) \log (x)-x \log (2 x-e)\right ) \left ((e-2 x) \log (9) \log ^2(x)-x (2 x+(e-2 x) \log (2 x-e)) \log (x)-(e-2 x) x \log (2 x-e)\right )}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (-\frac {\log ^2(x) \left (-2 x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x^3}{(e-2 x) (x \log (2 x-e)-\log (3) \log (x))^3}-\frac {(\log (x)+1) x}{x \log (2 x-e)-\log (3) \log (x)}+\frac {\log (x) \left (-2 \log (x) x^3-2 x^3+e \log (x) x^2-2 \left (1-\frac {e}{2}\right ) x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x \left (\left (x^2+\log (3)\right ) \log (x)-x \log (2 x-e)\right ) \left ((e-2 x) \log (9) \log ^2(x)-x (2 x+(e-2 x) \log (2 x-e)) \log (x)-(e-2 x) x \log (2 x-e)\right )}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (-\frac {\log ^2(x) \left (-2 x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x^3}{(e-2 x) (x \log (2 x-e)-\log (3) \log (x))^3}-\frac {(\log (x)+1) x}{x \log (2 x-e)-\log (3) \log (x)}+\frac {\log (x) \left (-2 \log (x) x^3-2 x^3+e \log (x) x^2-2 \left (1-\frac {e}{2}\right ) x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x \left (\left (x^2+\log (3)\right ) \log (x)-x \log (2 x-e)\right ) \left ((e-2 x) \log (9) \log ^2(x)-x (2 x+(e-2 x) \log (2 x-e)) \log (x)-(e-2 x) x \log (2 x-e)\right )}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (-\frac {\log ^2(x) \left (-2 x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x^3}{(e-2 x) (x \log (2 x-e)-\log (3) \log (x))^3}-\frac {(\log (x)+1) x}{x \log (2 x-e)-\log (3) \log (x)}+\frac {\log (x) \left (-2 \log (x) x^3-2 x^3+e \log (x) x^2-2 \left (1-\frac {e}{2}\right ) x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x \left (\left (x^2+\log (3)\right ) \log (x)-x \log (2 x-e)\right ) \left ((e-2 x) \log (9) \log ^2(x)-x (2 x+(e-2 x) \log (2 x-e)) \log (x)-(e-2 x) x \log (2 x-e)\right )}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (-\frac {\log ^2(x) \left (-2 x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x^3}{(e-2 x) (x \log (2 x-e)-\log (3) \log (x))^3}-\frac {(\log (x)+1) x}{x \log (2 x-e)-\log (3) \log (x)}+\frac {\log (x) \left (-2 \log (x) x^3-2 x^3+e \log (x) x^2-2 \left (1-\frac {e}{2}\right ) x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x \left (\left (x^2+\log (3)\right ) \log (x)-x \log (2 x-e)\right ) \left ((e-2 x) \log (9) \log ^2(x)-x (2 x+(e-2 x) \log (2 x-e)) \log (x)-(e-2 x) x \log (2 x-e)\right )}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (-\frac {\log ^2(x) \left (-2 x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x^3}{(e-2 x) (x \log (2 x-e)-\log (3) \log (x))^3}-\frac {(\log (x)+1) x}{x \log (2 x-e)-\log (3) \log (x)}+\frac {\log (x) \left (-2 \log (x) x^3-2 x^3+e \log (x) x^2-2 \left (1-\frac {e}{2}\right ) x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x \left (\left (x^2+\log (3)\right ) \log (x)-x \log (2 x-e)\right ) \left ((e-2 x) \log (9) \log ^2(x)-x (2 x+(e-2 x) \log (2 x-e)) \log (x)-(e-2 x) x \log (2 x-e)\right )}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (-\frac {\log ^2(x) \left (-2 x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x^3}{(e-2 x) (x \log (2 x-e)-\log (3) \log (x))^3}-\frac {(\log (x)+1) x}{x \log (2 x-e)-\log (3) \log (x)}+\frac {\log (x) \left (-2 \log (x) x^3-2 x^3+e \log (x) x^2-2 \left (1-\frac {e}{2}\right ) x^2-\log (9) \log (x) x+\log (9) x+e \log (3) \log (x)-e \log (3)\right ) x}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x \left (\left (x^2+\log (3)\right ) \log (x)-x \log (2 x-e)\right ) \left ((e-2 x) \log (9) \log ^2(x)-x (2 x+(e-2 x) \log (2 x-e)) \log (x)-(e-2 x) x \log (2 x-e)\right )}{(e-2 x) (\log (3) \log (x)-x \log (2 x-e))^3}dx\) |
Input:
Int[(((4*E*x^3 - 8*x^4)*Log[3] + (4*E*x - 8*x^2)*Log[3]^2)*Log[x]^3 + (2*E *x^3 - 4*x^4)*Log[-E + 2*x]^2 + Log[x]^2*(-4*x^5 - 4*x^3*Log[3] + (-2*E*x^ 4 + 4*x^5 + (-6*E*x^2 + 12*x^3)*Log[3])*Log[-E + 2*x]) + Log[x]*((4*x^4 - 2*E*x^4 + 4*x^5 + (-2*E*x^2 + 4*x^3)*Log[3])*Log[-E + 2*x] + (2*E*x^3 - 4* x^4)*Log[-E + 2*x]^2))/((E - 2*x)*Log[3]^3*Log[x]^3 + (-3*E*x + 6*x^2)*Log [3]^2*Log[x]^2*Log[-E + 2*x] + (3*E*x^2 - 6*x^3)*Log[3]*Log[x]*Log[-E + 2* x]^2 + (-(E*x^3) + 2*x^4)*Log[-E + 2*x]^3),x]
Output:
$Aborted
Time = 0.36 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.73
\[\frac {\left (x^{2} \ln \left (x \right )+2 \ln \left (3\right ) \ln \left (x \right )-2 \ln \left (-{\mathrm e}+2 x \right ) x \right ) x^{2} \ln \left (x \right )}{\left (\ln \left (3\right ) \ln \left (x \right )-\ln \left (-{\mathrm e}+2 x \right ) x \right )^{2}}\]
Input:
int((((4*x*exp(1)-8*x^2)*ln(3)^2+(4*x^3*exp(1)-8*x^4)*ln(3))*ln(x)^3+(((-6 *x^2*exp(1)+12*x^3)*ln(3)-2*x^4*exp(1)+4*x^5)*ln(-exp(1)+2*x)-4*x^3*ln(3)- 4*x^5)*ln(x)^2+((2*x^3*exp(1)-4*x^4)*ln(-exp(1)+2*x)^2+((-2*x^2*exp(1)+4*x ^3)*ln(3)-2*x^4*exp(1)+4*x^5+4*x^4)*ln(-exp(1)+2*x))*ln(x)+(2*x^3*exp(1)-4 *x^4)*ln(-exp(1)+2*x)^2)/((exp(1)-2*x)*ln(3)^3*ln(x)^3+(-3*x*exp(1)+6*x^2) *ln(3)^2*ln(-exp(1)+2*x)*ln(x)^2+(3*x^2*exp(1)-6*x^3)*ln(3)*ln(-exp(1)+2*x )^2*ln(x)+(-x^3*exp(1)+2*x^4)*ln(-exp(1)+2*x)^3),x)
Output:
(x^2*ln(x)+2*ln(3)*ln(x)-2*ln(-exp(1)+2*x)*x)*x^2*ln(x)/(ln(3)*ln(x)-ln(-e xp(1)+2*x)*x)^2
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (30) = 60\).
Time = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.63 \[ \int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx=-\frac {2 \, x^{3} \log \left (2 \, x - e\right ) \log \left (x\right ) - {\left (x^{4} + 2 \, x^{2} \log \left (3\right )\right )} \log \left (x\right )^{2}}{x^{2} \log \left (2 \, x - e\right )^{2} - 2 \, x \log \left (3\right ) \log \left (2 \, x - e\right ) \log \left (x\right ) + \log \left (3\right )^{2} \log \left (x\right )^{2}} \] Input:
integrate((((4*exp(1)*x-8*x^2)*log(3)^2+(4*x^3*exp(1)-8*x^4)*log(3))*log(x )^3+(((-6*x^2*exp(1)+12*x^3)*log(3)-2*x^4*exp(1)+4*x^5)*log(-exp(1)+2*x)-4 *x^3*log(3)-4*x^5)*log(x)^2+((2*x^3*exp(1)-4*x^4)*log(-exp(1)+2*x)^2+((-2* x^2*exp(1)+4*x^3)*log(3)-2*x^4*exp(1)+4*x^5+4*x^4)*log(-exp(1)+2*x))*log(x )+(2*x^3*exp(1)-4*x^4)*log(-exp(1)+2*x)^2)/((exp(1)-2*x)*log(3)^3*log(x)^3 +(-3*exp(1)*x+6*x^2)*log(3)^2*log(-exp(1)+2*x)*log(x)^2+(3*x^2*exp(1)-6*x^ 3)*log(3)*log(-exp(1)+2*x)^2*log(x)+(-x^3*exp(1)+2*x^4)*log(-exp(1)+2*x)^3 ),x, algorithm="fricas")
Output:
-(2*x^3*log(2*x - e)*log(x) - (x^4 + 2*x^2*log(3))*log(x)^2)/(x^2*log(2*x - e)^2 - 2*x*log(3)*log(2*x - e)*log(x) + log(3)^2*log(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (20) = 40\).
Time = 0.21 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.67 \[ \int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx=\frac {x^{4} \log {\left (x \right )}^{2} - 2 x^{3} \log {\left (x \right )} \log {\left (2 x - e \right )} + 2 x^{2} \log {\left (3 \right )} \log {\left (x \right )}^{2}}{x^{2} \log {\left (2 x - e \right )}^{2} - 2 x \log {\left (3 \right )} \log {\left (x \right )} \log {\left (2 x - e \right )} + \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2}} \] Input:
integrate((((4*exp(1)*x-8*x**2)*ln(3)**2+(4*x**3*exp(1)-8*x**4)*ln(3))*ln( x)**3+(((-6*x**2*exp(1)+12*x**3)*ln(3)-2*x**4*exp(1)+4*x**5)*ln(-exp(1)+2* x)-4*x**3*ln(3)-4*x**5)*ln(x)**2+((2*x**3*exp(1)-4*x**4)*ln(-exp(1)+2*x)** 2+((-2*x**2*exp(1)+4*x**3)*ln(3)-2*x**4*exp(1)+4*x**5+4*x**4)*ln(-exp(1)+2 *x))*ln(x)+(2*x**3*exp(1)-4*x**4)*ln(-exp(1)+2*x)**2)/((exp(1)-2*x)*ln(3)* *3*ln(x)**3+(-3*exp(1)*x+6*x**2)*ln(3)**2*ln(-exp(1)+2*x)*ln(x)**2+(3*x**2 *exp(1)-6*x**3)*ln(3)*ln(-exp(1)+2*x)**2*ln(x)+(-x**3*exp(1)+2*x**4)*ln(-e xp(1)+2*x)**3),x)
Output:
(x**4*log(x)**2 - 2*x**3*log(x)*log(2*x - E) + 2*x**2*log(3)*log(x)**2)/(x **2*log(2*x - E)**2 - 2*x*log(3)*log(x)*log(2*x - E) + log(3)**2*log(x)**2 )
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (30) = 60\).
Time = 0.43 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.63 \[ \int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx=-\frac {2 \, x^{3} \log \left (2 \, x - e\right ) \log \left (x\right ) - {\left (x^{4} + 2 \, x^{2} \log \left (3\right )\right )} \log \left (x\right )^{2}}{x^{2} \log \left (2 \, x - e\right )^{2} - 2 \, x \log \left (3\right ) \log \left (2 \, x - e\right ) \log \left (x\right ) + \log \left (3\right )^{2} \log \left (x\right )^{2}} \] Input:
integrate((((4*exp(1)*x-8*x^2)*log(3)^2+(4*x^3*exp(1)-8*x^4)*log(3))*log(x )^3+(((-6*x^2*exp(1)+12*x^3)*log(3)-2*x^4*exp(1)+4*x^5)*log(-exp(1)+2*x)-4 *x^3*log(3)-4*x^5)*log(x)^2+((2*x^3*exp(1)-4*x^4)*log(-exp(1)+2*x)^2+((-2* x^2*exp(1)+4*x^3)*log(3)-2*x^4*exp(1)+4*x^5+4*x^4)*log(-exp(1)+2*x))*log(x )+(2*x^3*exp(1)-4*x^4)*log(-exp(1)+2*x)^2)/((exp(1)-2*x)*log(3)^3*log(x)^3 +(-3*exp(1)*x+6*x^2)*log(3)^2*log(-exp(1)+2*x)*log(x)^2+(3*x^2*exp(1)-6*x^ 3)*log(3)*log(-exp(1)+2*x)^2*log(x)+(-x^3*exp(1)+2*x^4)*log(-exp(1)+2*x)^3 ),x, algorithm="maxima")
Output:
-(2*x^3*log(2*x - e)*log(x) - (x^4 + 2*x^2*log(3))*log(x)^2)/(x^2*log(2*x - e)^2 - 2*x*log(3)*log(2*x - e)*log(x) + log(3)^2*log(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (30) = 60\).
Time = 0.39 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.67 \[ \int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx=\frac {x^{4} \log \left (x\right )^{2} - 2 \, x^{3} \log \left (2 \, x - e\right ) \log \left (x\right ) + 2 \, x^{2} \log \left (3\right ) \log \left (x\right )^{2}}{x^{2} \log \left (2 \, x - e\right )^{2} - 2 \, x \log \left (3\right ) \log \left (2 \, x - e\right ) \log \left (x\right ) + \log \left (3\right )^{2} \log \left (x\right )^{2}} \] Input:
integrate((((4*exp(1)*x-8*x^2)*log(3)^2+(4*x^3*exp(1)-8*x^4)*log(3))*log(x )^3+(((-6*x^2*exp(1)+12*x^3)*log(3)-2*x^4*exp(1)+4*x^5)*log(-exp(1)+2*x)-4 *x^3*log(3)-4*x^5)*log(x)^2+((2*x^3*exp(1)-4*x^4)*log(-exp(1)+2*x)^2+((-2* x^2*exp(1)+4*x^3)*log(3)-2*x^4*exp(1)+4*x^5+4*x^4)*log(-exp(1)+2*x))*log(x )+(2*x^3*exp(1)-4*x^4)*log(-exp(1)+2*x)^2)/((exp(1)-2*x)*log(3)^3*log(x)^3 +(-3*exp(1)*x+6*x^2)*log(3)^2*log(-exp(1)+2*x)*log(x)^2+(3*x^2*exp(1)-6*x^ 3)*log(3)*log(-exp(1)+2*x)^2*log(x)+(-x^3*exp(1)+2*x^4)*log(-exp(1)+2*x)^3 ),x, algorithm="giac")
Output:
(x^4*log(x)^2 - 2*x^3*log(2*x - e)*log(x) + 2*x^2*log(3)*log(x)^2)/(x^2*lo g(2*x - e)^2 - 2*x*log(3)*log(2*x - e)*log(x) + log(3)^2*log(x)^2)
Timed out. \[ \int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx=\int \frac {\ln \left (x\right )\,\left (\ln \left (2\,x-\mathrm {e}\right )\,\left (2\,x^4\,\mathrm {e}-4\,x^4-4\,x^5+\ln \left (3\right )\,\left (2\,x^2\,\mathrm {e}-4\,x^3\right )\right )-{\ln \left (2\,x-\mathrm {e}\right )}^2\,\left (2\,x^3\,\mathrm {e}-4\,x^4\right )\right )-{\ln \left (2\,x-\mathrm {e}\right )}^2\,\left (2\,x^3\,\mathrm {e}-4\,x^4\right )+{\ln \left (x\right )}^2\,\left (4\,x^3\,\ln \left (3\right )+4\,x^5+\ln \left (2\,x-\mathrm {e}\right )\,\left (2\,x^4\,\mathrm {e}-4\,x^5+\ln \left (3\right )\,\left (6\,x^2\,\mathrm {e}-12\,x^3\right )\right )\right )-{\ln \left (x\right )}^3\,\left ({\ln \left (3\right )}^2\,\left (4\,x\,\mathrm {e}-8\,x^2\right )+\ln \left (3\right )\,\left (4\,x^3\,\mathrm {e}-8\,x^4\right )\right )}{\left (x^3\,\mathrm {e}-2\,x^4\right )\,{\ln \left (2\,x-\mathrm {e}\right )}^3-\ln \left (3\right )\,\left (3\,x^2\,\mathrm {e}-6\,x^3\right )\,{\ln \left (2\,x-\mathrm {e}\right )}^2\,\ln \left (x\right )+{\ln \left (3\right )}^2\,\left (3\,x\,\mathrm {e}-6\,x^2\right )\,\ln \left (2\,x-\mathrm {e}\right )\,{\ln \left (x\right )}^2+{\ln \left (3\right )}^3\,\left (2\,x-\mathrm {e}\right )\,{\ln \left (x\right )}^3} \,d x \] Input:
int((log(x)*(log(2*x - exp(1))*(2*x^4*exp(1) - 4*x^4 - 4*x^5 + log(3)*(2*x ^2*exp(1) - 4*x^3)) - log(2*x - exp(1))^2*(2*x^3*exp(1) - 4*x^4)) - log(2* x - exp(1))^2*(2*x^3*exp(1) - 4*x^4) + log(x)^2*(4*x^3*log(3) + 4*x^5 + lo g(2*x - exp(1))*(2*x^4*exp(1) - 4*x^5 + log(3)*(6*x^2*exp(1) - 12*x^3))) - log(x)^3*(log(3)^2*(4*x*exp(1) - 8*x^2) + log(3)*(4*x^3*exp(1) - 8*x^4))) /(log(2*x - exp(1))^3*(x^3*exp(1) - 2*x^4) + log(3)^3*log(x)^3*(2*x - exp( 1)) + log(2*x - exp(1))*log(3)^2*log(x)^2*(3*x*exp(1) - 6*x^2) - log(2*x - exp(1))^2*log(3)*log(x)*(3*x^2*exp(1) - 6*x^3)),x)
Output:
int((log(x)*(log(2*x - exp(1))*(2*x^4*exp(1) - 4*x^4 - 4*x^5 + log(3)*(2*x ^2*exp(1) - 4*x^3)) - log(2*x - exp(1))^2*(2*x^3*exp(1) - 4*x^4)) - log(2* x - exp(1))^2*(2*x^3*exp(1) - 4*x^4) + log(x)^2*(4*x^3*log(3) + 4*x^5 + lo g(2*x - exp(1))*(2*x^4*exp(1) - 4*x^5 + log(3)*(6*x^2*exp(1) - 12*x^3))) - log(x)^3*(log(3)^2*(4*x*exp(1) - 8*x^2) + log(3)*(4*x^3*exp(1) - 8*x^4))) /(log(2*x - exp(1))^3*(x^3*exp(1) - 2*x^4) + log(3)^3*log(x)^3*(2*x - exp( 1)) + log(2*x - exp(1))*log(3)^2*log(x)^2*(3*x*exp(1) - 6*x^2) - log(2*x - exp(1))^2*log(3)*log(x)*(3*x^2*exp(1) - 6*x^3)), x)
Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.37 \[ \int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx=\frac {\mathrm {log}\left (x \right ) x^{2} \left (-2 \,\mathrm {log}\left (-e +2 x \right ) x +2 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (3\right )+\mathrm {log}\left (x \right ) x^{2}\right )}{\mathrm {log}\left (-e +2 x \right )^{2} x^{2}-2 \,\mathrm {log}\left (-e +2 x \right ) \mathrm {log}\left (x \right ) \mathrm {log}\left (3\right ) x +\mathrm {log}\left (x \right )^{2} \mathrm {log}\left (3\right )^{2}} \] Input:
int((((4*exp(1)*x-8*x^2)*log(3)^2+(4*x^3*exp(1)-8*x^4)*log(3))*log(x)^3+(( (-6*x^2*exp(1)+12*x^3)*log(3)-2*x^4*exp(1)+4*x^5)*log(-exp(1)+2*x)-4*x^3*l og(3)-4*x^5)*log(x)^2+((2*x^3*exp(1)-4*x^4)*log(-exp(1)+2*x)^2+((-2*x^2*ex p(1)+4*x^3)*log(3)-2*x^4*exp(1)+4*x^5+4*x^4)*log(-exp(1)+2*x))*log(x)+(2*x ^3*exp(1)-4*x^4)*log(-exp(1)+2*x)^2)/((exp(1)-2*x)*log(3)^3*log(x)^3+(-3*e xp(1)*x+6*x^2)*log(3)^2*log(-exp(1)+2*x)*log(x)^2+(3*x^2*exp(1)-6*x^3)*log (3)*log(-exp(1)+2*x)^2*log(x)+(-x^3*exp(1)+2*x^4)*log(-exp(1)+2*x)^3),x)
Output:
(log(x)*x**2*( - 2*log( - e + 2*x)*x + 2*log(x)*log(3) + log(x)*x**2))/(lo g( - e + 2*x)**2*x**2 - 2*log( - e + 2*x)*log(x)*log(3)*x + log(x)**2*log( 3)**2)