Integrand size = 201, antiderivative size = 29 \[ \int \frac {-12 x-6 e^x x+3 x^2+\left (e^x (24-6 x)-6 x\right ) \log (x)+(24-6 x) \log ^2(x)+\left (12-3 x+e^x \left (12 x-3 x^2\right )\right ) \log \left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )}{-4 x^3+x^4+\left (e^x \left (8 x^2-2 x^3\right )+\left (8 x^2-2 x^3\right ) \log (x)\right ) \log \left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )+\left (e^{2 x} \left (-4 x+x^2\right )+e^x \left (-8 x+2 x^2\right ) \log (x)+\left (-4 x+x^2\right ) \log ^2(x)\right ) \log ^2\left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )} \, dx=\frac {3}{-x+\log \left (3 e^{\log ^2(x)} (-4+x)^2\right ) \left (e^x+\log (x)\right )} \] Output:
3/(ln(3*exp(ln(x)^2)*(-4+x)^2)*(ln(x)+exp(x))-x)
Time = 0.65 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {-12 x-6 e^x x+3 x^2+\left (e^x (24-6 x)-6 x\right ) \log (x)+(24-6 x) \log ^2(x)+\left (12-3 x+e^x \left (12 x-3 x^2\right )\right ) \log \left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )}{-4 x^3+x^4+\left (e^x \left (8 x^2-2 x^3\right )+\left (8 x^2-2 x^3\right ) \log (x)\right ) \log \left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )+\left (e^{2 x} \left (-4 x+x^2\right )+e^x \left (-8 x+2 x^2\right ) \log (x)+\left (-4 x+x^2\right ) \log ^2(x)\right ) \log ^2\left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )} \, dx=-\frac {3}{x-\log \left (3 e^{\log ^2(x)} (-4+x)^2\right ) \left (e^x+\log (x)\right )} \] Input:
Integrate[(-12*x - 6*E^x*x + 3*x^2 + (E^x*(24 - 6*x) - 6*x)*Log[x] + (24 - 6*x)*Log[x]^2 + (12 - 3*x + E^x*(12*x - 3*x^2))*Log[E^Log[x]^2*(48 - 24*x + 3*x^2)])/(-4*x^3 + x^4 + (E^x*(8*x^2 - 2*x^3) + (8*x^2 - 2*x^3)*Log[x]) *Log[E^Log[x]^2*(48 - 24*x + 3*x^2)] + (E^(2*x)*(-4*x + x^2) + E^x*(-8*x + 2*x^2)*Log[x] + (-4*x + x^2)*Log[x]^2)*Log[E^Log[x]^2*(48 - 24*x + 3*x^2) ]^2),x]
Output:
-3/(x - Log[3*E^Log[x]^2*(-4 + x)^2]*(E^x + Log[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.
| \(\displaystyle \int \frac {3 x^2+\left (e^x \left (12 x-3 x^2\right )-3 x+12\right ) \log \left (\left (3 x^2-24 x+48\right ) e^{\log ^2(x)}\right )-6 e^x x-12 x+(24-6 x) \log ^2(x)+\left (e^x (24-6 x)-6 x\right ) \log (x)}{x^4-4 x^3+\left (e^{2 x} \left (x^2-4 x\right )+\left (x^2-4 x\right ) \log ^2(x)+e^x \left (2 x^2-8 x\right ) \log (x)\right ) \log ^2\left (\left (3 x^2-24 x+48\right ) e^{\log ^2(x)}\right )+\left (e^x \left (8 x^2-2 x^3\right )+\left (8 x^2-2 x^3\right ) \log (x)\right ) \log \left (\left (3 x^2-24 x+48\right ) e^{\log ^2(x)}\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {3 \left (-x \left (x-2 e^x-4\right )+2 (x-4) \log ^2(x)+(x-4) \left (e^x x+1\right ) \log \left (3 (x-4)^2 e^{\log ^2(x)}\right )+2 \left (e^x (x-4)+x\right ) \log (x)\right )}{(4-x) x \left (x-\log \left (3 (x-4)^2 e^{\log ^2(x)}\right ) \left (e^x+\log (x)\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \int \frac {-2 (4-x) \log ^2(x)-2 \left (e^x (4-x)-x\right ) \log (x)+\left (-x+2 e^x+4\right ) x-(4-x) \left (e^x x+1\right ) \log \left (3 e^{\log ^2(x)} (4-x)^2\right )}{(4-x) x \left (x-\log \left (3 e^{\log ^2(x)} (4-x)^2\right ) \left (\log (x)+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+2 \log (x) x+2 x-8 \log (x)}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )}-\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^3-5 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x^2+2 \log (x) x^2+2 x^2+\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x-8 \log (x) x-4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right )}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {2 (x-4) \log ^2(x)+2 \left (e^x (x-4)+x\right ) \log (x)-x \left (x-2 e^x-4\right )+(x-4) \left (e^x x+1\right ) \log \left (3 e^{\log ^2(x)} (x-4)^2\right )}{(4-x) x \left (x-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (\log (x)+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+2 \log (x) x+2 x-8 \log (x)}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )}-\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^3-5 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x^2+2 \log (x) x^2+2 x^2+\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x-8 \log (x) x-4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right )}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {2 (x-4) \log ^2(x)+2 \left (e^x (x-4)+x\right ) \log (x)-x \left (x-2 e^x-4\right )+(x-4) \left (e^x x+1\right ) \log \left (3 e^{\log ^2(x)} (x-4)^2\right )}{(4-x) x \left (x-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (\log (x)+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+2 \log (x) x+2 x-8 \log (x)}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )}-\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^3-5 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x^2+2 \log (x) x^2+2 x^2+\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x-8 \log (x) x-4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right )}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {2 (x-4) \log ^2(x)+2 \left (e^x (x-4)+x\right ) \log (x)-x \left (x-2 e^x-4\right )+(x-4) \left (e^x x+1\right ) \log \left (3 e^{\log ^2(x)} (x-4)^2\right )}{(4-x) x \left (x-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (\log (x)+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+2 \log (x) x+2 x-8 \log (x)}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )}-\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^3-5 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x^2+2 \log (x) x^2+2 x^2+\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x-8 \log (x) x-4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right )}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {2 (x-4) \log ^2(x)+2 \left (e^x (x-4)+x\right ) \log (x)-x \left (x-2 e^x-4\right )+(x-4) \left (e^x x+1\right ) \log \left (3 e^{\log ^2(x)} (x-4)^2\right )}{(4-x) x \left (x-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (\log (x)+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+2 \log (x) x+2 x-8 \log (x)}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )}-\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^3-5 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x^2+2 \log (x) x^2+2 x^2+\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x-8 \log (x) x-4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right )}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {2 (x-4) \log ^2(x)+2 \left (e^x (x-4)+x\right ) \log (x)-x \left (x-2 e^x-4\right )+(x-4) \left (e^x x+1\right ) \log \left (3 e^{\log ^2(x)} (x-4)^2\right )}{(4-x) x \left (x-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (\log (x)+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+2 \log (x) x+2 x-8 \log (x)}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )}-\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^3-5 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x^2+2 \log (x) x^2+2 x^2+\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x-8 \log (x) x-4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right )}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {2 (x-4) \log ^2(x)+2 \left (e^x (x-4)+x\right ) \log (x)-x \left (x-2 e^x-4\right )+(x-4) \left (e^x x+1\right ) \log \left (3 e^{\log ^2(x)} (x-4)^2\right )}{(4-x) x \left (x-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (\log (x)+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+2 \log (x) x+2 x-8 \log (x)}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )}-\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^3-5 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x^2+2 \log (x) x^2+2 x^2+\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x-8 \log (x) x-4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right )}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {2 (x-4) \log ^2(x)+2 \left (e^x (x-4)+x\right ) \log (x)-x \left (x-2 e^x-4\right )+(x-4) \left (e^x x+1\right ) \log \left (3 e^{\log ^2(x)} (x-4)^2\right )}{(4-x) x \left (x-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (\log (x)+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+2 \log (x) x+2 x-8 \log (x)}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )}-\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^3-5 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x^2+2 \log (x) x^2+2 x^2+\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x-8 \log (x) x-4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right )}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {2 (x-4) \log ^2(x)+2 \left (e^x (x-4)+x\right ) \log (x)-x \left (x-2 e^x-4\right )+(x-4) \left (e^x x+1\right ) \log \left (3 e^{\log ^2(x)} (x-4)^2\right )}{(4-x) x \left (x-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (\log (x)+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+2 \log (x) x+2 x-8 \log (x)}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )}-\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^3-5 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x^2+2 \log (x) x^2+2 x^2+\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x-8 \log (x) x-4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right )}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {2 (x-4) \log ^2(x)+2 \left (e^x (x-4)+x\right ) \log (x)-x \left (x-2 e^x-4\right )+(x-4) \left (e^x x+1\right ) \log \left (3 e^{\log ^2(x)} (x-4)^2\right )}{(4-x) x \left (x-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (\log (x)+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+2 \log (x) x+2 x-8 \log (x)}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )}-\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^3-5 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x^2+2 \log (x) x^2+2 x^2+\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x-8 \log (x) x-4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right )}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {2 (x-4) \log ^2(x)+2 \left (e^x (x-4)+x\right ) \log (x)-x \left (x-2 e^x-4\right )+(x-4) \left (e^x x+1\right ) \log \left (3 e^{\log ^2(x)} (x-4)^2\right )}{(4-x) x \left (x-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (\log (x)+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+2 \log (x) x+2 x-8 \log (x)}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )}-\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^3-5 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x^2+2 \log (x) x^2+2 x^2+\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x-8 \log (x) x-4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right )}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {2 (x-4) \log ^2(x)+2 \left (e^x (x-4)+x\right ) \log (x)-x \left (x-2 e^x-4\right )+(x-4) \left (e^x x+1\right ) \log \left (3 e^{\log ^2(x)} (x-4)^2\right )}{(4-x) x \left (x-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (\log (x)+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+2 \log (x) x+2 x-8 \log (x)}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )}-\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^3-5 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x^2+2 \log (x) x^2+2 x^2+\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x-8 \log (x) x-4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right )}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {2 (x-4) \log ^2(x)+2 \left (e^x (x-4)+x\right ) \log (x)-x \left (x-2 e^x-4\right )+(x-4) \left (e^x x+1\right ) \log \left (3 e^{\log ^2(x)} (x-4)^2\right )}{(4-x) x \left (x-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (\log (x)+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+2 \log (x) x+2 x-8 \log (x)}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )}-\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^3-5 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x^2+2 \log (x) x^2+2 x^2+\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x-8 \log (x) x-4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right )}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {2 (x-4) \log ^2(x)+2 \left (e^x (x-4)+x\right ) \log (x)-x \left (x-2 e^x-4\right )+(x-4) \left (e^x x+1\right ) \log \left (3 e^{\log ^2(x)} (x-4)^2\right )}{(4-x) x \left (x-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (\log (x)+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+2 \log (x) x+2 x-8 \log (x)}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )}-\frac {\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^3-5 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x^2-\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x^2+2 \log (x) x^2+2 x^2+\log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) x+4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x) x-8 \log (x) x-4 \log ^2\left (3 e^{\log ^2(x)} (x-4)^2\right )}{(x-4) x \log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (x-e^x \log \left (3 e^{\log ^2(x)} (x-4)^2\right )-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {2 (x-4) \log ^2(x)+2 \left (e^x (x-4)+x\right ) \log (x)-x \left (x-2 e^x-4\right )+(x-4) \left (e^x x+1\right ) \log \left (3 e^{\log ^2(x)} (x-4)^2\right )}{(4-x) x \left (x-\log \left (3 e^{\log ^2(x)} (x-4)^2\right ) \left (\log (x)+e^x\right )\right )^2}dx\) |
Input:
Int[(-12*x - 6*E^x*x + 3*x^2 + (E^x*(24 - 6*x) - 6*x)*Log[x] + (24 - 6*x)* Log[x]^2 + (12 - 3*x + E^x*(12*x - 3*x^2))*Log[E^Log[x]^2*(48 - 24*x + 3*x ^2)])/(-4*x^3 + x^4 + (E^x*(8*x^2 - 2*x^3) + (8*x^2 - 2*x^3)*Log[x])*Log[E ^Log[x]^2*(48 - 24*x + 3*x^2)] + (E^(2*x)*(-4*x + x^2) + E^x*(-8*x + 2*x^2 )*Log[x] + (-4*x + x^2)*Log[x]^2)*Log[E^Log[x]^2*(48 - 24*x + 3*x^2)]^2),x ]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(95\) vs. \(2(27)=54\).
Time = 74.09 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.31
| method | result | size |
| parallelrisch | \(\frac {-24+3 \ln \left (x \right ) \ln \left (\left (3 x^{2}-24 x +48\right ) {\mathrm e}^{\ln \left (x \right )^{2}}\right )+3 \,{\mathrm e}^{x} \ln \left (\left (3 x^{2}-24 x +48\right ) {\mathrm e}^{\ln \left (x \right )^{2}}\right )-3 x}{-8 \,{\mathrm e}^{x} \ln \left (\left (3 x^{2}-24 x +48\right ) {\mathrm e}^{\ln \left (x \right )^{2}}\right )-8 \ln \left (x \right ) \ln \left (\left (3 x^{2}-24 x +48\right ) {\mathrm e}^{\ln \left (x \right )^{2}}\right )+8 x}\) | \(96\) |
| risch | \(\frac {6 i}{-2 i x +\pi \operatorname {csgn}\left (i {\mathrm e}^{\ln \left (x \right )^{2}} \left (x -4\right )^{2}\right )^{3} {\mathrm e}^{x}+\pi \operatorname {csgn}\left (i \left (x -4\right )^{2}\right )^{3} {\mathrm e}^{x}+\pi \operatorname {csgn}\left (i \left (x -4\right )^{2}\right )^{3} \ln \left (x \right )+\pi \operatorname {csgn}\left (i {\mathrm e}^{\ln \left (x \right )^{2}} \left (x -4\right )^{2}\right )^{3} \ln \left (x \right )+\pi \operatorname {csgn}\left (i \left (x -4\right )\right )^{2} \operatorname {csgn}\left (i \left (x -4\right )^{2}\right ) {\mathrm e}^{x}+\pi \operatorname {csgn}\left (i \left (x -4\right )\right )^{2} \operatorname {csgn}\left (i \left (x -4\right )^{2}\right ) \ln \left (x \right )-2 \pi \,\operatorname {csgn}\left (i \left (x -4\right )\right ) \operatorname {csgn}\left (i \left (x -4\right )^{2}\right )^{2} {\mathrm e}^{x}-2 \pi \,\operatorname {csgn}\left (i \left (x -4\right )\right ) \operatorname {csgn}\left (i \left (x -4\right )^{2}\right )^{2} \ln \left (x \right )-\pi \,\operatorname {csgn}\left (i \left (x -4\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{\ln \left (x \right )^{2}} \left (x -4\right )^{2}\right )^{2} {\mathrm e}^{x}-\pi \,\operatorname {csgn}\left (i \left (x -4\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{\ln \left (x \right )^{2}} \left (x -4\right )^{2}\right )^{2} \ln \left (x \right )-\pi \,\operatorname {csgn}\left (i {\mathrm e}^{\ln \left (x \right )^{2}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{\ln \left (x \right )^{2}} \left (x -4\right )^{2}\right )^{2} {\mathrm e}^{x}-\pi \,\operatorname {csgn}\left (i {\mathrm e}^{\ln \left (x \right )^{2}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{\ln \left (x \right )^{2}} \left (x -4\right )^{2}\right )^{2} \ln \left (x \right )+2 i {\mathrm e}^{x} \ln \left ({\mathrm e}^{\ln \left (x \right )^{2}}\right )+2 i \ln \left (x \right ) \ln \left ({\mathrm e}^{\ln \left (x \right )^{2}}\right )+2 i \ln \left (3\right ) {\mathrm e}^{x}+2 i \ln \left (3\right ) \ln \left (x \right )+4 i {\mathrm e}^{x} \ln \left (x -4\right )+4 i \ln \left (x \right ) \ln \left (x -4\right )+\pi \,\operatorname {csgn}\left (i \left (x -4\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{\ln \left (x \right )^{2}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{\ln \left (x \right )^{2}} \left (x -4\right )^{2}\right ) {\mathrm e}^{x}+\pi \,\operatorname {csgn}\left (i \left (x -4\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{\ln \left (x \right )^{2}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{\ln \left (x \right )^{2}} \left (x -4\right )^{2}\right ) \ln \left (x \right )}\) | \(417\) |
Input:
int((((-3*x^2+12*x)*exp(x)-3*x+12)*ln((3*x^2-24*x+48)*exp(ln(x)^2))+(-6*x+ 24)*ln(x)^2+((-6*x+24)*exp(x)-6*x)*ln(x)-6*exp(x)*x+3*x^2-12*x)/(((x^2-4*x )*ln(x)^2+(2*x^2-8*x)*exp(x)*ln(x)+(x^2-4*x)*exp(x)^2)*ln((3*x^2-24*x+48)* exp(ln(x)^2))^2+((-2*x^3+8*x^2)*ln(x)+(-2*x^3+8*x^2)*exp(x))*ln((3*x^2-24* x+48)*exp(ln(x)^2))+x^4-4*x^3),x,method=_RETURNVERBOSE)
Output:
1/8*(-24+3*ln(x)*ln((3*x^2-24*x+48)*exp(ln(x)^2))+3*exp(x)*ln((3*x^2-24*x+ 48)*exp(ln(x)^2))-3*x)/(-exp(x)*ln((3*x^2-24*x+48)*exp(ln(x)^2))-ln(x)*ln( (3*x^2-24*x+48)*exp(ln(x)^2))+x)
Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {-12 x-6 e^x x+3 x^2+\left (e^x (24-6 x)-6 x\right ) \log (x)+(24-6 x) \log ^2(x)+\left (12-3 x+e^x \left (12 x-3 x^2\right )\right ) \log \left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )}{-4 x^3+x^4+\left (e^x \left (8 x^2-2 x^3\right )+\left (8 x^2-2 x^3\right ) \log (x)\right ) \log \left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )+\left (e^{2 x} \left (-4 x+x^2\right )+e^x \left (-8 x+2 x^2\right ) \log (x)+\left (-4 x+x^2\right ) \log ^2(x)\right ) \log ^2\left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )} \, dx=\frac {3}{{\left (e^{x} + \log \left (x\right )\right )} \log \left (3 \, {\left (x^{2} - 8 \, x + 16\right )} e^{\left (\log \left (x\right )^{2}\right )}\right ) - x} \] Input:
integrate((((-3*x^2+12*x)*exp(x)-3*x+12)*log((3*x^2-24*x+48)*exp(log(x)^2) )+(-6*x+24)*log(x)^2+((-6*x+24)*exp(x)-6*x)*log(x)-6*exp(x)*x+3*x^2-12*x)/ (((x^2-4*x)*log(x)^2+(2*x^2-8*x)*exp(x)*log(x)+(x^2-4*x)*exp(x)^2)*log((3* x^2-24*x+48)*exp(log(x)^2))^2+((-2*x^3+8*x^2)*log(x)+(-2*x^3+8*x^2)*exp(x) )*log((3*x^2-24*x+48)*exp(log(x)^2))+x^4-4*x^3),x, algorithm="fricas")
Output:
3/((e^x + log(x))*log(3*(x^2 - 8*x + 16)*e^(log(x)^2)) - x)
Time = 0.41 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-12 x-6 e^x x+3 x^2+\left (e^x (24-6 x)-6 x\right ) \log (x)+(24-6 x) \log ^2(x)+\left (12-3 x+e^x \left (12 x-3 x^2\right )\right ) \log \left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )}{-4 x^3+x^4+\left (e^x \left (8 x^2-2 x^3\right )+\left (8 x^2-2 x^3\right ) \log (x)\right ) \log \left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )+\left (e^{2 x} \left (-4 x+x^2\right )+e^x \left (-8 x+2 x^2\right ) \log (x)+\left (-4 x+x^2\right ) \log ^2(x)\right ) \log ^2\left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )} \, dx=\frac {3}{- x + \left (e^{x} + \log {\left (x \right )}\right ) \log {\left (\left (3 x^{2} - 24 x + 48\right ) e^{\log {\left (x \right )}^{2}} \right )}} \] Input:
integrate((((-3*x**2+12*x)*exp(x)-3*x+12)*ln((3*x**2-24*x+48)*exp(ln(x)**2 ))+(-6*x+24)*ln(x)**2+((-6*x+24)*exp(x)-6*x)*ln(x)-6*exp(x)*x+3*x**2-12*x) /(((x**2-4*x)*ln(x)**2+(2*x**2-8*x)*exp(x)*ln(x)+(x**2-4*x)*exp(x)**2)*ln( (3*x**2-24*x+48)*exp(ln(x)**2))**2+((-2*x**3+8*x**2)*ln(x)+(-2*x**3+8*x**2 )*exp(x))*ln((3*x**2-24*x+48)*exp(ln(x)**2))+x**4-4*x**3),x)
Output:
3/(-x + (exp(x) + log(x))*log((3*x**2 - 24*x + 48)*exp(log(x)**2)))
\[ \int \frac {-12 x-6 e^x x+3 x^2+\left (e^x (24-6 x)-6 x\right ) \log (x)+(24-6 x) \log ^2(x)+\left (12-3 x+e^x \left (12 x-3 x^2\right )\right ) \log \left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )}{-4 x^3+x^4+\left (e^x \left (8 x^2-2 x^3\right )+\left (8 x^2-2 x^3\right ) \log (x)\right ) \log \left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )+\left (e^{2 x} \left (-4 x+x^2\right )+e^x \left (-8 x+2 x^2\right ) \log (x)+\left (-4 x+x^2\right ) \log ^2(x)\right ) \log ^2\left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )} \, dx=\int { -\frac {3 \, {\left (2 \, {\left (x - 4\right )} \log \left (x\right )^{2} - x^{2} + 2 \, x e^{x} + {\left ({\left (x^{2} - 4 \, x\right )} e^{x} + x - 4\right )} \log \left (3 \, {\left (x^{2} - 8 \, x + 16\right )} e^{\left (\log \left (x\right )^{2}\right )}\right ) + 2 \, {\left ({\left (x - 4\right )} e^{x} + x\right )} \log \left (x\right ) + 4 \, x\right )}}{x^{4} - 4 \, x^{3} + {\left (2 \, {\left (x^{2} - 4 \, x\right )} e^{x} \log \left (x\right ) + {\left (x^{2} - 4 \, x\right )} \log \left (x\right )^{2} + {\left (x^{2} - 4 \, x\right )} e^{\left (2 \, x\right )}\right )} \log \left (3 \, {\left (x^{2} - 8 \, x + 16\right )} e^{\left (\log \left (x\right )^{2}\right )}\right )^{2} - 2 \, {\left ({\left (x^{3} - 4 \, x^{2}\right )} e^{x} + {\left (x^{3} - 4 \, x^{2}\right )} \log \left (x\right )\right )} \log \left (3 \, {\left (x^{2} - 8 \, x + 16\right )} e^{\left (\log \left (x\right )^{2}\right )}\right )} \,d x } \] Input:
integrate((((-3*x^2+12*x)*exp(x)-3*x+12)*log((3*x^2-24*x+48)*exp(log(x)^2) )+(-6*x+24)*log(x)^2+((-6*x+24)*exp(x)-6*x)*log(x)-6*exp(x)*x+3*x^2-12*x)/ (((x^2-4*x)*log(x)^2+(2*x^2-8*x)*exp(x)*log(x)+(x^2-4*x)*exp(x)^2)*log((3* x^2-24*x+48)*exp(log(x)^2))^2+((-2*x^3+8*x^2)*log(x)+(-2*x^3+8*x^2)*exp(x) )*log((3*x^2-24*x+48)*exp(log(x)^2))+x^4-4*x^3),x, algorithm="maxima")
Output:
3*(x - 4)/(e^x*log(3) + 2*(e^x + log(x))*log(x - 4) + log(3)*log(x) + (e^x + log(x))*log(e^(log(x)^2)) - x) - 3*integrate(1/(e^x*log(3) + 2*(e^x + l og(x))*log(x - 4) + log(3)*log(x) + (e^x + log(x))*log(e^(log(x)^2)) - x), x)
Time = 1.38 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {-12 x-6 e^x x+3 x^2+\left (e^x (24-6 x)-6 x\right ) \log (x)+(24-6 x) \log ^2(x)+\left (12-3 x+e^x \left (12 x-3 x^2\right )\right ) \log \left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )}{-4 x^3+x^4+\left (e^x \left (8 x^2-2 x^3\right )+\left (8 x^2-2 x^3\right ) \log (x)\right ) \log \left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )+\left (e^{2 x} \left (-4 x+x^2\right )+e^x \left (-8 x+2 x^2\right ) \log (x)+\left (-4 x+x^2\right ) \log ^2(x)\right ) \log ^2\left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )} \, dx=\frac {3}{e^{x} \log \left (x\right )^{2} + \log \left (x\right )^{3} + e^{x} \log \left (3 \, x^{2} - 24 \, x + 48\right ) + \log \left (3 \, x^{2} - 24 \, x + 48\right ) \log \left (x\right ) - x} \] Input:
integrate((((-3*x^2+12*x)*exp(x)-3*x+12)*log((3*x^2-24*x+48)*exp(log(x)^2) )+(-6*x+24)*log(x)^2+((-6*x+24)*exp(x)-6*x)*log(x)-6*exp(x)*x+3*x^2-12*x)/ (((x^2-4*x)*log(x)^2+(2*x^2-8*x)*exp(x)*log(x)+(x^2-4*x)*exp(x)^2)*log((3* x^2-24*x+48)*exp(log(x)^2))^2+((-2*x^3+8*x^2)*log(x)+(-2*x^3+8*x^2)*exp(x) )*log((3*x^2-24*x+48)*exp(log(x)^2))+x^4-4*x^3),x, algorithm="giac")
Output:
3/(e^x*log(x)^2 + log(x)^3 + e^x*log(3*x^2 - 24*x + 48) + log(3*x^2 - 24*x + 48)*log(x) - x)
Time = 3.35 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {-12 x-6 e^x x+3 x^2+\left (e^x (24-6 x)-6 x\right ) \log (x)+(24-6 x) \log ^2(x)+\left (12-3 x+e^x \left (12 x-3 x^2\right )\right ) \log \left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )}{-4 x^3+x^4+\left (e^x \left (8 x^2-2 x^3\right )+\left (8 x^2-2 x^3\right ) \log (x)\right ) \log \left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )+\left (e^{2 x} \left (-4 x+x^2\right )+e^x \left (-8 x+2 x^2\right ) \log (x)+\left (-4 x+x^2\right ) \log ^2(x)\right ) \log ^2\left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )} \, dx=-\frac {3}{x-\ln \left ({\mathrm {e}}^{{\ln \left (x\right )}^2}\,\left (3\,x^2-24\,x+48\right )\right )\,\left ({\mathrm {e}}^x+\ln \left (x\right )\right )} \] Input:
int((12*x - log(exp(log(x)^2)*(3*x^2 - 24*x + 48))*(exp(x)*(12*x - 3*x^2) - 3*x + 12) + 6*x*exp(x) - 3*x^2 + log(x)^2*(6*x - 24) + log(x)*(6*x + exp (x)*(6*x - 24)))/(log(exp(log(x)^2)*(3*x^2 - 24*x + 48))^2*(exp(2*x)*(4*x - x^2) + log(x)^2*(4*x - x^2) + exp(x)*log(x)*(8*x - 2*x^2)) - log(exp(log (x)^2)*(3*x^2 - 24*x + 48))*(exp(x)*(8*x^2 - 2*x^3) + log(x)*(8*x^2 - 2*x^ 3)) + 4*x^3 - x^4),x)
Output:
-3/(x - log(exp(log(x)^2)*(3*x^2 - 24*x + 48))*(exp(x) + log(x)))
Time = 0.18 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.59 \[ \int \frac {-12 x-6 e^x x+3 x^2+\left (e^x (24-6 x)-6 x\right ) \log (x)+(24-6 x) \log ^2(x)+\left (12-3 x+e^x \left (12 x-3 x^2\right )\right ) \log \left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )}{-4 x^3+x^4+\left (e^x \left (8 x^2-2 x^3\right )+\left (8 x^2-2 x^3\right ) \log (x)\right ) \log \left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )+\left (e^{2 x} \left (-4 x+x^2\right )+e^x \left (-8 x+2 x^2\right ) \log (x)+\left (-4 x+x^2\right ) \log ^2(x)\right ) \log ^2\left (e^{\log ^2(x)} \left (48-24 x+3 x^2\right )\right )} \, dx=\frac {3}{e^{x} \mathrm {log}\left (3 e^{\mathrm {log}\left (x \right )^{2}} x^{2}-24 e^{\mathrm {log}\left (x \right )^{2}} x +48 e^{\mathrm {log}\left (x \right )^{2}}\right )+\mathrm {log}\left (3 e^{\mathrm {log}\left (x \right )^{2}} x^{2}-24 e^{\mathrm {log}\left (x \right )^{2}} x +48 e^{\mathrm {log}\left (x \right )^{2}}\right ) \mathrm {log}\left (x \right )-x} \] Input:
int((((-3*x^2+12*x)*exp(x)-3*x+12)*log((3*x^2-24*x+48)*exp(log(x)^2))+(-6* x+24)*log(x)^2+((-6*x+24)*exp(x)-6*x)*log(x)-6*exp(x)*x+3*x^2-12*x)/(((x^2 -4*x)*log(x)^2+(2*x^2-8*x)*exp(x)*log(x)+(x^2-4*x)*exp(x)^2)*log((3*x^2-24 *x+48)*exp(log(x)^2))^2+((-2*x^3+8*x^2)*log(x)+(-2*x^3+8*x^2)*exp(x))*log( (3*x^2-24*x+48)*exp(log(x)^2))+x^4-4*x^3),x)
Output:
3/(e**x*log(3*e**(log(x)**2)*x**2 - 24*e**(log(x)**2)*x + 48*e**(log(x)**2 )) + log(3*e**(log(x)**2)*x**2 - 24*e**(log(x)**2)*x + 48*e**(log(x)**2))* log(x) - x)