Integrand size = 189, antiderivative size = 28 \[ \int \frac {-1-x+e^{3 x} \left (-3 x-3 x^2\right )+e^x \left (1+2 x+x^2+x^3\right )+\left (1+3 x+2 x^2+x^3+e^{2 x} \left (-3 x-3 x^2\right )\right ) \log (x)+\left (e^x \left (x+x^2\right )+\left (x+x^2\right ) \log (x)\right ) \log \left (\frac {x+x^2}{e^x+\log (x)}\right )}{e^{3 x} \left (-x-x^2\right )+e^x \left (x^2+x^3\right )+\left (x^2+x^3+e^{2 x} \left (-x-x^2\right )\right ) \log (x)+\left (e^x \left (x+x^2\right )+\left (x+x^2\right ) \log (x)\right ) \log \left (\frac {x+x^2}{e^x+\log (x)}\right )} \, dx=\log \left (e^x \left (-e^{2 x}+x+\log \left (\frac {x (1+x)}{e^x+\log (x)}\right )\right )\right ) \] Output:
ln((ln(x/(ln(x)+exp(x))*(1+x))+x-exp(x)^2)*exp(x))
Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {-1-x+e^{3 x} \left (-3 x-3 x^2\right )+e^x \left (1+2 x+x^2+x^3\right )+\left (1+3 x+2 x^2+x^3+e^{2 x} \left (-3 x-3 x^2\right )\right ) \log (x)+\left (e^x \left (x+x^2\right )+\left (x+x^2\right ) \log (x)\right ) \log \left (\frac {x+x^2}{e^x+\log (x)}\right )}{e^{3 x} \left (-x-x^2\right )+e^x \left (x^2+x^3\right )+\left (x^2+x^3+e^{2 x} \left (-x-x^2\right )\right ) \log (x)+\left (e^x \left (x+x^2\right )+\left (x+x^2\right ) \log (x)\right ) \log \left (\frac {x+x^2}{e^x+\log (x)}\right )} \, dx=x+\log \left (e^{2 x}-x-\log \left (\frac {x (1+x)}{e^x+\log (x)}\right )\right ) \] Input:
Integrate[(-1 - x + E^(3*x)*(-3*x - 3*x^2) + E^x*(1 + 2*x + x^2 + x^3) + ( 1 + 3*x + 2*x^2 + x^3 + E^(2*x)*(-3*x - 3*x^2))*Log[x] + (E^x*(x + x^2) + (x + x^2)*Log[x])*Log[(x + x^2)/(E^x + Log[x])])/(E^(3*x)*(-x - x^2) + E^x *(x^2 + x^3) + (x^2 + x^3 + E^(2*x)*(-x - x^2))*Log[x] + (E^x*(x + x^2) + (x + x^2)*Log[x])*Log[(x + x^2)/(E^x + Log[x])]),x]
Output:
x + Log[E^(2*x) - x - Log[(x*(1 + x))/(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.
| \(\displaystyle \int \frac {e^{3 x} \left (-3 x^2-3 x\right )+\left (e^x \left (x^2+x\right )+\left (x^2+x\right ) \log (x)\right ) \log \left (\frac {x^2+x}{e^x+\log (x)}\right )+e^x \left (x^3+x^2+2 x+1\right )+\left (x^3+2 x^2+e^{2 x} \left (-3 x^2-3 x\right )+3 x+1\right ) \log (x)-x-1}{e^{3 x} \left (-x^2-x\right )+\left (e^x \left (x^2+x\right )+\left (x^2+x\right ) \log (x)\right ) \log \left (\frac {x^2+x}{e^x+\log (x)}\right )+e^x \left (x^3+x^2\right )+\left (x^3+x^2+e^{2 x} \left (-x^2-x\right )\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-e^{3 x} \left (-3 x^2-3 x\right )-\left (e^x \left (x^2+x\right )+\left (x^2+x\right ) \log (x)\right ) \log \left (\frac {x^2+x}{e^x+\log (x)}\right )-e^x \left (x^3+x^2+2 x+1\right )-\left (x^3+2 x^2+e^{2 x} \left (-3 x^2-3 x\right )+3 x+1\right ) \log (x)+x+1}{x (x+1) \left (e^x+\log (x)\right ) \left (-x+e^{2 x}-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x^4+2 x^3-2 x^3 \log ^2(x)+4 x^3 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2-x^2 \log ^2(x)+2 x^2 \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2 \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-e^x x^2 \log (x)+4 x^2 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+e^x x-x+e^x+3 x \log ^2(x)+2 x \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+\log ^2(x)-e^x x \log (x)-x \log (x)-2 x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-\log (x)-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )}{x (x+1) \left (x-e^{2 x}+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+\frac {x \log (x)-1}{x \left (e^x+\log (x)\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+3\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-e^x x^3-e^x x^2+3 e^{3 x} x^2-\log (x) \left (x^3+\left (2-3 e^{2 x}\right ) x^2-3 \left (e^{2 x}-1\right ) x+(x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1\right )-2 e^x x+3 e^{3 x} x+x-e^x-e^x (x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1}{x (x+1) \left (e^x+\log (x)\right ) \left (-x+e^{2 x}-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x^4+2 x^3-2 x^3 \log ^2(x)+4 x^3 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2-x^2 \log ^2(x)+2 x^2 \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2 \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-e^x x^2 \log (x)+4 x^2 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+e^x x-x+e^x+3 x \log ^2(x)+2 x \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+\log ^2(x)-e^x x \log (x)-x \log (x)-2 x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-\log (x)-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )}{x (x+1) \left (x-e^{2 x}+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+\frac {x \log (x)-1}{x \left (e^x+\log (x)\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+3\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-e^x x^3-e^x x^2+3 e^{3 x} x^2-\log (x) \left (x^3+\left (2-3 e^{2 x}\right ) x^2-3 \left (e^{2 x}-1\right ) x+(x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1\right )-2 e^x x+3 e^{3 x} x+x-e^x-e^x (x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1}{x (x+1) \left (e^x+\log (x)\right ) \left (-x+e^{2 x}-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x^4+2 x^3-2 x^3 \log ^2(x)+4 x^3 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2-x^2 \log ^2(x)+2 x^2 \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2 \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-e^x x^2 \log (x)+4 x^2 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+e^x x-x+e^x+3 x \log ^2(x)+2 x \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+\log ^2(x)-e^x x \log (x)-x \log (x)-2 x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-\log (x)-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )}{x (x+1) \left (x-e^{2 x}+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+\frac {x \log (x)-1}{x \left (e^x+\log (x)\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+3\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-e^x x^3-e^x x^2+3 e^{3 x} x^2-\log (x) \left (x^3+\left (2-3 e^{2 x}\right ) x^2-3 \left (e^{2 x}-1\right ) x+(x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1\right )-2 e^x x+3 e^{3 x} x+x-e^x-e^x (x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1}{x (x+1) \left (e^x+\log (x)\right ) \left (-x+e^{2 x}-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x^4+2 x^3-2 x^3 \log ^2(x)+4 x^3 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2-x^2 \log ^2(x)+2 x^2 \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2 \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-e^x x^2 \log (x)+4 x^2 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+e^x x-x+e^x+3 x \log ^2(x)+2 x \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+\log ^2(x)-e^x x \log (x)-x \log (x)-2 x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-\log (x)-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )}{x (x+1) \left (x-e^{2 x}+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+\frac {x \log (x)-1}{x \left (e^x+\log (x)\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+3\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-e^x x^3-e^x x^2+3 e^{3 x} x^2-\log (x) \left (x^3+\left (2-3 e^{2 x}\right ) x^2-3 \left (e^{2 x}-1\right ) x+(x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1\right )-2 e^x x+3 e^{3 x} x+x-e^x-e^x (x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1}{x (x+1) \left (e^x+\log (x)\right ) \left (-x+e^{2 x}-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x^4+2 x^3-2 x^3 \log ^2(x)+4 x^3 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2-x^2 \log ^2(x)+2 x^2 \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2 \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-e^x x^2 \log (x)+4 x^2 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+e^x x-x+e^x+3 x \log ^2(x)+2 x \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+\log ^2(x)-e^x x \log (x)-x \log (x)-2 x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-\log (x)-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )}{x (x+1) \left (x-e^{2 x}+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+\frac {x \log (x)-1}{x \left (e^x+\log (x)\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+3\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-e^x x^3-e^x x^2+3 e^{3 x} x^2-\log (x) \left (x^3+\left (2-3 e^{2 x}\right ) x^2-3 \left (e^{2 x}-1\right ) x+(x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1\right )-2 e^x x+3 e^{3 x} x+x-e^x-e^x (x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1}{x (x+1) \left (e^x+\log (x)\right ) \left (-x+e^{2 x}-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x^4+2 x^3-2 x^3 \log ^2(x)+4 x^3 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2-x^2 \log ^2(x)+2 x^2 \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2 \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-e^x x^2 \log (x)+4 x^2 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+e^x x-x+e^x+3 x \log ^2(x)+2 x \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+\log ^2(x)-e^x x \log (x)-x \log (x)-2 x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-\log (x)-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )}{x (x+1) \left (x-e^{2 x}+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+\frac {x \log (x)-1}{x \left (e^x+\log (x)\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+3\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-e^x x^3-e^x x^2+3 e^{3 x} x^2-\log (x) \left (x^3+\left (2-3 e^{2 x}\right ) x^2-3 \left (e^{2 x}-1\right ) x+(x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1\right )-2 e^x x+3 e^{3 x} x+x-e^x-e^x (x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1}{x (x+1) \left (e^x+\log (x)\right ) \left (-x+e^{2 x}-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x^4+2 x^3-2 x^3 \log ^2(x)+4 x^3 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2-x^2 \log ^2(x)+2 x^2 \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2 \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-e^x x^2 \log (x)+4 x^2 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+e^x x-x+e^x+3 x \log ^2(x)+2 x \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+\log ^2(x)-e^x x \log (x)-x \log (x)-2 x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-\log (x)-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )}{x (x+1) \left (x-e^{2 x}+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+\frac {x \log (x)-1}{x \left (e^x+\log (x)\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+3\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-e^x x^3-e^x x^2+3 e^{3 x} x^2-\log (x) \left (x^3+\left (2-3 e^{2 x}\right ) x^2-3 \left (e^{2 x}-1\right ) x+(x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1\right )-2 e^x x+3 e^{3 x} x+x-e^x-e^x (x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1}{x (x+1) \left (e^x+\log (x)\right ) \left (-x+e^{2 x}-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x^4+2 x^3-2 x^3 \log ^2(x)+4 x^3 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2-x^2 \log ^2(x)+2 x^2 \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2 \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-e^x x^2 \log (x)+4 x^2 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+e^x x-x+e^x+3 x \log ^2(x)+2 x \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+\log ^2(x)-e^x x \log (x)-x \log (x)-2 x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-\log (x)-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )}{x (x+1) \left (x-e^{2 x}+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+\frac {x \log (x)-1}{x \left (e^x+\log (x)\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+3\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-e^x x^3-e^x x^2+3 e^{3 x} x^2-\log (x) \left (x^3+\left (2-3 e^{2 x}\right ) x^2-3 \left (e^{2 x}-1\right ) x+(x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1\right )-2 e^x x+3 e^{3 x} x+x-e^x-e^x (x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1}{x (x+1) \left (e^x+\log (x)\right ) \left (-x+e^{2 x}-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x^4+2 x^3-2 x^3 \log ^2(x)+4 x^3 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2-x^2 \log ^2(x)+2 x^2 \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2 \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-e^x x^2 \log (x)+4 x^2 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+e^x x-x+e^x+3 x \log ^2(x)+2 x \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+\log ^2(x)-e^x x \log (x)-x \log (x)-2 x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-\log (x)-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )}{x (x+1) \left (x-e^{2 x}+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+\frac {x \log (x)-1}{x \left (e^x+\log (x)\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+3\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-e^x x^3-e^x x^2+3 e^{3 x} x^2-\log (x) \left (x^3+\left (2-3 e^{2 x}\right ) x^2-3 \left (e^{2 x}-1\right ) x+(x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1\right )-2 e^x x+3 e^{3 x} x+x-e^x-e^x (x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1}{x (x+1) \left (e^x+\log (x)\right ) \left (-x+e^{2 x}-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x^4+2 x^3-2 x^3 \log ^2(x)+4 x^3 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2-x^2 \log ^2(x)+2 x^2 \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2 \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-e^x x^2 \log (x)+4 x^2 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+e^x x-x+e^x+3 x \log ^2(x)+2 x \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+\log ^2(x)-e^x x \log (x)-x \log (x)-2 x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-\log (x)-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )}{x (x+1) \left (x-e^{2 x}+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+\frac {x \log (x)-1}{x \left (e^x+\log (x)\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+3\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-e^x x^3-e^x x^2+3 e^{3 x} x^2-\log (x) \left (x^3+\left (2-3 e^{2 x}\right ) x^2-3 \left (e^{2 x}-1\right ) x+(x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1\right )-2 e^x x+3 e^{3 x} x+x-e^x-e^x (x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1}{x (x+1) \left (e^x+\log (x)\right ) \left (-x+e^{2 x}-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x^4+2 x^3-2 x^3 \log ^2(x)+4 x^3 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2-x^2 \log ^2(x)+2 x^2 \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2 \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-e^x x^2 \log (x)+4 x^2 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+e^x x-x+e^x+3 x \log ^2(x)+2 x \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+\log ^2(x)-e^x x \log (x)-x \log (x)-2 x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-\log (x)-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )}{x (x+1) \left (x-e^{2 x}+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+\frac {x \log (x)-1}{x \left (e^x+\log (x)\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+3\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-e^x x^3-e^x x^2+3 e^{3 x} x^2-\log (x) \left (x^3+\left (2-3 e^{2 x}\right ) x^2-3 \left (e^{2 x}-1\right ) x+(x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1\right )-2 e^x x+3 e^{3 x} x+x-e^x-e^x (x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1}{x (x+1) \left (e^x+\log (x)\right ) \left (-x+e^{2 x}-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x^4+2 x^3-2 x^3 \log ^2(x)+4 x^3 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2-x^2 \log ^2(x)+2 x^2 \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2 \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-e^x x^2 \log (x)+4 x^2 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+e^x x-x+e^x+3 x \log ^2(x)+2 x \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+\log ^2(x)-e^x x \log (x)-x \log (x)-2 x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-\log (x)-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )}{x (x+1) \left (x-e^{2 x}+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+\frac {x \log (x)-1}{x \left (e^x+\log (x)\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+3\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-e^x x^3-e^x x^2+3 e^{3 x} x^2-\log (x) \left (x^3+\left (2-3 e^{2 x}\right ) x^2-3 \left (e^{2 x}-1\right ) x+(x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1\right )-2 e^x x+3 e^{3 x} x+x-e^x-e^x (x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1}{x (x+1) \left (e^x+\log (x)\right ) \left (-x+e^{2 x}-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x^4+2 x^3-2 x^3 \log ^2(x)+4 x^3 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2-x^2 \log ^2(x)+2 x^2 \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2 \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-e^x x^2 \log (x)+4 x^2 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+e^x x-x+e^x+3 x \log ^2(x)+2 x \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+\log ^2(x)-e^x x \log (x)-x \log (x)-2 x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-\log (x)-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )}{x (x+1) \left (x-e^{2 x}+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+\frac {x \log (x)-1}{x \left (e^x+\log (x)\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+3\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-e^x x^3-e^x x^2+3 e^{3 x} x^2-\log (x) \left (x^3+\left (2-3 e^{2 x}\right ) x^2-3 \left (e^{2 x}-1\right ) x+(x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1\right )-2 e^x x+3 e^{3 x} x+x-e^x-e^x (x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1}{x (x+1) \left (e^x+\log (x)\right ) \left (-x+e^{2 x}-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x^4+2 x^3-2 x^3 \log ^2(x)+4 x^3 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2-x^2 \log ^2(x)+2 x^2 \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2 \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-e^x x^2 \log (x)+4 x^2 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+e^x x-x+e^x+3 x \log ^2(x)+2 x \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+\log ^2(x)-e^x x \log (x)-x \log (x)-2 x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-\log (x)-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )}{x (x+1) \left (x-e^{2 x}+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+\frac {x \log (x)-1}{x \left (e^x+\log (x)\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+3\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-e^x x^3-e^x x^2+3 e^{3 x} x^2-\log (x) \left (x^3+\left (2-3 e^{2 x}\right ) x^2-3 \left (e^{2 x}-1\right ) x+(x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1\right )-2 e^x x+3 e^{3 x} x+x-e^x-e^x (x+1) x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+1}{x (x+1) \left (e^x+\log (x)\right ) \left (-x+e^{2 x}-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x^4+2 x^3-2 x^3 \log ^2(x)+4 x^3 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2-x^2 \log ^2(x)+2 x^2 \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x^2 \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-e^x x^2 \log (x)+4 x^2 \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+e^x x-x+e^x+3 x \log ^2(x)+2 x \log ^2\left (\frac {x (x+1)}{e^x+\log (x)}\right )-2 x \log ^2(x) \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )+\log ^2(x)-e^x x \log (x)-x \log (x)-2 x \log \left (\frac {x (x+1)}{e^x+\log (x)}\right )-\log (x)-\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )}{x (x+1) \left (x-e^{2 x}+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+\frac {x \log (x)-1}{x \left (e^x+\log (x)\right ) \left (x-\log ^2(x)+\log \left (\frac {x (x+1)}{e^x+\log (x)}\right )\right )}+3\right )dx\) |
Input:
Int[(-1 - x + E^(3*x)*(-3*x - 3*x^2) + E^x*(1 + 2*x + x^2 + x^3) + (1 + 3* x + 2*x^2 + x^3 + E^(2*x)*(-3*x - 3*x^2))*Log[x] + (E^x*(x + x^2) + (x + x ^2)*Log[x])*Log[(x + x^2)/(E^x + Log[x])])/(E^(3*x)*(-x - x^2) + E^x*(x^2 + x^3) + (x^2 + x^3 + E^(2*x)*(-x - x^2))*Log[x] + (E^x*(x + x^2) + (x + x ^2)*Log[x])*Log[(x + x^2)/(E^x + Log[x])]),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 225.49 (sec) , antiderivative size = 260, normalized size of antiderivative = 9.29
| method | result | size |
| risch | \(x +\ln \left (\ln \left (\ln \left (x \right )+{\mathrm e}^{x}\right )+\frac {i \left (\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i \left (1+x \right )}{\ln \left (x \right )+{\mathrm e}^{x}}\right ) \operatorname {csgn}\left (\frac {i x \left (1+x \right )}{\ln \left (x \right )+{\mathrm e}^{x}}\right )-\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x \left (1+x \right )}{\ln \left (x \right )+{\mathrm e}^{x}}\right )^{2}+\pi \,\operatorname {csgn}\left (i \left (1+x \right )\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )+{\mathrm e}^{x}}\right ) \operatorname {csgn}\left (\frac {i \left (1+x \right )}{\ln \left (x \right )+{\mathrm e}^{x}}\right )-\pi \,\operatorname {csgn}\left (i \left (1+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (1+x \right )}{\ln \left (x \right )+{\mathrm e}^{x}}\right )^{2}-\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )+{\mathrm e}^{x}}\right ) \operatorname {csgn}\left (\frac {i \left (1+x \right )}{\ln \left (x \right )+{\mathrm e}^{x}}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i \left (1+x \right )}{\ln \left (x \right )+{\mathrm e}^{x}}\right )^{3}-\pi \,\operatorname {csgn}\left (\frac {i \left (1+x \right )}{\ln \left (x \right )+{\mathrm e}^{x}}\right ) \operatorname {csgn}\left (\frac {i x \left (1+x \right )}{\ln \left (x \right )+{\mathrm e}^{x}}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i x \left (1+x \right )}{\ln \left (x \right )+{\mathrm e}^{x}}\right )^{3}-2 i {\mathrm e}^{2 x}+2 i x +2 i \ln \left (x \right )+2 i \ln \left (1+x \right )\right )}{2}\right )\) | \(260\) |
Input:
int((((x^2+x)*ln(x)+(x^2+x)*exp(x))*ln((x^2+x)/(ln(x)+exp(x)))+((-3*x^2-3* x)*exp(x)^2+x^3+2*x^2+3*x+1)*ln(x)+(-3*x^2-3*x)*exp(x)^3+(x^3+x^2+2*x+1)*e xp(x)-x-1)/(((x^2+x)*ln(x)+(x^2+x)*exp(x))*ln((x^2+x)/(ln(x)+exp(x)))+((-x ^2-x)*exp(x)^2+x^3+x^2)*ln(x)+(-x^2-x)*exp(x)^3+(x^3+x^2)*exp(x)),x,method =_RETURNVERBOSE)
Output:
x+ln(ln(ln(x)+exp(x))+1/2*I*(Pi*csgn(I*x)*csgn(I*(1+x)/(ln(x)+exp(x)))*csg n(I*x/(ln(x)+exp(x))*(1+x))-Pi*csgn(I*x)*csgn(I*x/(ln(x)+exp(x))*(1+x))^2+ Pi*csgn(I*(1+x))*csgn(I/(ln(x)+exp(x)))*csgn(I*(1+x)/(ln(x)+exp(x)))-Pi*cs gn(I*(1+x))*csgn(I*(1+x)/(ln(x)+exp(x)))^2-Pi*csgn(I/(ln(x)+exp(x)))*csgn( I*(1+x)/(ln(x)+exp(x)))^2+Pi*csgn(I*(1+x)/(ln(x)+exp(x)))^3-Pi*csgn(I*(1+x )/(ln(x)+exp(x)))*csgn(I*x/(ln(x)+exp(x))*(1+x))^2+Pi*csgn(I*x/(ln(x)+exp( x))*(1+x))^3-2*I*exp(2*x)+2*I*x+2*I*ln(x)+2*I*ln(1+x)))
Time = 0.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {-1-x+e^{3 x} \left (-3 x-3 x^2\right )+e^x \left (1+2 x+x^2+x^3\right )+\left (1+3 x+2 x^2+x^3+e^{2 x} \left (-3 x-3 x^2\right )\right ) \log (x)+\left (e^x \left (x+x^2\right )+\left (x+x^2\right ) \log (x)\right ) \log \left (\frac {x+x^2}{e^x+\log (x)}\right )}{e^{3 x} \left (-x-x^2\right )+e^x \left (x^2+x^3\right )+\left (x^2+x^3+e^{2 x} \left (-x-x^2\right )\right ) \log (x)+\left (e^x \left (x+x^2\right )+\left (x+x^2\right ) \log (x)\right ) \log \left (\frac {x+x^2}{e^x+\log (x)}\right )} \, dx=x + \log \left (x - e^{\left (2 \, x\right )} + \log \left (\frac {x^{2} + x}{e^{x} + \log \left (x\right )}\right )\right ) \] Input:
integrate((((x^2+x)*log(x)+(x^2+x)*exp(x))*log((x^2+x)/(log(x)+exp(x)))+(( -3*x^2-3*x)*exp(x)^2+x^3+2*x^2+3*x+1)*log(x)+(-3*x^2-3*x)*exp(x)^3+(x^3+x^ 2+2*x+1)*exp(x)-x-1)/(((x^2+x)*log(x)+(x^2+x)*exp(x))*log((x^2+x)/(log(x)+ exp(x)))+((-x^2-x)*exp(x)^2+x^3+x^2)*log(x)+(-x^2-x)*exp(x)^3+(x^3+x^2)*ex p(x)),x, algorithm="fricas")
Output:
x + log(x - e^(2*x) + log((x^2 + x)/(e^x + log(x))))
Time = 0.97 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {-1-x+e^{3 x} \left (-3 x-3 x^2\right )+e^x \left (1+2 x+x^2+x^3\right )+\left (1+3 x+2 x^2+x^3+e^{2 x} \left (-3 x-3 x^2\right )\right ) \log (x)+\left (e^x \left (x+x^2\right )+\left (x+x^2\right ) \log (x)\right ) \log \left (\frac {x+x^2}{e^x+\log (x)}\right )}{e^{3 x} \left (-x-x^2\right )+e^x \left (x^2+x^3\right )+\left (x^2+x^3+e^{2 x} \left (-x-x^2\right )\right ) \log (x)+\left (e^x \left (x+x^2\right )+\left (x+x^2\right ) \log (x)\right ) \log \left (\frac {x+x^2}{e^x+\log (x)}\right )} \, dx=x + \log {\left (x - e^{2 x} + \log {\left (\frac {x^{2} + x}{e^{x} + \log {\left (x \right )}} \right )} \right )} \] Input:
integrate((((x**2+x)*ln(x)+(x**2+x)*exp(x))*ln((x**2+x)/(ln(x)+exp(x)))+(( -3*x**2-3*x)*exp(x)**2+x**3+2*x**2+3*x+1)*ln(x)+(-3*x**2-3*x)*exp(x)**3+(x **3+x**2+2*x+1)*exp(x)-x-1)/(((x**2+x)*ln(x)+(x**2+x)*exp(x))*ln((x**2+x)/ (ln(x)+exp(x)))+((-x**2-x)*exp(x)**2+x**3+x**2)*ln(x)+(-x**2-x)*exp(x)**3+ (x**3+x**2)*exp(x)),x)
Output:
x + log(x - exp(2*x) + log((x**2 + x)/(exp(x) + log(x))))
Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {-1-x+e^{3 x} \left (-3 x-3 x^2\right )+e^x \left (1+2 x+x^2+x^3\right )+\left (1+3 x+2 x^2+x^3+e^{2 x} \left (-3 x-3 x^2\right )\right ) \log (x)+\left (e^x \left (x+x^2\right )+\left (x+x^2\right ) \log (x)\right ) \log \left (\frac {x+x^2}{e^x+\log (x)}\right )}{e^{3 x} \left (-x-x^2\right )+e^x \left (x^2+x^3\right )+\left (x^2+x^3+e^{2 x} \left (-x-x^2\right )\right ) \log (x)+\left (e^x \left (x+x^2\right )+\left (x+x^2\right ) \log (x)\right ) \log \left (\frac {x+x^2}{e^x+\log (x)}\right )} \, dx=x + \log \left (-x + e^{\left (2 \, x\right )} - \log \left (x + 1\right ) - \log \left (x\right ) + \log \left (e^{x} + \log \left (x\right )\right )\right ) \] Input:
integrate((((x^2+x)*log(x)+(x^2+x)*exp(x))*log((x^2+x)/(log(x)+exp(x)))+(( -3*x^2-3*x)*exp(x)^2+x^3+2*x^2+3*x+1)*log(x)+(-3*x^2-3*x)*exp(x)^3+(x^3+x^ 2+2*x+1)*exp(x)-x-1)/(((x^2+x)*log(x)+(x^2+x)*exp(x))*log((x^2+x)/(log(x)+ exp(x)))+((-x^2-x)*exp(x)^2+x^3+x^2)*log(x)+(-x^2-x)*exp(x)^3+(x^3+x^2)*ex p(x)),x, algorithm="maxima")
Output:
x + log(-x + e^(2*x) - log(x + 1) - log(x) + log(e^x + log(x)))
Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {-1-x+e^{3 x} \left (-3 x-3 x^2\right )+e^x \left (1+2 x+x^2+x^3\right )+\left (1+3 x+2 x^2+x^3+e^{2 x} \left (-3 x-3 x^2\right )\right ) \log (x)+\left (e^x \left (x+x^2\right )+\left (x+x^2\right ) \log (x)\right ) \log \left (\frac {x+x^2}{e^x+\log (x)}\right )}{e^{3 x} \left (-x-x^2\right )+e^x \left (x^2+x^3\right )+\left (x^2+x^3+e^{2 x} \left (-x-x^2\right )\right ) \log (x)+\left (e^x \left (x+x^2\right )+\left (x+x^2\right ) \log (x)\right ) \log \left (\frac {x+x^2}{e^x+\log (x)}\right )} \, dx=x + \log \left (x - e^{\left (2 \, x\right )} + \log \left (x + 1\right ) + \log \left (x\right ) - \log \left (e^{x} + \log \left (x\right )\right )\right ) \] Input:
integrate((((x^2+x)*log(x)+(x^2+x)*exp(x))*log((x^2+x)/(log(x)+exp(x)))+(( -3*x^2-3*x)*exp(x)^2+x^3+2*x^2+3*x+1)*log(x)+(-3*x^2-3*x)*exp(x)^3+(x^3+x^ 2+2*x+1)*exp(x)-x-1)/(((x^2+x)*log(x)+(x^2+x)*exp(x))*log((x^2+x)/(log(x)+ exp(x)))+((-x^2-x)*exp(x)^2+x^3+x^2)*log(x)+(-x^2-x)*exp(x)^3+(x^3+x^2)*ex p(x)),x, algorithm="giac")
Output:
x + log(x - e^(2*x) + log(x + 1) + log(x) - log(e^x + log(x)))
Time = 3.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {-1-x+e^{3 x} \left (-3 x-3 x^2\right )+e^x \left (1+2 x+x^2+x^3\right )+\left (1+3 x+2 x^2+x^3+e^{2 x} \left (-3 x-3 x^2\right )\right ) \log (x)+\left (e^x \left (x+x^2\right )+\left (x+x^2\right ) \log (x)\right ) \log \left (\frac {x+x^2}{e^x+\log (x)}\right )}{e^{3 x} \left (-x-x^2\right )+e^x \left (x^2+x^3\right )+\left (x^2+x^3+e^{2 x} \left (-x-x^2\right )\right ) \log (x)+\left (e^x \left (x+x^2\right )+\left (x+x^2\right ) \log (x)\right ) \log \left (\frac {x+x^2}{e^x+\log (x)}\right )} \, dx=x+\ln \left (x-{\mathrm {e}}^{2\,x}+\ln \left (\frac {x\,\left (x+1\right )}{{\mathrm {e}}^x+\ln \left (x\right )}\right )\right ) \] Input:
int(-(x + exp(3*x)*(3*x + 3*x^2) - exp(x)*(2*x + x^2 + x^3 + 1) - log(x)*( 3*x - exp(2*x)*(3*x + 3*x^2) + 2*x^2 + x^3 + 1) - log((x + x^2)/(exp(x) + log(x)))*(exp(x)*(x + x^2) + log(x)*(x + x^2)) + 1)/(exp(x)*(x^2 + x^3) - exp(3*x)*(x + x^2) + log((x + x^2)/(exp(x) + log(x)))*(exp(x)*(x + x^2) + log(x)*(x + x^2)) + log(x)*(x^2 - exp(2*x)*(x + x^2) + x^3)),x)
Output:
x + log(x - exp(2*x) + log((x*(x + 1))/(exp(x) + log(x))))
Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.29 \[ \int \frac {-1-x+e^{3 x} \left (-3 x-3 x^2\right )+e^x \left (1+2 x+x^2+x^3\right )+\left (1+3 x+2 x^2+x^3+e^{2 x} \left (-3 x-3 x^2\right )\right ) \log (x)+\left (e^x \left (x+x^2\right )+\left (x+x^2\right ) \log (x)\right ) \log \left (\frac {x+x^2}{e^x+\log (x)}\right )}{e^{3 x} \left (-x-x^2\right )+e^x \left (x^2+x^3\right )+\left (x^2+x^3+e^{2 x} \left (-x-x^2\right )\right ) \log (x)+\left (e^x \left (x+x^2\right )+\left (x+x^2\right ) \log (x)\right ) \log \left (\frac {x+x^2}{e^x+\log (x)}\right )} \, dx=\mathrm {log}\left (e^{3 x}+e^{2 x} \mathrm {log}\left (x \right )-e^{x} \mathrm {log}\left (\frac {x^{2}+x}{e^{x}+\mathrm {log}\left (x \right )}\right )-e^{x} x -\mathrm {log}\left (\frac {x^{2}+x}{e^{x}+\mathrm {log}\left (x \right )}\right ) \mathrm {log}\left (x \right )-\mathrm {log}\left (x \right ) x \right )-\mathrm {log}\left (x +1\right )+\mathrm {log}\left (\frac {x^{2}+x}{e^{x}+\mathrm {log}\left (x \right )}\right )-\mathrm {log}\left (x \right )+x \] Input:
int((((x^2+x)*log(x)+(x^2+x)*exp(x))*log((x^2+x)/(log(x)+exp(x)))+((-3*x^2 -3*x)*exp(x)^2+x^3+2*x^2+3*x+1)*log(x)+(-3*x^2-3*x)*exp(x)^3+(x^3+x^2+2*x+ 1)*exp(x)-x-1)/(((x^2+x)*log(x)+(x^2+x)*exp(x))*log((x^2+x)/(log(x)+exp(x) ))+((-x^2-x)*exp(x)^2+x^3+x^2)*log(x)+(-x^2-x)*exp(x)^3+(x^3+x^2)*exp(x)), x)
Output:
log(e**(3*x) + e**(2*x)*log(x) - e**x*log((x**2 + x)/(e**x + log(x))) - e* *x*x - log((x**2 + x)/(e**x + log(x)))*log(x) - log(x)*x) - log(x + 1) + l og((x**2 + x)/(e**x + log(x))) - log(x) + x