Integrand size = 317, antiderivative size = 30 \[ \int \frac {e^{\frac {2 \left (4 x^2+e^x \left (4 x^2-4 x^3\right )+e^{2 x} \left (-e^3+x^2-2 x^3+x^4\right )+\left (4 x^2-4 x^3+e^x \left (-2 e^3+2 x^2-4 x^3+2 x^4\right )\right ) \log (x)+\left (-e^3+x^2-2 x^3+x^4\right ) \log ^2(x)\right )}{e^{2 x}+2 e^x \log (x)+\log ^2(x)}} \left (-16 x+e^x \left (8 x-8 x^2\right )+e^{2 x} \left (16 x-32 x^2+8 x^3\right )+e^{3 x} \left (4 x-12 x^2+8 x^3\right )+\left (8 x+8 x^2+e^x \left (32 x-56 x^2+8 x^3\right )+e^{2 x} \left (12 x-36 x^2+24 x^3\right )\right ) \log (x)+\left (16 x-24 x^2+e^x \left (12 x-36 x^2+24 x^3\right )\right ) \log ^2(x)+\left (4 x-12 x^2+8 x^3\right ) \log ^3(x)\right )}{e^{3 x}+3 e^{2 x} \log (x)+3 e^x \log ^2(x)+\log ^3(x)} \, dx=e^{-2 e^3+2 \left (x-x^2+\frac {2 x}{e^x+\log (x)}\right )^2} \] Output:
exp((2*x/(ln(x)+exp(x))-x^2+x)^2-exp(3))^2
Leaf count is larger than twice the leaf count of optimal. \(110\) vs. \(2(30)=60\).
Time = 1.02 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.67 \[ \int \frac {e^{\frac {2 \left (4 x^2+e^x \left (4 x^2-4 x^3\right )+e^{2 x} \left (-e^3+x^2-2 x^3+x^4\right )+\left (4 x^2-4 x^3+e^x \left (-2 e^3+2 x^2-4 x^3+2 x^4\right )\right ) \log (x)+\left (-e^3+x^2-2 x^3+x^4\right ) \log ^2(x)\right )}{e^{2 x}+2 e^x \log (x)+\log ^2(x)}} \left (-16 x+e^x \left (8 x-8 x^2\right )+e^{2 x} \left (16 x-32 x^2+8 x^3\right )+e^{3 x} \left (4 x-12 x^2+8 x^3\right )+\left (8 x+8 x^2+e^x \left (32 x-56 x^2+8 x^3\right )+e^{2 x} \left (12 x-36 x^2+24 x^3\right )\right ) \log (x)+\left (16 x-24 x^2+e^x \left (12 x-36 x^2+24 x^3\right )\right ) \log ^2(x)+\left (4 x-12 x^2+8 x^3\right ) \log ^3(x)\right )}{e^{3 x}+3 e^{2 x} \log (x)+3 e^x \log ^2(x)+\log ^3(x)} \, dx=e^{-\frac {2 \left (e^{3+2 x}-4 x^2+4 e^x (-1+x) x^2-e^{2 x} (-1+x)^2 x^2+\left (e^3-(-1+x)^2 x^2\right ) \log ^2(x)\right )}{\left (e^x+\log (x)\right )^2}} x^{-\frac {4 \left (e^{3+x}+2 (-1+x) x^2-e^x (-1+x)^2 x^2\right )}{\left (e^x+\log (x)\right )^2}} \] Input:
Integrate[(E^((2*(4*x^2 + E^x*(4*x^2 - 4*x^3) + E^(2*x)*(-E^3 + x^2 - 2*x^ 3 + x^4) + (4*x^2 - 4*x^3 + E^x*(-2*E^3 + 2*x^2 - 4*x^3 + 2*x^4))*Log[x] + (-E^3 + x^2 - 2*x^3 + x^4)*Log[x]^2))/(E^(2*x) + 2*E^x*Log[x] + Log[x]^2) )*(-16*x + E^x*(8*x - 8*x^2) + E^(2*x)*(16*x - 32*x^2 + 8*x^3) + E^(3*x)*( 4*x - 12*x^2 + 8*x^3) + (8*x + 8*x^2 + E^x*(32*x - 56*x^2 + 8*x^3) + E^(2* x)*(12*x - 36*x^2 + 24*x^3))*Log[x] + (16*x - 24*x^2 + E^x*(12*x - 36*x^2 + 24*x^3))*Log[x]^2 + (4*x - 12*x^2 + 8*x^3)*Log[x]^3))/(E^(3*x) + 3*E^(2* x)*Log[x] + 3*E^x*Log[x]^2 + Log[x]^3),x]
Output:
1/(E^((2*(E^(3 + 2*x) - 4*x^2 + 4*E^x*(-1 + x)*x^2 - E^(2*x)*(-1 + x)^2*x^ 2 + (E^3 - (-1 + x)^2*x^2)*Log[x]^2))/(E^x + Log[x])^2)*x^((4*(E^(3 + x) + 2*(-1 + x)*x^2 - E^x*(-1 + x)^2*x^2))/(E^x + Log[x])^2))
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.
| \(\displaystyle \int \frac {\left (e^x \left (8 x-8 x^2\right )+e^{2 x} \left (8 x^3-32 x^2+16 x\right )+e^{3 x} \left (8 x^3-12 x^2+4 x\right )+\left (8 x^3-12 x^2+4 x\right ) \log ^3(x)+\left (-24 x^2+e^x \left (24 x^3-36 x^2+12 x\right )+16 x\right ) \log ^2(x)+\left (8 x^2+e^x \left (8 x^3-56 x^2+32 x\right )+e^{2 x} \left (24 x^3-36 x^2+12 x\right )+8 x\right ) \log (x)-16 x\right ) \exp \left (\frac {2 \left (4 x^2+e^x \left (4 x^2-4 x^3\right )+e^{2 x} \left (x^4-2 x^3+x^2-e^3\right )+\left (x^4-2 x^3+x^2-e^3\right ) \log ^2(x)+\left (-4 x^3+4 x^2+e^x \left (2 x^4-4 x^3+2 x^2-2 e^3\right )\right ) \log (x)\right )}{e^{2 x}+\log ^2(x)+2 e^x \log (x)}\right )}{e^{3 x}+\log ^3(x)+3 e^x \log ^2(x)+3 e^{2 x} \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (e^x \left (8 x-8 x^2\right )+e^{2 x} \left (8 x^3-32 x^2+16 x\right )+e^{3 x} \left (8 x^3-12 x^2+4 x\right )+\left (8 x^3-12 x^2+4 x\right ) \log ^3(x)+\left (-24 x^2+e^x \left (24 x^3-36 x^2+12 x\right )+16 x\right ) \log ^2(x)+\left (8 x^2+e^x \left (8 x^3-56 x^2+32 x\right )+e^{2 x} \left (24 x^3-36 x^2+12 x\right )+8 x\right ) \log (x)-16 x\right ) \exp \left (\frac {2 \left (4 x^2+e^x \left (4 x^2-4 x^3\right )+e^{2 x} \left (x^4-2 x^3+x^2-e^3\right )+\left (x^4-2 x^3+x^2-e^3\right ) \log ^2(x)+\left (-4 x^3+4 x^2+e^x \left (2 x^4-4 x^3+2 x^2-2 e^3\right )\right ) \log (x)\right )}{\left (e^x+\log (x)\right )^2}\right )}{\left (e^x+\log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {8 x \left (x^2-4 x+2\right ) \exp \left (\frac {2 \left (4 x^2+e^x \left (4 x^2-4 x^3\right )+e^{2 x} \left (x^4-2 x^3+x^2-e^3\right )+\left (x^4-2 x^3+x^2-e^3\right ) \log ^2(x)+\left (-4 x^3+4 x^2+e^x \left (2 x^4-4 x^3+2 x^2-2 e^3\right )\right ) \log (x)\right )}{\left (e^x+\log (x)\right )^2}\right )}{e^x+\log (x)}+4 x \left (2 x^2-3 x+1\right ) \exp \left (\frac {2 \left (4 x^2+e^x \left (4 x^2-4 x^3\right )+e^{2 x} \left (x^4-2 x^3+x^2-e^3\right )+\left (x^4-2 x^3+x^2-e^3\right ) \log ^2(x)+\left (-4 x^3+4 x^2+e^x \left (2 x^4-4 x^3+2 x^2-2 e^3\right )\right ) \log (x)\right )}{\left (e^x+\log (x)\right )^2}\right )+\frac {16 x (x \log (x)-1) \exp \left (\frac {2 \left (4 x^2+e^x \left (4 x^2-4 x^3\right )+e^{2 x} \left (x^4-2 x^3+x^2-e^3\right )+\left (x^4-2 x^3+x^2-e^3\right ) \log ^2(x)+\left (-4 x^3+4 x^2+e^x \left (2 x^4-4 x^3+2 x^2-2 e^3\right )\right ) \log (x)\right )}{\left (e^x+\log (x)\right )^2}\right )}{\left (e^x+\log (x)\right )^3}-\frac {8 (x-1) x (x \log (x)+1) \exp \left (\frac {2 \left (4 x^2+e^x \left (4 x^2-4 x^3\right )+e^{2 x} \left (x^4-2 x^3+x^2-e^3\right )+\left (x^4-2 x^3+x^2-e^3\right ) \log ^2(x)+\left (-4 x^3+4 x^2+e^x \left (2 x^4-4 x^3+2 x^2-2 e^3\right )\right ) \log (x)\right )}{\left (e^x+\log (x)\right )^2}\right )}{\left (e^x+\log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 4 x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2-4 e^{x+3}}{\left (e^x+\log (x)\right )^2}+1} \left (2 x^2+\frac {2 \left (x^2-4 x+2\right )}{e^x+\log (x)}-3 x+\frac {4 (x \log (x)-1)}{\left (e^x+\log (x)\right )^3}-\frac {2 (x-1) (x \log (x)+1)}{\left (e^x+\log (x)\right )^2}+1\right ) \exp \left (-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)+e^{2 x+3}\right )}{\left (e^x+\log (x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int \exp \left (-\frac {2 \left (-e^{2 x} (1-x)^2 x^2-4 e^x (1-x) x^2-4 x^2+e^{2 x+3}+\left (e^3-(1-x)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) x^{1-\frac {4 \left (-e^x (1-x)^2 x^2-2 (1-x) x^2+e^{x+3}\right )}{\left (\log (x)+e^x\right )^2}} \left (2 x^2-3 x-\frac {4 (1-x \log (x))}{\left (\log (x)+e^x\right )^3}+\frac {2 (1-x) (x \log (x)+1)}{\left (\log (x)+e^x\right )^2}+\frac {2 \left (x^2-4 x+2\right )}{\log (x)+e^x}+1\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (\exp \left (-\frac {2 \left (-e^{2 x} (1-x)^2 x^2-4 e^x (1-x) x^2-4 x^2+e^{2 x+3}+\left (e^3-(1-x)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) x^{1-\frac {4 \left (-e^x (1-x)^2 x^2-2 (1-x) x^2+e^{x+3}\right )}{\left (\log (x)+e^x\right )^2}}+\frac {4 \exp \left (-\frac {2 \left (-e^{2 x} (1-x)^2 x^2-4 e^x (1-x) x^2-4 x^2+e^{2 x+3}+\left (e^3-(1-x)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) (x \log (x)-1) x^{1-\frac {4 \left (-e^x (1-x)^2 x^2-2 (1-x) x^2+e^{x+3}\right )}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^3}-\frac {2 \exp \left (-\frac {2 \left (-e^{2 x} (1-x)^2 x^2-4 e^x (1-x) x^2-4 x^2+e^{2 x+3}+\left (e^3-(1-x)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) (x-1) (x \log (x)+1) x^{1-\frac {4 \left (-e^x (1-x)^2 x^2-2 (1-x) x^2+e^{x+3}\right )}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^2}+\frac {2 \exp \left (-\frac {2 \left (-e^{2 x} (1-x)^2 x^2-4 e^x (1-x) x^2-4 x^2+e^{2 x+3}+\left (e^3-(1-x)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) \left (x^2-4 x+2\right ) x^{1-\frac {4 \left (-e^x (1-x)^2 x^2-2 (1-x) x^2+e^{x+3}\right )}{\left (\log (x)+e^x\right )^2}}}{\log (x)+e^x}-3 \exp \left (-\frac {2 \left (-e^{2 x} (1-x)^2 x^2-4 e^x (1-x) x^2-4 x^2+e^{2 x+3}+\left (e^3-(1-x)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) x^{2-\frac {4 \left (-e^x (1-x)^2 x^2-2 (1-x) x^2+e^{x+3}\right )}{\left (\log (x)+e^x\right )^2}}+2 \exp \left (-\frac {2 \left (-e^{2 x} (1-x)^2 x^2-4 e^x (1-x) x^2-4 x^2+e^{2 x+3}+\left (e^3-(1-x)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) x^{3-\frac {4 \left (-e^x (1-x)^2 x^2-2 (1-x) x^2+e^{x+3}\right )}{\left (\log (x)+e^x\right )^2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \exp \left (-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}} \left (2 x^2-3 x+\frac {4 (x \log (x)-1)}{\left (\log (x)+e^x\right )^3}-\frac {2 (x-1) (x \log (x)+1)}{\left (\log (x)+e^x\right )^2}+\frac {2 \left (x^2-4 x+2\right )}{\log (x)+e^x}+1\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}+\frac {4 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x \log (x)-1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^3}-\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x-1) (x \log (x)+1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^2}+\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} \left (x^2-4 x+2\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\log (x)+e^x}-3 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+1}+2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}} \left (\left (2 x^2-3 x+1\right ) \log ^3(x)+\left (-6 x+e^x \left (6 x^2-9 x+3\right )+4\right ) \log ^2(x)+\left (2 (x+1)+2 e^x \left (x^2-7 x+4\right )+e^{2 x} \left (6 x^2-9 x+3\right )\right ) \log (x)-2 e^x (x-1)+2 e^{2 x} \left (x^2-4 x+2\right )+e^{3 x} \left (2 x^2-3 x+1\right )-4\right )}{\left (\log (x)+e^x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}+\frac {4 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x \log (x)-1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^3}-\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x-1) (x \log (x)+1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^2}+\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} \left (x^2-4 x+2\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\log (x)+e^x}-3 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+1}+2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}} \left (\left (2 x^2-3 x+1\right ) \log ^3(x)+\left (-6 x+e^x \left (6 x^2-9 x+3\right )+4\right ) \log ^2(x)+\left (2 (x+1)+2 e^x \left (x^2-7 x+4\right )+e^{2 x} \left (6 x^2-9 x+3\right )\right ) \log (x)-2 e^x (x-1)+2 e^{2 x} \left (x^2-4 x+2\right )+e^{3 x} \left (2 x^2-3 x+1\right )-4\right )}{\left (\log (x)+e^x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}+\frac {4 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x \log (x)-1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^3}-\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x-1) (x \log (x)+1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^2}+\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} \left (x^2-4 x+2\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\log (x)+e^x}-3 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+1}+2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}} \left (\left (2 x^2-3 x+1\right ) \log ^3(x)+\left (-6 x+e^x \left (6 x^2-9 x+3\right )+4\right ) \log ^2(x)+\left (2 (x+1)+2 e^x \left (x^2-7 x+4\right )+e^{2 x} \left (6 x^2-9 x+3\right )\right ) \log (x)-2 e^x (x-1)+2 e^{2 x} \left (x^2-4 x+2\right )+e^{3 x} \left (2 x^2-3 x+1\right )-4\right )}{\left (\log (x)+e^x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}+\frac {4 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x \log (x)-1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^3}-\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x-1) (x \log (x)+1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^2}+\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} \left (x^2-4 x+2\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\log (x)+e^x}-3 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+1}+2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}} \left (\left (2 x^2-3 x+1\right ) \log ^3(x)+\left (-6 x+e^x \left (6 x^2-9 x+3\right )+4\right ) \log ^2(x)+\left (2 (x+1)+2 e^x \left (x^2-7 x+4\right )+e^{2 x} \left (6 x^2-9 x+3\right )\right ) \log (x)-2 e^x (x-1)+2 e^{2 x} \left (x^2-4 x+2\right )+e^{3 x} \left (2 x^2-3 x+1\right )-4\right )}{\left (\log (x)+e^x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}+\frac {4 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x \log (x)-1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^3}-\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x-1) (x \log (x)+1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^2}+\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} \left (x^2-4 x+2\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\log (x)+e^x}-3 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+1}+2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}} \left (\left (2 x^2-3 x+1\right ) \log ^3(x)+\left (-6 x+e^x \left (6 x^2-9 x+3\right )+4\right ) \log ^2(x)+\left (2 (x+1)+2 e^x \left (x^2-7 x+4\right )+e^{2 x} \left (6 x^2-9 x+3\right )\right ) \log (x)-2 e^x (x-1)+2 e^{2 x} \left (x^2-4 x+2\right )+e^{3 x} \left (2 x^2-3 x+1\right )-4\right )}{\left (\log (x)+e^x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}+\frac {4 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x \log (x)-1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^3}-\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x-1) (x \log (x)+1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^2}+\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} \left (x^2-4 x+2\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\log (x)+e^x}-3 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+1}+2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}} \left (\left (2 x^2-3 x+1\right ) \log ^3(x)+\left (-6 x+e^x \left (6 x^2-9 x+3\right )+4\right ) \log ^2(x)+\left (2 (x+1)+2 e^x \left (x^2-7 x+4\right )+e^{2 x} \left (6 x^2-9 x+3\right )\right ) \log (x)-2 e^x (x-1)+2 e^{2 x} \left (x^2-4 x+2\right )+e^{3 x} \left (2 x^2-3 x+1\right )-4\right )}{\left (\log (x)+e^x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}+\frac {4 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x \log (x)-1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^3}-\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x-1) (x \log (x)+1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^2}+\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} \left (x^2-4 x+2\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\log (x)+e^x}-3 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+1}+2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}} \left (\left (2 x^2-3 x+1\right ) \log ^3(x)+\left (-6 x+e^x \left (6 x^2-9 x+3\right )+4\right ) \log ^2(x)+\left (2 (x+1)+2 e^x \left (x^2-7 x+4\right )+e^{2 x} \left (6 x^2-9 x+3\right )\right ) \log (x)-2 e^x (x-1)+2 e^{2 x} \left (x^2-4 x+2\right )+e^{3 x} \left (2 x^2-3 x+1\right )-4\right )}{\left (\log (x)+e^x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}+\frac {4 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x \log (x)-1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^3}-\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x-1) (x \log (x)+1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^2}+\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} \left (x^2-4 x+2\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\log (x)+e^x}-3 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+1}+2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}} \left (\left (2 x^2-3 x+1\right ) \log ^3(x)+\left (-6 x+e^x \left (6 x^2-9 x+3\right )+4\right ) \log ^2(x)+\left (2 (x+1)+2 e^x \left (x^2-7 x+4\right )+e^{2 x} \left (6 x^2-9 x+3\right )\right ) \log (x)-2 e^x (x-1)+2 e^{2 x} \left (x^2-4 x+2\right )+e^{3 x} \left (2 x^2-3 x+1\right )-4\right )}{\left (\log (x)+e^x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}+\frac {4 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x \log (x)-1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^3}-\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x-1) (x \log (x)+1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^2}+\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} \left (x^2-4 x+2\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\log (x)+e^x}-3 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+1}+2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}} \left (\left (2 x^2-3 x+1\right ) \log ^3(x)+\left (-6 x+e^x \left (6 x^2-9 x+3\right )+4\right ) \log ^2(x)+\left (2 (x+1)+2 e^x \left (x^2-7 x+4\right )+e^{2 x} \left (6 x^2-9 x+3\right )\right ) \log (x)-2 e^x (x-1)+2 e^{2 x} \left (x^2-4 x+2\right )+e^{3 x} \left (2 x^2-3 x+1\right )-4\right )}{\left (\log (x)+e^x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}+\frac {4 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x \log (x)-1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^3}-\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x-1) (x \log (x)+1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^2}+\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} \left (x^2-4 x+2\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\log (x)+e^x}-3 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+1}+2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}} \left (\left (2 x^2-3 x+1\right ) \log ^3(x)+\left (-6 x+e^x \left (6 x^2-9 x+3\right )+4\right ) \log ^2(x)+\left (2 (x+1)+2 e^x \left (x^2-7 x+4\right )+e^{2 x} \left (6 x^2-9 x+3\right )\right ) \log (x)-2 e^x (x-1)+2 e^{2 x} \left (x^2-4 x+2\right )+e^{3 x} \left (2 x^2-3 x+1\right )-4\right )}{\left (\log (x)+e^x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}+\frac {4 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x \log (x)-1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^3}-\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x-1) (x \log (x)+1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^2}+\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} \left (x^2-4 x+2\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\log (x)+e^x}-3 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+1}+2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}} \left (\left (2 x^2-3 x+1\right ) \log ^3(x)+\left (-6 x+e^x \left (6 x^2-9 x+3\right )+4\right ) \log ^2(x)+\left (2 (x+1)+2 e^x \left (x^2-7 x+4\right )+e^{2 x} \left (6 x^2-9 x+3\right )\right ) \log (x)-2 e^x (x-1)+2 e^{2 x} \left (x^2-4 x+2\right )+e^{3 x} \left (2 x^2-3 x+1\right )-4\right )}{\left (\log (x)+e^x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}+\frac {4 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x \log (x)-1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^3}-\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} (x-1) (x \log (x)+1) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\left (\log (x)+e^x\right )^2}+\frac {2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} \left (x^2-4 x+2\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}}}{\log (x)+e^x}-3 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+1}+2 e^{-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}} x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}+2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {2 \left (-e^{2 x} (x-1)^2 x^2+4 e^x (x-1) x^2-4 x^2+e^{2 x+3}+\left (e^3-(x-1)^2 x^2\right ) \log ^2(x)\right )}{\left (\log (x)+e^x\right )^2}\right ) x^{\frac {4 e^x (x-1)^2 x^2-8 (x-1) x^2+e^{2 x}-4 e^{x+3}+\log ^2(x)+2 e^x \log (x)}{\left (\log (x)+e^x\right )^2}} \left (\left (2 x^2-3 x+1\right ) \log ^3(x)+\left (-6 x+e^x \left (6 x^2-9 x+3\right )+4\right ) \log ^2(x)+\left (2 (x+1)+2 e^x \left (x^2-7 x+4\right )+e^{2 x} \left (6 x^2-9 x+3\right )\right ) \log (x)-2 e^x (x-1)+2 e^{2 x} \left (x^2-4 x+2\right )+e^{3 x} \left (2 x^2-3 x+1\right )-4\right )}{\left (\log (x)+e^x\right )^3}dx\) |
Input:
Int[(E^((2*(4*x^2 + E^x*(4*x^2 - 4*x^3) + E^(2*x)*(-E^3 + x^2 - 2*x^3 + x^ 4) + (4*x^2 - 4*x^3 + E^x*(-2*E^3 + 2*x^2 - 4*x^3 + 2*x^4))*Log[x] + (-E^3 + x^2 - 2*x^3 + x^4)*Log[x]^2))/(E^(2*x) + 2*E^x*Log[x] + Log[x]^2))*(-16 *x + E^x*(8*x - 8*x^2) + E^(2*x)*(16*x - 32*x^2 + 8*x^3) + E^(3*x)*(4*x - 12*x^2 + 8*x^3) + (8*x + 8*x^2 + E^x*(32*x - 56*x^2 + 8*x^3) + E^(2*x)*(12 *x - 36*x^2 + 24*x^3))*Log[x] + (16*x - 24*x^2 + E^x*(12*x - 36*x^2 + 24*x ^3))*Log[x]^2 + (4*x - 12*x^2 + 8*x^3)*Log[x]^3))/(E^(3*x) + 3*E^(2*x)*Log [x] + 3*E^x*Log[x]^2 + Log[x]^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(156\) vs. \(2(27)=54\).
Time = 0.11 (sec) , antiderivative size = 157, normalized size of antiderivative = 5.23
\[{\mathrm e}^{-\frac {2 \left (-x^{4} \ln \left (x \right )^{2}-2 \,{\mathrm e}^{x} \ln \left (x \right ) x^{4}+2 x^{3} \ln \left (x \right )^{2}+4 \,{\mathrm e}^{x} \ln \left (x \right ) x^{3}-x^{4} {\mathrm e}^{2 x}-x^{2} \ln \left (x \right )^{2}-2 x^{2} {\mathrm e}^{x} \ln \left (x \right )+4 x^{3} \ln \left (x \right )+4 \,{\mathrm e}^{x} x^{3}+2 \,{\mathrm e}^{2 x} x^{3}+\ln \left (x \right )^{2} {\mathrm e}^{3}+2 \ln \left (x \right ) {\mathrm e}^{3+x}-4 x^{2} \ln \left (x \right )-4 \,{\mathrm e}^{x} x^{2}-{\mathrm e}^{2 x} x^{2}+{\mathrm e}^{3+2 x}-4 x^{2}\right )}{\ln \left (x \right )^{2}+2 \,{\mathrm e}^{x} \ln \left (x \right )+{\mathrm e}^{2 x}}}\]
Input:
int(((8*x^3-12*x^2+4*x)*ln(x)^3+((24*x^3-36*x^2+12*x)*exp(x)-24*x^2+16*x)* ln(x)^2+((24*x^3-36*x^2+12*x)*exp(x)^2+(8*x^3-56*x^2+32*x)*exp(x)+8*x^2+8* x)*ln(x)+(8*x^3-12*x^2+4*x)*exp(x)^3+(8*x^3-32*x^2+16*x)*exp(x)^2+(-8*x^2+ 8*x)*exp(x)-16*x)*exp(((-exp(3)+x^4-2*x^3+x^2)*ln(x)^2+((-2*exp(3)+2*x^4-4 *x^3+2*x^2)*exp(x)-4*x^3+4*x^2)*ln(x)+(-exp(3)+x^4-2*x^3+x^2)*exp(x)^2+(-4 *x^3+4*x^2)*exp(x)+4*x^2)/(ln(x)^2+2*exp(x)*ln(x)+exp(x)^2))^2/(ln(x)^3+3* exp(x)*ln(x)^2+3*exp(x)^2*ln(x)+exp(x)^3),x)
Output:
exp(-2*(-x^4*ln(x)^2-2*exp(x)*ln(x)*x^4+2*x^3*ln(x)^2+4*exp(x)*ln(x)*x^3-x ^4*exp(2*x)-x^2*ln(x)^2-2*x^2*exp(x)*ln(x)+4*x^3*ln(x)+4*exp(x)*x^3+2*exp( 2*x)*x^3+ln(x)^2*exp(3)+2*ln(x)*exp(3+x)-4*x^2*ln(x)-4*exp(x)*x^2-exp(2*x) *x^2+exp(3+2*x)-4*x^2)/(ln(x)^2+2*exp(x)*ln(x)+exp(2*x)))
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (27) = 54\).
Time = 0.09 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.87 \[ \int \frac {e^{\frac {2 \left (4 x^2+e^x \left (4 x^2-4 x^3\right )+e^{2 x} \left (-e^3+x^2-2 x^3+x^4\right )+\left (4 x^2-4 x^3+e^x \left (-2 e^3+2 x^2-4 x^3+2 x^4\right )\right ) \log (x)+\left (-e^3+x^2-2 x^3+x^4\right ) \log ^2(x)\right )}{e^{2 x}+2 e^x \log (x)+\log ^2(x)}} \left (-16 x+e^x \left (8 x-8 x^2\right )+e^{2 x} \left (16 x-32 x^2+8 x^3\right )+e^{3 x} \left (4 x-12 x^2+8 x^3\right )+\left (8 x+8 x^2+e^x \left (32 x-56 x^2+8 x^3\right )+e^{2 x} \left (12 x-36 x^2+24 x^3\right )\right ) \log (x)+\left (16 x-24 x^2+e^x \left (12 x-36 x^2+24 x^3\right )\right ) \log ^2(x)+\left (4 x-12 x^2+8 x^3\right ) \log ^3(x)\right )}{e^{3 x}+3 e^{2 x} \log (x)+3 e^x \log ^2(x)+\log ^3(x)} \, dx=e^{\left (\frac {2 \, {\left ({\left (x^{4} - 2 \, x^{3} + x^{2} - e^{3}\right )} \log \left (x\right )^{2} + 4 \, x^{2} + {\left (x^{4} - 2 \, x^{3} + x^{2} - e^{3}\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{3} - x^{2}\right )} e^{x} - 2 \, {\left (2 \, x^{3} - 2 \, x^{2} - {\left (x^{4} - 2 \, x^{3} + x^{2} - e^{3}\right )} e^{x}\right )} \log \left (x\right )\right )}}{2 \, e^{x} \log \left (x\right ) + \log \left (x\right )^{2} + e^{\left (2 \, x\right )}}\right )} \] Input:
integrate(((8*x^3-12*x^2+4*x)*log(x)^3+((24*x^3-36*x^2+12*x)*exp(x)-24*x^2 +16*x)*log(x)^2+((24*x^3-36*x^2+12*x)*exp(x)^2+(8*x^3-56*x^2+32*x)*exp(x)+ 8*x^2+8*x)*log(x)+(8*x^3-12*x^2+4*x)*exp(x)^3+(8*x^3-32*x^2+16*x)*exp(x)^2 +(-8*x^2+8*x)*exp(x)-16*x)*exp(((-exp(3)+x^4-2*x^3+x^2)*log(x)^2+((-2*exp( 3)+2*x^4-4*x^3+2*x^2)*exp(x)-4*x^3+4*x^2)*log(x)+(-exp(3)+x^4-2*x^3+x^2)*e xp(x)^2+(-4*x^3+4*x^2)*exp(x)+4*x^2)/(log(x)^2+2*exp(x)*log(x)+exp(x)^2))^ 2/(log(x)^3+3*exp(x)*log(x)^2+3*exp(x)^2*log(x)+exp(x)^3),x, algorithm="fr icas")
Output:
e^(2*((x^4 - 2*x^3 + x^2 - e^3)*log(x)^2 + 4*x^2 + (x^4 - 2*x^3 + x^2 - e^ 3)*e^(2*x) - 4*(x^3 - x^2)*e^x - 2*(2*x^3 - 2*x^2 - (x^4 - 2*x^3 + x^2 - e ^3)*e^x)*log(x))/(2*e^x*log(x) + log(x)^2 + e^(2*x)))
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (24) = 48\).
Time = 1.31 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.07 \[ \int \frac {e^{\frac {2 \left (4 x^2+e^x \left (4 x^2-4 x^3\right )+e^{2 x} \left (-e^3+x^2-2 x^3+x^4\right )+\left (4 x^2-4 x^3+e^x \left (-2 e^3+2 x^2-4 x^3+2 x^4\right )\right ) \log (x)+\left (-e^3+x^2-2 x^3+x^4\right ) \log ^2(x)\right )}{e^{2 x}+2 e^x \log (x)+\log ^2(x)}} \left (-16 x+e^x \left (8 x-8 x^2\right )+e^{2 x} \left (16 x-32 x^2+8 x^3\right )+e^{3 x} \left (4 x-12 x^2+8 x^3\right )+\left (8 x+8 x^2+e^x \left (32 x-56 x^2+8 x^3\right )+e^{2 x} \left (12 x-36 x^2+24 x^3\right )\right ) \log (x)+\left (16 x-24 x^2+e^x \left (12 x-36 x^2+24 x^3\right )\right ) \log ^2(x)+\left (4 x-12 x^2+8 x^3\right ) \log ^3(x)\right )}{e^{3 x}+3 e^{2 x} \log (x)+3 e^x \log ^2(x)+\log ^3(x)} \, dx=e^{\frac {2 \cdot \left (4 x^{2} + \left (- 4 x^{3} + 4 x^{2}\right ) e^{x} + \left (- 4 x^{3} + 4 x^{2} + \left (2 x^{4} - 4 x^{3} + 2 x^{2} - 2 e^{3}\right ) e^{x}\right ) \log {\left (x \right )} + \left (x^{4} - 2 x^{3} + x^{2} - e^{3}\right ) e^{2 x} + \left (x^{4} - 2 x^{3} + x^{2} - e^{3}\right ) \log {\left (x \right )}^{2}\right )}{e^{2 x} + 2 e^{x} \log {\left (x \right )} + \log {\left (x \right )}^{2}}} \] Input:
integrate(((8*x**3-12*x**2+4*x)*ln(x)**3+((24*x**3-36*x**2+12*x)*exp(x)-24 *x**2+16*x)*ln(x)**2+((24*x**3-36*x**2+12*x)*exp(x)**2+(8*x**3-56*x**2+32* x)*exp(x)+8*x**2+8*x)*ln(x)+(8*x**3-12*x**2+4*x)*exp(x)**3+(8*x**3-32*x**2 +16*x)*exp(x)**2+(-8*x**2+8*x)*exp(x)-16*x)*exp(((-exp(3)+x**4-2*x**3+x**2 )*ln(x)**2+((-2*exp(3)+2*x**4-4*x**3+2*x**2)*exp(x)-4*x**3+4*x**2)*ln(x)+( -exp(3)+x**4-2*x**3+x**2)*exp(x)**2+(-4*x**3+4*x**2)*exp(x)+4*x**2)/(ln(x) **2+2*exp(x)*ln(x)+exp(x)**2))**2/(ln(x)**3+3*exp(x)*ln(x)**2+3*exp(x)**2* ln(x)+exp(x)**3),x)
Output:
exp(2*(4*x**2 + (-4*x**3 + 4*x**2)*exp(x) + (-4*x**3 + 4*x**2 + (2*x**4 - 4*x**3 + 2*x**2 - 2*exp(3))*exp(x))*log(x) + (x**4 - 2*x**3 + x**2 - exp(3 ))*exp(2*x) + (x**4 - 2*x**3 + x**2 - exp(3))*log(x)**2)/(exp(2*x) + 2*exp (x)*log(x) + log(x)**2))
Leaf count of result is larger than twice the leaf count of optimal. 429 vs. \(2 (27) = 54\).
Time = 1.60 (sec) , antiderivative size = 429, normalized size of antiderivative = 14.30 \[ \int \frac {e^{\frac {2 \left (4 x^2+e^x \left (4 x^2-4 x^3\right )+e^{2 x} \left (-e^3+x^2-2 x^3+x^4\right )+\left (4 x^2-4 x^3+e^x \left (-2 e^3+2 x^2-4 x^3+2 x^4\right )\right ) \log (x)+\left (-e^3+x^2-2 x^3+x^4\right ) \log ^2(x)\right )}{e^{2 x}+2 e^x \log (x)+\log ^2(x)}} \left (-16 x+e^x \left (8 x-8 x^2\right )+e^{2 x} \left (16 x-32 x^2+8 x^3\right )+e^{3 x} \left (4 x-12 x^2+8 x^3\right )+\left (8 x+8 x^2+e^x \left (32 x-56 x^2+8 x^3\right )+e^{2 x} \left (12 x-36 x^2+24 x^3\right )\right ) \log (x)+\left (16 x-24 x^2+e^x \left (12 x-36 x^2+24 x^3\right )\right ) \log ^2(x)+\left (4 x-12 x^2+8 x^3\right ) \log ^3(x)\right )}{e^{3 x}+3 e^{2 x} \log (x)+3 e^x \log ^2(x)+\log ^3(x)} \, dx =\text {Too large to display} \] Input:
integrate(((8*x^3-12*x^2+4*x)*log(x)^3+((24*x^3-36*x^2+12*x)*exp(x)-24*x^2 +16*x)*log(x)^2+((24*x^3-36*x^2+12*x)*exp(x)^2+(8*x^3-56*x^2+32*x)*exp(x)+ 8*x^2+8*x)*log(x)+(8*x^3-12*x^2+4*x)*exp(x)^3+(8*x^3-32*x^2+16*x)*exp(x)^2 +(-8*x^2+8*x)*exp(x)-16*x)*exp(((-exp(3)+x^4-2*x^3+x^2)*log(x)^2+((-2*exp( 3)+2*x^4-4*x^3+2*x^2)*exp(x)-4*x^3+4*x^2)*log(x)+(-exp(3)+x^4-2*x^3+x^2)*e xp(x)^2+(-4*x^3+4*x^2)*exp(x)+4*x^2)/(log(x)^2+2*exp(x)*log(x)+exp(x)^2))^ 2/(log(x)^3+3*exp(x)*log(x)^2+3*exp(x)^2*log(x)+exp(x)^3),x, algorithm="ma xima")
Output:
e^(4*x^4*e^x*log(x)/(2*e^x*log(x) + log(x)^2 + e^(2*x)) + 2*x^4*log(x)^2/( 2*e^x*log(x) + log(x)^2 + e^(2*x)) + 2*x^4*e^(2*x)/(2*e^x*log(x) + log(x)^ 2 + e^(2*x)) - 8*x^3*e^x*log(x)/(2*e^x*log(x) + log(x)^2 + e^(2*x)) - 4*x^ 3*log(x)^2/(2*e^x*log(x) + log(x)^2 + e^(2*x)) - 4*x^3*e^(2*x)/(2*e^x*log( x) + log(x)^2 + e^(2*x)) - 8*x^3*e^x/(2*e^x*log(x) + log(x)^2 + e^(2*x)) - 8*x^3*log(x)/(2*e^x*log(x) + log(x)^2 + e^(2*x)) + 4*x^2*e^x*log(x)/(2*e^ x*log(x) + log(x)^2 + e^(2*x)) + 2*x^2*log(x)^2/(2*e^x*log(x) + log(x)^2 + e^(2*x)) + 2*x^2*e^(2*x)/(2*e^x*log(x) + log(x)^2 + e^(2*x)) + 8*x^2*e^x/ (2*e^x*log(x) + log(x)^2 + e^(2*x)) + 8*x^2*log(x)/(2*e^x*log(x) + log(x)^ 2 + e^(2*x)) - 2*e^3*log(x)^2/(2*e^x*log(x) + log(x)^2 + e^(2*x)) + 8*x^2/ (2*e^x*log(x) + log(x)^2 + e^(2*x)) - 4*e^(x + 3)*log(x)/(2*e^x*log(x) + l og(x)^2 + e^(2*x)) - 2*e^(2*x + 3)/(2*e^x*log(x) + log(x)^2 + e^(2*x)))
Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (27) = 54\).
Time = 0.93 (sec) , antiderivative size = 155, normalized size of antiderivative = 5.17 \[ \int \frac {e^{\frac {2 \left (4 x^2+e^x \left (4 x^2-4 x^3\right )+e^{2 x} \left (-e^3+x^2-2 x^3+x^4\right )+\left (4 x^2-4 x^3+e^x \left (-2 e^3+2 x^2-4 x^3+2 x^4\right )\right ) \log (x)+\left (-e^3+x^2-2 x^3+x^4\right ) \log ^2(x)\right )}{e^{2 x}+2 e^x \log (x)+\log ^2(x)}} \left (-16 x+e^x \left (8 x-8 x^2\right )+e^{2 x} \left (16 x-32 x^2+8 x^3\right )+e^{3 x} \left (4 x-12 x^2+8 x^3\right )+\left (8 x+8 x^2+e^x \left (32 x-56 x^2+8 x^3\right )+e^{2 x} \left (12 x-36 x^2+24 x^3\right )\right ) \log (x)+\left (16 x-24 x^2+e^x \left (12 x-36 x^2+24 x^3\right )\right ) \log ^2(x)+\left (4 x-12 x^2+8 x^3\right ) \log ^3(x)\right )}{e^{3 x}+3 e^{2 x} \log (x)+3 e^x \log ^2(x)+\log ^3(x)} \, dx=e^{\left (\frac {2 \, {\left (2 \, x^{4} e^{x} \log \left (x\right ) + x^{4} \log \left (x\right )^{2} + x^{4} e^{\left (2 \, x\right )} - 4 \, x^{3} e^{x} \log \left (x\right ) - 2 \, x^{3} \log \left (x\right )^{2} - 2 \, x^{3} e^{\left (2 \, x\right )} - 4 \, x^{3} e^{x} - 4 \, x^{3} \log \left (x\right ) + 2 \, x^{2} e^{x} \log \left (x\right ) + x^{2} \log \left (x\right )^{2} + x^{2} e^{\left (2 \, x\right )} + 4 \, x^{2} e^{x} + 4 \, x^{2} \log \left (x\right ) - e^{3} \log \left (x\right )^{2} + 4 \, x^{2} - 2 \, e^{\left (x + 3\right )} \log \left (x\right ) - e^{\left (2 \, x + 3\right )}\right )}}{2 \, e^{x} \log \left (x\right ) + \log \left (x\right )^{2} + e^{\left (2 \, x\right )}}\right )} \] Input:
integrate(((8*x^3-12*x^2+4*x)*log(x)^3+((24*x^3-36*x^2+12*x)*exp(x)-24*x^2 +16*x)*log(x)^2+((24*x^3-36*x^2+12*x)*exp(x)^2+(8*x^3-56*x^2+32*x)*exp(x)+ 8*x^2+8*x)*log(x)+(8*x^3-12*x^2+4*x)*exp(x)^3+(8*x^3-32*x^2+16*x)*exp(x)^2 +(-8*x^2+8*x)*exp(x)-16*x)*exp(((-exp(3)+x^4-2*x^3+x^2)*log(x)^2+((-2*exp( 3)+2*x^4-4*x^3+2*x^2)*exp(x)-4*x^3+4*x^2)*log(x)+(-exp(3)+x^4-2*x^3+x^2)*e xp(x)^2+(-4*x^3+4*x^2)*exp(x)+4*x^2)/(log(x)^2+2*exp(x)*log(x)+exp(x)^2))^ 2/(log(x)^3+3*exp(x)*log(x)^2+3*exp(x)^2*log(x)+exp(x)^3),x, algorithm="gi ac")
Output:
e^(2*(2*x^4*e^x*log(x) + x^4*log(x)^2 + x^4*e^(2*x) - 4*x^3*e^x*log(x) - 2 *x^3*log(x)^2 - 2*x^3*e^(2*x) - 4*x^3*e^x - 4*x^3*log(x) + 2*x^2*e^x*log(x ) + x^2*log(x)^2 + x^2*e^(2*x) + 4*x^2*e^x + 4*x^2*log(x) - e^3*log(x)^2 + 4*x^2 - 2*e^(x + 3)*log(x) - e^(2*x + 3))/(2*e^x*log(x) + log(x)^2 + e^(2 *x)))
Time = 4.72 (sec) , antiderivative size = 345, normalized size of antiderivative = 11.50 \[ \int \frac {e^{\frac {2 \left (4 x^2+e^x \left (4 x^2-4 x^3\right )+e^{2 x} \left (-e^3+x^2-2 x^3+x^4\right )+\left (4 x^2-4 x^3+e^x \left (-2 e^3+2 x^2-4 x^3+2 x^4\right )\right ) \log (x)+\left (-e^3+x^2-2 x^3+x^4\right ) \log ^2(x)\right )}{e^{2 x}+2 e^x \log (x)+\log ^2(x)}} \left (-16 x+e^x \left (8 x-8 x^2\right )+e^{2 x} \left (16 x-32 x^2+8 x^3\right )+e^{3 x} \left (4 x-12 x^2+8 x^3\right )+\left (8 x+8 x^2+e^x \left (32 x-56 x^2+8 x^3\right )+e^{2 x} \left (12 x-36 x^2+24 x^3\right )\right ) \log (x)+\left (16 x-24 x^2+e^x \left (12 x-36 x^2+24 x^3\right )\right ) \log ^2(x)+\left (4 x-12 x^2+8 x^3\right ) \log ^3(x)\right )}{e^{3 x}+3 e^{2 x} \log (x)+3 e^x \log ^2(x)+\log ^3(x)} \, dx=x^{\frac {4\,\left (x^2\,{\mathrm {e}}^x-2\,x^3\,{\mathrm {e}}^x+x^4\,{\mathrm {e}}^x-{\mathrm {e}}^3\,{\mathrm {e}}^x+2\,x^2-2\,x^3\right )}{{\ln \left (x\right )}^2+2\,{\mathrm {e}}^x\,\ln \left (x\right )+{\mathrm {e}}^{2\,x}}}\,{\mathrm {e}}^{\frac {2\,x^2\,{\mathrm {e}}^{2\,x}}{{\ln \left (x\right )}^2+2\,{\mathrm {e}}^x\,\ln \left (x\right )+{\mathrm {e}}^{2\,x}}}\,{\mathrm {e}}^{\frac {2\,x^4\,{\mathrm {e}}^{2\,x}}{{\ln \left (x\right )}^2+2\,{\mathrm {e}}^x\,\ln \left (x\right )+{\mathrm {e}}^{2\,x}}}\,{\mathrm {e}}^{-\frac {4\,x^3\,{\mathrm {e}}^{2\,x}}{{\ln \left (x\right )}^2+2\,{\mathrm {e}}^x\,\ln \left (x\right )+{\mathrm {e}}^{2\,x}}}\,{\mathrm {e}}^{\frac {2\,x^2\,{\ln \left (x\right )}^2}{{\ln \left (x\right )}^2+2\,{\mathrm {e}}^x\,\ln \left (x\right )+{\mathrm {e}}^{2\,x}}}\,{\mathrm {e}}^{\frac {2\,x^4\,{\ln \left (x\right )}^2}{{\ln \left (x\right )}^2+2\,{\mathrm {e}}^x\,\ln \left (x\right )+{\mathrm {e}}^{2\,x}}}\,{\mathrm {e}}^{-\frac {4\,x^3\,{\ln \left (x\right )}^2}{{\ln \left (x\right )}^2+2\,{\mathrm {e}}^x\,\ln \left (x\right )+{\mathrm {e}}^{2\,x}}}\,{\mathrm {e}}^{-\frac {2\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^3}{{\ln \left (x\right )}^2+2\,{\mathrm {e}}^x\,\ln \left (x\right )+{\mathrm {e}}^{2\,x}}}\,{\mathrm {e}}^{-\frac {2\,{\mathrm {e}}^3\,{\ln \left (x\right )}^2}{{\ln \left (x\right )}^2+2\,{\mathrm {e}}^x\,\ln \left (x\right )+{\mathrm {e}}^{2\,x}}}\,{\mathrm {e}}^{\frac {8\,x^2\,{\mathrm {e}}^x}{{\ln \left (x\right )}^2+2\,{\mathrm {e}}^x\,\ln \left (x\right )+{\mathrm {e}}^{2\,x}}}\,{\mathrm {e}}^{-\frac {8\,x^3\,{\mathrm {e}}^x}{{\ln \left (x\right )}^2+2\,{\mathrm {e}}^x\,\ln \left (x\right )+{\mathrm {e}}^{2\,x}}}\,{\mathrm {e}}^{\frac {8\,x^2}{{\ln \left (x\right )}^2+2\,{\mathrm {e}}^x\,\ln \left (x\right )+{\mathrm {e}}^{2\,x}}} \] Input:
int((exp(-(2*(exp(2*x)*(exp(3) - x^2 + 2*x^3 - x^4) - exp(x)*(4*x^2 - 4*x^ 3) + log(x)^2*(exp(3) - x^2 + 2*x^3 - x^4) + log(x)*(exp(x)*(2*exp(3) - 2* x^2 + 4*x^3 - 2*x^4) - 4*x^2 + 4*x^3) - 4*x^2))/(exp(2*x) + 2*exp(x)*log(x ) + log(x)^2))*(exp(3*x)*(4*x - 12*x^2 + 8*x^3) - 16*x + exp(2*x)*(16*x - 32*x^2 + 8*x^3) + log(x)^3*(4*x - 12*x^2 + 8*x^3) + exp(x)*(8*x - 8*x^2) + log(x)*(8*x + exp(2*x)*(12*x - 36*x^2 + 24*x^3) + 8*x^2 + exp(x)*(32*x - 56*x^2 + 8*x^3)) + log(x)^2*(16*x - 24*x^2 + exp(x)*(12*x - 36*x^2 + 24*x^ 3))))/(exp(3*x) + log(x)^3 + 3*exp(2*x)*log(x) + 3*exp(x)*log(x)^2),x)
Output:
x^((4*(x^2*exp(x) - 2*x^3*exp(x) + x^4*exp(x) - exp(3)*exp(x) + 2*x^2 - 2* x^3))/(exp(2*x) + 2*exp(x)*log(x) + log(x)^2))*exp((2*x^2*exp(2*x))/(exp(2 *x) + 2*exp(x)*log(x) + log(x)^2))*exp((2*x^4*exp(2*x))/(exp(2*x) + 2*exp( x)*log(x) + log(x)^2))*exp(-(4*x^3*exp(2*x))/(exp(2*x) + 2*exp(x)*log(x) + log(x)^2))*exp((2*x^2*log(x)^2)/(exp(2*x) + 2*exp(x)*log(x) + log(x)^2))* exp((2*x^4*log(x)^2)/(exp(2*x) + 2*exp(x)*log(x) + log(x)^2))*exp(-(4*x^3* log(x)^2)/(exp(2*x) + 2*exp(x)*log(x) + log(x)^2))*exp(-(2*exp(2*x)*exp(3) )/(exp(2*x) + 2*exp(x)*log(x) + log(x)^2))*exp(-(2*exp(3)*log(x)^2)/(exp(2 *x) + 2*exp(x)*log(x) + log(x)^2))*exp((8*x^2*exp(x))/(exp(2*x) + 2*exp(x) *log(x) + log(x)^2))*exp(-(8*x^3*exp(x))/(exp(2*x) + 2*exp(x)*log(x) + log (x)^2))*exp((8*x^2)/(exp(2*x) + 2*exp(x)*log(x) + log(x)^2))
\[ \int \frac {e^{\frac {2 \left (4 x^2+e^x \left (4 x^2-4 x^3\right )+e^{2 x} \left (-e^3+x^2-2 x^3+x^4\right )+\left (4 x^2-4 x^3+e^x \left (-2 e^3+2 x^2-4 x^3+2 x^4\right )\right ) \log (x)+\left (-e^3+x^2-2 x^3+x^4\right ) \log ^2(x)\right )}{e^{2 x}+2 e^x \log (x)+\log ^2(x)}} \left (-16 x+e^x \left (8 x-8 x^2\right )+e^{2 x} \left (16 x-32 x^2+8 x^3\right )+e^{3 x} \left (4 x-12 x^2+8 x^3\right )+\left (8 x+8 x^2+e^x \left (32 x-56 x^2+8 x^3\right )+e^{2 x} \left (12 x-36 x^2+24 x^3\right )\right ) \log (x)+\left (16 x-24 x^2+e^x \left (12 x-36 x^2+24 x^3\right )\right ) \log ^2(x)+\left (4 x-12 x^2+8 x^3\right ) \log ^3(x)\right )}{e^{3 x}+3 e^{2 x} \log (x)+3 e^x \log ^2(x)+\log ^3(x)} \, dx=\int \frac {\left (\left (8 x^{3}-12 x^{2}+4 x \right ) \mathrm {log}\left (x \right )^{3}+\left (\left (24 x^{3}-36 x^{2}+12 x \right ) {\mathrm e}^{x}-24 x^{2}+16 x \right ) \mathrm {log}\left (x \right )^{2}+\left (\left (24 x^{3}-36 x^{2}+12 x \right ) \left ({\mathrm e}^{x}\right )^{2}+\left (8 x^{3}-56 x^{2}+32 x \right ) {\mathrm e}^{x}+8 x^{2}+8 x \right ) \mathrm {log}\left (x \right )+\left (8 x^{3}-12 x^{2}+4 x \right ) \left ({\mathrm e}^{x}\right )^{3}+\left (8 x^{3}-32 x^{2}+16 x \right ) \left ({\mathrm e}^{x}\right )^{2}+\left (-8 x^{2}+8 x \right ) {\mathrm e}^{x}-16 x \right ) \left ({\mathrm e}^{\frac {\left (-{\mathrm e}^{3}+x^{4}-2 x^{3}+x^{2}\right ) \mathrm {log}\left (x \right )^{2}+\left (\left (-2 \,{\mathrm e}^{3}+2 x^{4}-4 x^{3}+2 x^{2}\right ) {\mathrm e}^{x}-4 x^{3}+4 x^{2}\right ) \mathrm {log}\left (x \right )+\left (-{\mathrm e}^{3}+x^{4}-2 x^{3}+x^{2}\right ) \left ({\mathrm e}^{x}\right )^{2}+\left (-4 x^{3}+4 x^{2}\right ) {\mathrm e}^{x}+4 x^{2}}{\mathrm {log}\left (x \right )^{2}+2 \,{\mathrm e}^{x} \mathrm {log}\left (x \right )+\left ({\mathrm e}^{x}\right )^{2}}}\right )^{2}}{\mathrm {log}\left (x \right )^{3}+3 \,{\mathrm e}^{x} \mathrm {log}\left (x \right )^{2}+3 \left ({\mathrm e}^{x}\right )^{2} \mathrm {log}\left (x \right )+\left ({\mathrm e}^{x}\right )^{3}}d x \] Input:
int(((8*x^3-12*x^2+4*x)*log(x)^3+((24*x^3-36*x^2+12*x)*exp(x)-24*x^2+16*x) *log(x)^2+((24*x^3-36*x^2+12*x)*exp(x)^2+(8*x^3-56*x^2+32*x)*exp(x)+8*x^2+ 8*x)*log(x)+(8*x^3-12*x^2+4*x)*exp(x)^3+(8*x^3-32*x^2+16*x)*exp(x)^2+(-8*x ^2+8*x)*exp(x)-16*x)*exp(((-exp(3)+x^4-2*x^3+x^2)*log(x)^2+((-2*exp(3)+2*x ^4-4*x^3+2*x^2)*exp(x)-4*x^3+4*x^2)*log(x)+(-exp(3)+x^4-2*x^3+x^2)*exp(x)^ 2+(-4*x^3+4*x^2)*exp(x)+4*x^2)/(log(x)^2+2*exp(x)*log(x)+exp(x)^2))^2/(log (x)^3+3*exp(x)*log(x)^2+3*exp(x)^2*log(x)+exp(x)^3),x)
Output:
int(((8*x^3-12*x^2+4*x)*log(x)^3+((24*x^3-36*x^2+12*x)*exp(x)-24*x^2+16*x) *log(x)^2+((24*x^3-36*x^2+12*x)*exp(x)^2+(8*x^3-56*x^2+32*x)*exp(x)+8*x^2+ 8*x)*log(x)+(8*x^3-12*x^2+4*x)*exp(x)^3+(8*x^3-32*x^2+16*x)*exp(x)^2+(-8*x ^2+8*x)*exp(x)-16*x)*exp(((-exp(3)+x^4-2*x^3+x^2)*log(x)^2+((-2*exp(3)+2*x ^4-4*x^3+2*x^2)*exp(x)-4*x^3+4*x^2)*log(x)+(-exp(3)+x^4-2*x^3+x^2)*exp(x)^ 2+(-4*x^3+4*x^2)*exp(x)+4*x^2)/(log(x)^2+2*exp(x)*log(x)+exp(x)^2))^2/(log (x)^3+3*exp(x)*log(x)^2+3*exp(x)^2*log(x)+exp(x)^3),x)