Integrand size = 211, antiderivative size = 29 \begin {dmath*} \int \frac {4 x^3-4 x^4+x^5+\left (8 x^2-8 x^3+2 x^4\right ) \log (x)+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3+\left (8-8 x+2 x^2\right ) \log (x)\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (20-15 x-12 x^2+9 x^3-2 x^4+\left (-2-14 x+16 x^2-4 x^3\right ) \log (x)\right )}{4 x^3-4 x^4+x^5+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (-8 x^2+8 x^3-2 x^4\right )} \, dx=x+\frac {x}{e^{\frac {-9+\log (x)}{(-2+x) x}}-x}+\log ^2(x) \end {dmath*}
\begin {dmath*} \int \frac {4 x^3-4 x^4+x^5+\left (8 x^2-8 x^3+2 x^4\right ) \log (x)+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3+\left (8-8 x+2 x^2\right ) \log (x)\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (20-15 x-12 x^2+9 x^3-2 x^4+\left (-2-14 x+16 x^2-4 x^3\right ) \log (x)\right )}{4 x^3-4 x^4+x^5+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (-8 x^2+8 x^3-2 x^4\right )} \, dx=\int \frac {4 x^3-4 x^4+x^5+\left (8 x^2-8 x^3+2 x^4\right ) \log (x)+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3+\left (8-8 x+2 x^2\right ) \log (x)\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (20-15 x-12 x^2+9 x^3-2 x^4+\left (-2-14 x+16 x^2-4 x^3\right ) \log (x)\right )}{4 x^3-4 x^4+x^5+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (-8 x^2+8 x^3-2 x^4\right )} \, dx \end {dmath*}
Integrate[(4*x^3 - 4*x^4 + x^5 + (8*x^2 - 8*x^3 + 2*x^4)*Log[x] + E^((2*(- 9 + Log[x]))/(-2*x + x^2))*(4*x - 4*x^2 + x^3 + (8 - 8*x + 2*x^2)*Log[x]) + E^((-9 + Log[x])/(-2*x + x^2))*(20 - 15*x - 12*x^2 + 9*x^3 - 2*x^4 + (-2 - 14*x + 16*x^2 - 4*x^3)*Log[x]))/(4*x^3 - 4*x^4 + x^5 + E^((2*(-9 + Log[ x]))/(-2*x + x^2))*(4*x - 4*x^2 + x^3) + E^((-9 + Log[x])/(-2*x + x^2))*(- 8*x^2 + 8*x^3 - 2*x^4)),x]
Integrate[(4*x^3 - 4*x^4 + x^5 + (8*x^2 - 8*x^3 + 2*x^4)*Log[x] + E^((2*(- 9 + Log[x]))/(-2*x + x^2))*(4*x - 4*x^2 + x^3 + (8 - 8*x + 2*x^2)*Log[x]) + E^((-9 + Log[x])/(-2*x + x^2))*(20 - 15*x - 12*x^2 + 9*x^3 - 2*x^4 + (-2 - 14*x + 16*x^2 - 4*x^3)*Log[x]))/(4*x^3 - 4*x^4 + x^5 + E^((2*(-9 + Log[ x]))/(-2*x + x^2))*(4*x - 4*x^2 + x^3) + E^((-9 + Log[x])/(-2*x + x^2))*(- 8*x^2 + 8*x^3 - 2*x^4)), 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.
| \(\displaystyle \int \frac {x^5-4 x^4+4 x^3+e^{\frac {2 (\log (x)-9)}{x^2-2 x}} \left (x^3-4 x^2+\left (2 x^2-8 x+8\right ) \log (x)+4 x\right )+\left (2 x^4-8 x^3+8 x^2\right ) \log (x)+e^{\frac {\log (x)-9}{x^2-2 x}} \left (-2 x^4+9 x^3-12 x^2+\left (-4 x^3+16 x^2-14 x-2\right ) \log (x)-15 x+20\right )}{x^5-4 x^4+4 x^3+\left (x^3-4 x^2+4 x\right ) e^{\frac {2 (\log (x)-9)}{x^2-2 x}}+\left (-2 x^4+8 x^3-8 x^2\right ) e^{\frac {\log (x)-9}{x^2-2 x}}} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{\frac {18}{(x-2) x}} \left (x^5-4 x^4+4 x^3+e^{\frac {2 (\log (x)-9)}{x^2-2 x}} \left (x^3-4 x^2+\left (2 x^2-8 x+8\right ) \log (x)+4 x\right )+\left (2 x^4-8 x^3+8 x^2\right ) \log (x)+e^{\frac {\log (x)-9}{x^2-2 x}} \left (-2 x^4+9 x^3-12 x^2+\left (-4 x^3+16 x^2-14 x-2\right ) \log (x)-15 x+20\right )\right )}{(2-x)^2 x \left (e^{\frac {9}{(x-2) x}} x-x^{\frac {1}{x^2-2 x}}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {-8 \log (x) x^{\frac {2}{x^2-2 x}}+15 e^{\frac {9}{(x-2) x}} x^{1+\frac {1}{x^2-2 x}}+14 e^{\frac {9}{(x-2) x}} \log (x) x^{1+\frac {1}{x^2-2 x}}+12 e^{\frac {9}{(x-2) x}} x^{2+\frac {1}{x^2-2 x}}-16 e^{\frac {9}{(x-2) x}} \log (x) x^{2+\frac {1}{x^2-2 x}}-9 e^{\frac {9}{(x-2) x}} x^{3+\frac {1}{x^2-2 x}}+4 e^{\frac {9}{(x-2) x}} \log (x) x^{3+\frac {1}{x^2-2 x}}+2 e^{\frac {9}{(x-2) x}} x^{4+\frac {1}{x^2-2 x}}+8 \log (x) x^{1+\frac {2}{x^2-2 x}}-4 x^{1+\frac {2}{x^2-2 x}}-2 \log (x) x^{2+\frac {2}{x^2-2 x}}+4 x^{2+\frac {2}{x^2-2 x}}-x^{3+\frac {2}{x^2-2 x}}-20 e^{\frac {9}{(x-2) x}} x^{\frac {1}{x^2-2 x}}+2 e^{\frac {9}{(x-2) x}} \log (x) x^{\frac {1}{x^2-2 x}}-e^{\frac {18}{(x-2) x}} x^5+4 e^{\frac {18}{(x-2) x}} x^4-2 e^{\frac {18}{(x-2) x}} \log (x) x^4-4 e^{\frac {18}{(x-2) x}} x^3+8 e^{\frac {18}{(x-2) x}} \log (x) x^3-8 e^{\frac {18}{(x-2) x}} \log (x) x^2}{4 (x-2) \left (e^{\frac {9}{(x-2) x}} x-x^{\frac {1}{x^2-2 x}}\right )^2}+\frac {8 \log (x) x^{\frac {2}{x^2-2 x}}-15 e^{\frac {9}{(x-2) x}} x^{1+\frac {1}{x^2-2 x}}-14 e^{\frac {9}{(x-2) x}} \log (x) x^{1+\frac {1}{x^2-2 x}}-12 e^{\frac {9}{(x-2) x}} x^{2+\frac {1}{x^2-2 x}}+16 e^{\frac {9}{(x-2) x}} \log (x) x^{2+\frac {1}{x^2-2 x}}+9 e^{\frac {9}{(x-2) x}} x^{3+\frac {1}{x^2-2 x}}-4 e^{\frac {9}{(x-2) x}} \log (x) x^{3+\frac {1}{x^2-2 x}}-2 e^{\frac {9}{(x-2) x}} x^{4+\frac {1}{x^2-2 x}}-8 \log (x) x^{1+\frac {2}{x^2-2 x}}+4 x^{1+\frac {2}{x^2-2 x}}+2 \log (x) x^{2+\frac {2}{x^2-2 x}}-4 x^{2+\frac {2}{x^2-2 x}}+x^{3+\frac {2}{x^2-2 x}}+20 e^{\frac {9}{(x-2) x}} x^{\frac {1}{x^2-2 x}}-2 e^{\frac {9}{(x-2) x}} \log (x) x^{\frac {1}{x^2-2 x}}+e^{\frac {18}{(x-2) x}} x^5-4 e^{\frac {18}{(x-2) x}} x^4+2 e^{\frac {18}{(x-2) x}} \log (x) x^4+4 e^{\frac {18}{(x-2) x}} x^3-8 e^{\frac {18}{(x-2) x}} \log (x) x^3+8 e^{\frac {18}{(x-2) x}} \log (x) x^2}{4 x \left (e^{\frac {9}{(x-2) x}} x-x^{\frac {1}{x^2-2 x}}\right )^2}+\frac {8 \log (x) x^{\frac {2}{x^2-2 x}}-15 e^{\frac {9}{(x-2) x}} x^{1+\frac {1}{x^2-2 x}}-14 e^{\frac {9}{(x-2) x}} \log (x) x^{1+\frac {1}{x^2-2 x}}-12 e^{\frac {9}{(x-2) x}} x^{2+\frac {1}{x^2-2 x}}+16 e^{\frac {9}{(x-2) x}} \log (x) x^{2+\frac {1}{x^2-2 x}}+9 e^{\frac {9}{(x-2) x}} x^{3+\frac {1}{x^2-2 x}}-4 e^{\frac {9}{(x-2) x}} \log (x) x^{3+\frac {1}{x^2-2 x}}-2 e^{\frac {9}{(x-2) x}} x^{4+\frac {1}{x^2-2 x}}-8 \log (x) x^{1+\frac {2}{x^2-2 x}}+4 x^{1+\frac {2}{x^2-2 x}}+2 \log (x) x^{2+\frac {2}{x^2-2 x}}-4 x^{2+\frac {2}{x^2-2 x}}+x^{3+\frac {2}{x^2-2 x}}+20 e^{\frac {9}{(x-2) x}} x^{\frac {1}{x^2-2 x}}-2 e^{\frac {9}{(x-2) x}} \log (x) x^{\frac {1}{x^2-2 x}}+e^{\frac {18}{(x-2) x}} x^5-4 e^{\frac {18}{(x-2) x}} x^4+2 e^{\frac {18}{(x-2) x}} \log (x) x^4+4 e^{\frac {18}{(x-2) x}} x^3-8 e^{\frac {18}{(x-2) x}} \log (x) x^3+8 e^{\frac {18}{(x-2) x}} \log (x) x^2}{2 (x-2)^2 \left (e^{\frac {9}{(x-2) x}} x-x^{\frac {1}{x^2-2 x}}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x-2)^2 x^{\frac {2}{(x-2) x}+1}+e^{\frac {18}{(x-2) x}} (x-2)^2 x^3+2 \left ((x-2)^2 x^{\frac {2}{(x-2) x}}+e^{\frac {18}{(x-2) x}} (x-2)^2 x^2-e^{\frac {9}{(x-2) x}} \left (2 x^3-8 x^2+7 x+1\right ) x^{\frac {1}{x^2-2 x}}\right ) \log (x)+e^{\frac {9}{(x-2) x}} \left (-2 x^4+9 x^3-12 x^2-15 x+20\right ) x^{\frac {1}{x^2-2 x}}}{(2-x)^2 x \left (e^{\frac {9}{(x-2) x}} x-x^{\frac {1}{x^2-2 x}}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {e^{\frac {9}{(x-2) x}} \left (x^3-4 x^2-15 x+2 x \log (x)-2 \log (x)+20\right )}{(x-2)^2 x \left (e^{\frac {9}{(x-2) x}} x-x^{\frac {1}{x^2-2 x}}\right )}+\frac {e^{\frac {18}{(x-2) x}} \left (x^3-4 x^2-15 x+2 x \log (x)-2 \log (x)+20\right )}{(x-2)^2 \left (e^{\frac {9}{(x-2) x}} x-x^{\frac {1}{x^2-2 x}}\right )^2}+\frac {x+2 \log (x)}{x}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (-\frac {e^{\frac {9}{(x-2) x}} \left (x^3-4 x^2-15 x+2 x \log (x)-2 \log (x)+20\right )}{(x-2)^2 x \left (e^{\frac {9}{(x-2) x}} x-x^{\frac {1}{x^2-2 x}}\right )}+\frac {e^{\frac {18}{(x-2) x}} \left (x^3-4 x^2-15 x+2 x \log (x)-2 \log (x)+20\right )}{(x-2)^2 \left (e^{\frac {9}{(x-2) x}} x-x^{\frac {1}{x^2-2 x}}\right )^2}+\frac {x+2 \log (x)}{x}\right )dx\) |
Int[(4*x^3 - 4*x^4 + x^5 + (8*x^2 - 8*x^3 + 2*x^4)*Log[x] + E^((2*(-9 + Lo g[x]))/(-2*x + x^2))*(4*x - 4*x^2 + x^3 + (8 - 8*x + 2*x^2)*Log[x]) + E^(( -9 + Log[x])/(-2*x + x^2))*(20 - 15*x - 12*x^2 + 9*x^3 - 2*x^4 + (-2 - 14* x + 16*x^2 - 4*x^3)*Log[x]))/(4*x^3 - 4*x^4 + x^5 + E^((2*(-9 + Log[x]))/( -2*x + x^2))*(4*x - 4*x^2 + x^3) + E^((-9 + Log[x])/(-2*x + x^2))*(-8*x^2 + 8*x^3 - 2*x^4)),x]
3.8.31.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.35 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03
| method | result | size |
| risch | \(\ln \left (x \right )^{2}+x -\frac {x}{x -{\mathrm e}^{\frac {\ln \left (x \right )-9}{\left (-2+x \right ) x}}}\) | \(30\) |
| parallelrisch | \(\frac {x \ln \left (x \right )^{2}-\ln \left (x \right )^{2} {\mathrm e}^{\frac {\ln \left (x \right )-9}{\left (-2+x \right ) x}}+x^{2}-{\mathrm e}^{\frac {\ln \left (x \right )-9}{\left (-2+x \right ) x}} x +7 x -8 \,{\mathrm e}^{\frac {\ln \left (x \right )-9}{\left (-2+x \right ) x}}}{x -{\mathrm e}^{\frac {\ln \left (x \right )-9}{\left (-2+x \right ) x}}}\) | \(88\) |
int((((2*x^2-8*x+8)*ln(x)+x^3-4*x^2+4*x)*exp((ln(x)-9)/(x^2-2*x))^2+((-4*x ^3+16*x^2-14*x-2)*ln(x)-2*x^4+9*x^3-12*x^2-15*x+20)*exp((ln(x)-9)/(x^2-2*x ))+(2*x^4-8*x^3+8*x^2)*ln(x)+x^5-4*x^4+4*x^3)/((x^3-4*x^2+4*x)*exp((ln(x)- 9)/(x^2-2*x))^2+(-2*x^4+8*x^3-8*x^2)*exp((ln(x)-9)/(x^2-2*x))+x^5-4*x^4+4* x^3),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.00 \begin {dmath*} \int \frac {4 x^3-4 x^4+x^5+\left (8 x^2-8 x^3+2 x^4\right ) \log (x)+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3+\left (8-8 x+2 x^2\right ) \log (x)\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (20-15 x-12 x^2+9 x^3-2 x^4+\left (-2-14 x+16 x^2-4 x^3\right ) \log (x)\right )}{4 x^3-4 x^4+x^5+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (-8 x^2+8 x^3-2 x^4\right )} \, dx=\frac {x \log \left (x\right )^{2} + x^{2} - {\left (\log \left (x\right )^{2} + x\right )} e^{\left (\frac {\log \left (x\right ) - 9}{x^{2} - 2 \, x}\right )} - x}{x - e^{\left (\frac {\log \left (x\right ) - 9}{x^{2} - 2 \, x}\right )}} \end {dmath*}
integrate((((2*x^2-8*x+8)*log(x)+x^3-4*x^2+4*x)*exp((log(x)-9)/(x^2-2*x))^ 2+((-4*x^3+16*x^2-14*x-2)*log(x)-2*x^4+9*x^3-12*x^2-15*x+20)*exp((log(x)-9 )/(x^2-2*x))+(2*x^4-8*x^3+8*x^2)*log(x)+x^5-4*x^4+4*x^3)/((x^3-4*x^2+4*x)* exp((log(x)-9)/(x^2-2*x))^2+(-2*x^4+8*x^3-8*x^2)*exp((log(x)-9)/(x^2-2*x)) +x^5-4*x^4+4*x^3),x, algorithm=\
(x*log(x)^2 + x^2 - (log(x)^2 + x)*e^((log(x) - 9)/(x^2 - 2*x)) - x)/(x - e^((log(x) - 9)/(x^2 - 2*x)))
Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \begin {dmath*} \int \frac {4 x^3-4 x^4+x^5+\left (8 x^2-8 x^3+2 x^4\right ) \log (x)+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3+\left (8-8 x+2 x^2\right ) \log (x)\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (20-15 x-12 x^2+9 x^3-2 x^4+\left (-2-14 x+16 x^2-4 x^3\right ) \log (x)\right )}{4 x^3-4 x^4+x^5+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (-8 x^2+8 x^3-2 x^4\right )} \, dx=x + \frac {x}{- x + e^{\frac {\log {\left (x \right )} - 9}{x^{2} - 2 x}}} + \log {\left (x \right )}^{2} \end {dmath*}
integrate((((2*x**2-8*x+8)*ln(x)+x**3-4*x**2+4*x)*exp((ln(x)-9)/(x**2-2*x) )**2+((-4*x**3+16*x**2-14*x-2)*ln(x)-2*x**4+9*x**3-12*x**2-15*x+20)*exp((l n(x)-9)/(x**2-2*x))+(2*x**4-8*x**3+8*x**2)*ln(x)+x**5-4*x**4+4*x**3)/((x** 3-4*x**2+4*x)*exp((ln(x)-9)/(x**2-2*x))**2+(-2*x**4+8*x**3-8*x**2)*exp((ln (x)-9)/(x**2-2*x))+x**5-4*x**4+4*x**3),x)
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (29) = 58\).
Time = 0.33 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.31 \begin {dmath*} \int \frac {4 x^3-4 x^4+x^5+\left (8 x^2-8 x^3+2 x^4\right ) \log (x)+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3+\left (8-8 x+2 x^2\right ) \log (x)\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (20-15 x-12 x^2+9 x^3-2 x^4+\left (-2-14 x+16 x^2-4 x^3\right ) \log (x)\right )}{4 x^3-4 x^4+x^5+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (-8 x^2+8 x^3-2 x^4\right )} \, dx=-\frac {{\left (\log \left (x\right )^{2} + x\right )} e^{\left (\frac {\log \left (x\right )}{2 \, {\left (x - 2\right )}} + \frac {9}{2 \, x}\right )} - {\left (x \log \left (x\right )^{2} + x^{2} - x\right )} e^{\left (\frac {\log \left (x\right )}{2 \, x} + \frac {9}{2 \, {\left (x - 2\right )}}\right )}}{x e^{\left (\frac {\log \left (x\right )}{2 \, x} + \frac {9}{2 \, {\left (x - 2\right )}}\right )} - e^{\left (\frac {\log \left (x\right )}{2 \, {\left (x - 2\right )}} + \frac {9}{2 \, x}\right )}} \end {dmath*}
integrate((((2*x^2-8*x+8)*log(x)+x^3-4*x^2+4*x)*exp((log(x)-9)/(x^2-2*x))^ 2+((-4*x^3+16*x^2-14*x-2)*log(x)-2*x^4+9*x^3-12*x^2-15*x+20)*exp((log(x)-9 )/(x^2-2*x))+(2*x^4-8*x^3+8*x^2)*log(x)+x^5-4*x^4+4*x^3)/((x^3-4*x^2+4*x)* exp((log(x)-9)/(x^2-2*x))^2+(-2*x^4+8*x^3-8*x^2)*exp((log(x)-9)/(x^2-2*x)) +x^5-4*x^4+4*x^3),x, algorithm=\
-((log(x)^2 + x)*e^(1/2*log(x)/(x - 2) + 9/2/x) - (x*log(x)^2 + x^2 - x)*e ^(1/2*log(x)/x + 9/2/(x - 2)))/(x*e^(1/2*log(x)/x + 9/2/(x - 2)) - e^(1/2* log(x)/(x - 2) + 9/2/x))
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (29) = 58\).
Time = 0.67 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.55 \begin {dmath*} \int \frac {4 x^3-4 x^4+x^5+\left (8 x^2-8 x^3+2 x^4\right ) \log (x)+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3+\left (8-8 x+2 x^2\right ) \log (x)\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (20-15 x-12 x^2+9 x^3-2 x^4+\left (-2-14 x+16 x^2-4 x^3\right ) \log (x)\right )}{4 x^3-4 x^4+x^5+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (-8 x^2+8 x^3-2 x^4\right )} \, dx=\frac {x \log \left (x\right )^{2} - e^{\left (\frac {\log \left (x\right ) - 9}{x^{2} - 2 \, x}\right )} \log \left (x\right )^{2} + x^{2} - x e^{\left (\frac {\log \left (x\right ) - 9}{x^{2} - 2 \, x}\right )} - x}{x - e^{\left (\frac {\log \left (x\right ) - 9}{x^{2} - 2 \, x}\right )}} \end {dmath*}
integrate((((2*x^2-8*x+8)*log(x)+x^3-4*x^2+4*x)*exp((log(x)-9)/(x^2-2*x))^ 2+((-4*x^3+16*x^2-14*x-2)*log(x)-2*x^4+9*x^3-12*x^2-15*x+20)*exp((log(x)-9 )/(x^2-2*x))+(2*x^4-8*x^3+8*x^2)*log(x)+x^5-4*x^4+4*x^3)/((x^3-4*x^2+4*x)* exp((log(x)-9)/(x^2-2*x))^2+(-2*x^4+8*x^3-8*x^2)*exp((log(x)-9)/(x^2-2*x)) +x^5-4*x^4+4*x^3),x, algorithm=\
(x*log(x)^2 - e^((log(x) - 9)/(x^2 - 2*x))*log(x)^2 + x^2 - x*e^((log(x) - 9)/(x^2 - 2*x)) - x)/(x - e^((log(x) - 9)/(x^2 - 2*x)))
Time = 16.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 4.24 \begin {dmath*} \int \frac {4 x^3-4 x^4+x^5+\left (8 x^2-8 x^3+2 x^4\right ) \log (x)+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3+\left (8-8 x+2 x^2\right ) \log (x)\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (20-15 x-12 x^2+9 x^3-2 x^4+\left (-2-14 x+16 x^2-4 x^3\right ) \log (x)\right )}{4 x^3-4 x^4+x^5+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (-8 x^2+8 x^3-2 x^4\right )} \, dx=\frac {{\mathrm {e}}^{\frac {9}{2\,x-x^2}}\,{\ln \left (x\right )}^2+x\,{\mathrm {e}}^{\frac {9}{2\,x-x^2}}+x\,x^{\frac {1}{2\,x-x^2}}-x^{\frac {1}{2\,x-x^2}}\,x^2-x\,x^{\frac {1}{2\,x-x^2}}\,{\ln \left (x\right )}^2}{{\mathrm {e}}^{\frac {9}{2\,x-x^2}}-x\,x^{\frac {1}{2\,x-x^2}}} \end {dmath*}
int((exp(-(2*(log(x) - 9))/(2*x - x^2))*(4*x + log(x)*(2*x^2 - 8*x + 8) - 4*x^2 + x^3) + log(x)*(8*x^2 - 8*x^3 + 2*x^4) + 4*x^3 - 4*x^4 + x^5 - exp( -(log(x) - 9)/(2*x - x^2))*(15*x + 12*x^2 - 9*x^3 + 2*x^4 + log(x)*(14*x - 16*x^2 + 4*x^3 + 2) - 20))/(exp(-(2*(log(x) - 9))/(2*x - x^2))*(4*x - 4*x ^2 + x^3) - exp(-(log(x) - 9)/(2*x - x^2))*(8*x^2 - 8*x^3 + 2*x^4) + 4*x^3 - 4*x^4 + x^5),x)