Integrand size = 97, antiderivative size = 26 \[ \int \frac {e^{-\frac {1+e^x (2-x)-2 x+x^2}{-2+x}} \left (-6+8 x-2 x^2+e^{\frac {1+e^x (2-x)-2 x+x^2}{-2+x}} \left (-4+4 x-x^2\right )+e^x \left (8-8 x+2 x^2\right )\right )}{4-4 x+x^2} \, dx=2 e^{e^x-\frac {5}{5+5 (-3+x)}-x}-x \]
Time = 0.51 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-\frac {1+e^x (2-x)-2 x+x^2}{-2+x}} \left (-6+8 x-2 x^2+e^{\frac {1+e^x (2-x)-2 x+x^2}{-2+x}} \left (-4+4 x-x^2\right )+e^x \left (8-8 x+2 x^2\right )\right )}{4-4 x+x^2} \, dx=2 e^{e^x-\frac {1}{-2+x}-x}-x \]
Integrate[(-6 + 8*x - 2*x^2 + E^((1 + E^x*(2 - x) - 2*x + x^2)/(-2 + x))*( -4 + 4*x - x^2) + E^x*(8 - 8*x + 2*x^2))/(E^((1 + E^x*(2 - x) - 2*x + x^2) /(-2 + x))*(4 - 4*x + x^2)),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.
\(\displaystyle \int \frac {e^{-\frac {x^2-2 x+e^x (2-x)+1}{x-2}} \left (-2 x^2+e^{\frac {x^2-2 x+e^x (2-x)+1}{x-2}} \left (-x^2+4 x-4\right )+e^x \left (2 x^2-8 x+8\right )+8 x-6\right )}{x^2-4 x+4} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (-\frac {2 e^{e^x-\frac {(x-1)^2}{x-2}} \left (x^2-4 x+3\right )}{(x-2)^2}+2 e^{e^x+\frac {1}{2-x}}-1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int e^{e^x+\frac {1}{2-x}}dx-2 \int e^{e^x-\frac {(x-1)^2}{x-2}}dx+2 \int \frac {e^{e^x-\frac {(x-1)^2}{x-2}}}{(x-2)^2}dx-x\) |
Int[(-6 + 8*x - 2*x^2 + E^((1 + E^x*(2 - x) - 2*x + x^2)/(-2 + x))*(-4 + 4 *x - x^2) + E^x*(8 - 8*x + 2*x^2))/(E^((1 + E^x*(2 - x) - 2*x + x^2)/(-2 + x))*(4 - 4*x + x^2)),x]
3.7.35.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23
method | result | size |
risch | \(-x +2 \,{\mathrm e}^{\frac {{\mathrm e}^{x} x -x^{2}-2 \,{\mathrm e}^{x}+2 x -1}{-2+x}}\) | \(32\) |
norman | \(\frac {\left (-4+4 \,{\mathrm e}^{\frac {\left (2-x \right ) {\mathrm e}^{x}+x^{2}-2 x +1}{-2+x}}+2 x -x^{2} {\mathrm e}^{\frac {\left (2-x \right ) {\mathrm e}^{x}+x^{2}-2 x +1}{-2+x}}\right ) {\mathrm e}^{-\frac {\left (2-x \right ) {\mathrm e}^{x}+x^{2}-2 x +1}{-2+x}}}{-2+x}\) | \(90\) |
parallelrisch | \(\frac {\left (-4-x^{2} {\mathrm e}^{\frac {\left (2-x \right ) {\mathrm e}^{x}+x^{2}-2 x +1}{-2+x}}-6 \,{\mathrm e}^{\frac {\left (2-x \right ) {\mathrm e}^{x}+x^{2}-2 x +1}{-2+x}} x +2 x +16 \,{\mathrm e}^{\frac {\left (2-x \right ) {\mathrm e}^{x}+x^{2}-2 x +1}{-2+x}}\right ) {\mathrm e}^{-\frac {\left (2-x \right ) {\mathrm e}^{x}+x^{2}-2 x +1}{-2+x}}}{-2+x}\) | \(116\) |
int(((-x^2+4*x-4)*exp(((2-x)*exp(x)+x^2-2*x+1)/(-2+x))+(2*x^2-8*x+8)*exp(x )-2*x^2+8*x-6)/(x^2-4*x+4)/exp(((2-x)*exp(x)+x^2-2*x+1)/(-2+x)),x,method=_ RETURNVERBOSE)
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-\frac {1+e^x (2-x)-2 x+x^2}{-2+x}} \left (-6+8 x-2 x^2+e^{\frac {1+e^x (2-x)-2 x+x^2}{-2+x}} \left (-4+4 x-x^2\right )+e^x \left (8-8 x+2 x^2\right )\right )}{4-4 x+x^2} \, dx=-x + 2 \, e^{\left (-\frac {x^{2} - {\left (x - 2\right )} e^{x} - 2 \, x + 1}{x - 2}\right )} \]
integrate(((-x^2+4*x-4)*exp(((2-x)*exp(x)+x^2-2*x+1)/(-2+x))+(2*x^2-8*x+8) *exp(x)-2*x^2+8*x-6)/(x^2-4*x+4)/exp(((2-x)*exp(x)+x^2-2*x+1)/(-2+x)),x, a lgorithm=\
Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-\frac {1+e^x (2-x)-2 x+x^2}{-2+x}} \left (-6+8 x-2 x^2+e^{\frac {1+e^x (2-x)-2 x+x^2}{-2+x}} \left (-4+4 x-x^2\right )+e^x \left (8-8 x+2 x^2\right )\right )}{4-4 x+x^2} \, dx=- x + 2 e^{- \frac {x^{2} - 2 x + \left (2 - x\right ) e^{x} + 1}{x - 2}} \]
integrate(((-x**2+4*x-4)*exp(((2-x)*exp(x)+x**2-2*x+1)/(-2+x))+(2*x**2-8*x +8)*exp(x)-2*x**2+8*x-6)/(x**2-4*x+4)/exp(((2-x)*exp(x)+x**2-2*x+1)/(-2+x) ),x)
Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-\frac {1+e^x (2-x)-2 x+x^2}{-2+x}} \left (-6+8 x-2 x^2+e^{\frac {1+e^x (2-x)-2 x+x^2}{-2+x}} \left (-4+4 x-x^2\right )+e^x \left (8-8 x+2 x^2\right )\right )}{4-4 x+x^2} \, dx=-x + 2 \, e^{\left (-x - \frac {1}{x - 2} + e^{x}\right )} \]
integrate(((-x^2+4*x-4)*exp(((2-x)*exp(x)+x^2-2*x+1)/(-2+x))+(2*x^2-8*x+8) *exp(x)-2*x^2+8*x-6)/(x^2-4*x+4)/exp(((2-x)*exp(x)+x^2-2*x+1)/(-2+x)),x, a lgorithm=\
\[ \int \frac {e^{-\frac {1+e^x (2-x)-2 x+x^2}{-2+x}} \left (-6+8 x-2 x^2+e^{\frac {1+e^x (2-x)-2 x+x^2}{-2+x}} \left (-4+4 x-x^2\right )+e^x \left (8-8 x+2 x^2\right )\right )}{4-4 x+x^2} \, dx=\int { -\frac {{\left (2 \, x^{2} - 2 \, {\left (x^{2} - 4 \, x + 4\right )} e^{x} + {\left (x^{2} - 4 \, x + 4\right )} e^{\left (\frac {x^{2} - {\left (x - 2\right )} e^{x} - 2 \, x + 1}{x - 2}\right )} - 8 \, x + 6\right )} e^{\left (-\frac {x^{2} - {\left (x - 2\right )} e^{x} - 2 \, x + 1}{x - 2}\right )}}{x^{2} - 4 \, x + 4} \,d x } \]
integrate(((-x^2+4*x-4)*exp(((2-x)*exp(x)+x^2-2*x+1)/(-2+x))+(2*x^2-8*x+8) *exp(x)-2*x^2+8*x-6)/(x^2-4*x+4)/exp(((2-x)*exp(x)+x^2-2*x+1)/(-2+x)),x, a lgorithm=\
integrate(-(2*x^2 - 2*(x^2 - 4*x + 4)*e^x + (x^2 - 4*x + 4)*e^((x^2 - (x - 2)*e^x - 2*x + 1)/(x - 2)) - 8*x + 6)*e^(-(x^2 - (x - 2)*e^x - 2*x + 1)/( x - 2))/(x^2 - 4*x + 4), x)
Time = 8.90 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {e^{-\frac {1+e^x (2-x)-2 x+x^2}{-2+x}} \left (-6+8 x-2 x^2+e^{\frac {1+e^x (2-x)-2 x+x^2}{-2+x}} \left (-4+4 x-x^2\right )+e^x \left (8-8 x+2 x^2\right )\right )}{4-4 x+x^2} \, dx=2\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^x}{x-2}}\,{\mathrm {e}}^{\frac {2\,x}{x-2}}\,{\mathrm {e}}^{-\frac {x^2}{x-2}}\,{\mathrm {e}}^{-\frac {2\,{\mathrm {e}}^x}{x-2}}\,{\mathrm {e}}^{-\frac {1}{x-2}}-x \]
int(-(exp((2*x + exp(x)*(x - 2) - x^2 - 1)/(x - 2))*(2*x^2 - exp(x)*(2*x^2 - 8*x + 8) - 8*x + exp(-(2*x + exp(x)*(x - 2) - x^2 - 1)/(x - 2))*(x^2 - 4*x + 4) + 6))/(x^2 - 4*x + 4),x)