Integrand size = 52, antiderivative size = 22 \[ \int \frac {1}{2} e^{-2 x-\frac {1}{2} e^{-2 x} \left (1-x+4 e^{2 x} x\right )} \left (e^{2 x} (2-4 x)+3 x-2 x^2\right ) \, dx=e^{-\frac {1}{2} e^{-2 x} (1-x)-2 x} x \] Output:
x/exp(1/4*(1-x)/exp(x)^2+x)^2
Time = 0.56 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{2} e^{-2 x-\frac {1}{2} e^{-2 x} \left (1-x+4 e^{2 x} x\right )} \left (e^{2 x} (2-4 x)+3 x-2 x^2\right ) \, dx=e^{\frac {1}{2} e^{-2 x} \left (-1+x-4 e^{2 x} x\right )} x \] Input:
Integrate[(E^(-2*x - (1 - x + 4*E^(2*x)*x)/(2*E^(2*x)))*(E^(2*x)*(2 - 4*x) + 3*x - 2*x^2))/2,x]
Output:
E^((-1 + x - 4*E^(2*x)*x)/(2*E^(2*x)))*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.
\(\displaystyle \int \frac {1}{2} \left (-2 x^2+3 x+e^{2 x} (2-4 x)\right ) \exp \left (-2 x-\frac {1}{2} e^{-2 x} \left (4 e^{2 x} x-x+1\right )\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \exp \left (-2 x-\frac {1}{2} e^{-2 x} \left (4 e^{2 x} x-x+1\right )\right ) \left (-2 x^2+3 x+2 e^{2 x} (1-2 x)\right )dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \frac {1}{2} \int e^{-\frac {1}{2} e^{-2 x} \left (8 e^{2 x} x-x+1\right )} \left (-2 x^2+3 x+2 e^{2 x} (1-2 x)\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} \int \left (-2 e^{-\frac {1}{2} e^{-2 x} \left (8 e^{2 x} x-x+1\right )} x^2+3 e^{-\frac {1}{2} e^{-2 x} \left (8 e^{2 x} x-x+1\right )} x-2 e^{2 x-\frac {1}{2} e^{-2 x} \left (8 e^{2 x} x-x+1\right )} (2 x-1)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-2 \int e^{-\frac {1}{2} e^{-2 x} \left (8 e^{2 x} x-x+1\right )} x^2dx+2 \int e^{-\frac {1}{2} e^{-2 x} \left (4 e^{2 x} x-x+1\right )}dx-4 \int e^{-\frac {1}{2} e^{-2 x} \left (4 e^{2 x} x-x+1\right )} xdx+3 \int e^{-\frac {1}{2} e^{-2 x} \left (8 e^{2 x} x-x+1\right )} xdx\right )\) |
Input:
Int[(E^(-2*x - (1 - x + 4*E^(2*x)*x)/(2*E^(2*x)))*(E^(2*x)*(2 - 4*x) + 3*x - 2*x^2))/2,x]
Output:
$Aborted
Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
method | result | size |
risch | \(x \,{\mathrm e}^{-\frac {\left (4 x \,{\mathrm e}^{2 x}-x +1\right ) {\mathrm e}^{-2 x}}{2}}\) | \(22\) |
norman | \(x \,{\mathrm e}^{-\frac {\left (4 x \,{\mathrm e}^{2 x}-x +1\right ) {\mathrm e}^{-2 x}}{2}}\) | \(24\) |
parallelrisch | \(x \,{\mathrm e}^{-\frac {\left (4 x \,{\mathrm e}^{2 x}-x +1\right ) {\mathrm e}^{-2 x}}{2}}\) | \(24\) |
Input:
int(1/2*((-4*x+2)*exp(x)^2-2*x^2+3*x)/exp(x)^2/exp(1/4*(4*x*exp(x)^2-x+1)/ exp(x)^2)^2,x,method=_RETURNVERBOSE)
Output:
x*exp(-1/2*(4*x*exp(2*x)-x+1)*exp(-2*x))
Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {1}{2} e^{-2 x-\frac {1}{2} e^{-2 x} \left (1-x+4 e^{2 x} x\right )} \left (e^{2 x} (2-4 x)+3 x-2 x^2\right ) \, dx=x e^{\left (-\frac {1}{2} \, {\left (8 \, x e^{\left (2 \, x\right )} - x + 1\right )} e^{\left (-2 \, x\right )} + 2 \, x\right )} \] Input:
integrate(1/2*((-4*x+2)*exp(x)^2-2*x^2+3*x)/exp(x)^2/exp(1/4*(4*x*exp(x)^2 -x+1)/exp(x)^2)^2,x, algorithm="fricas")
Output:
x*e^(-1/2*(8*x*e^(2*x) - x + 1)*e^(-2*x) + 2*x)
Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2} e^{-2 x-\frac {1}{2} e^{-2 x} \left (1-x+4 e^{2 x} x\right )} \left (e^{2 x} (2-4 x)+3 x-2 x^2\right ) \, dx=x e^{- 2 \left (x e^{2 x} - \frac {x}{4} + \frac {1}{4}\right ) e^{- 2 x}} \] Input:
integrate(1/2*((-4*x+2)*exp(x)**2-2*x**2+3*x)/exp(x)**2/exp(1/4*(4*x*exp(x )**2-x+1)/exp(x)**2)**2,x)
Output:
x*exp(-2*(x*exp(2*x) - x/4 + 1/4)*exp(-2*x))
Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{2} e^{-2 x-\frac {1}{2} e^{-2 x} \left (1-x+4 e^{2 x} x\right )} \left (e^{2 x} (2-4 x)+3 x-2 x^2\right ) \, dx=x e^{\left (\frac {1}{2} \, x e^{\left (-2 \, x\right )} - 2 \, x - \frac {1}{2} \, e^{\left (-2 \, x\right )}\right )} \] Input:
integrate(1/2*((-4*x+2)*exp(x)^2-2*x^2+3*x)/exp(x)^2/exp(1/4*(4*x*exp(x)^2 -x+1)/exp(x)^2)^2,x, algorithm="maxima")
Output:
x*e^(1/2*x*e^(-2*x) - 2*x - 1/2*e^(-2*x))
\[ \int \frac {1}{2} e^{-2 x-\frac {1}{2} e^{-2 x} \left (1-x+4 e^{2 x} x\right )} \left (e^{2 x} (2-4 x)+3 x-2 x^2\right ) \, dx=\int { -\frac {1}{2} \, {\left (2 \, x^{2} + 2 \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} - 3 \, x\right )} e^{\left (-\frac {1}{2} \, {\left (4 \, x e^{\left (2 \, x\right )} - x + 1\right )} e^{\left (-2 \, x\right )} - 2 \, x\right )} \,d x } \] Input:
integrate(1/2*((-4*x+2)*exp(x)^2-2*x^2+3*x)/exp(x)^2/exp(1/4*(4*x*exp(x)^2 -x+1)/exp(x)^2)^2,x, algorithm="giac")
Output:
integrate(-1/2*(2*x^2 + 2*(2*x - 1)*e^(2*x) - 3*x)*e^(-1/2*(4*x*e^(2*x) - x + 1)*e^(-2*x) - 2*x), x)
Time = 2.80 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {1}{2} e^{-2 x-\frac {1}{2} e^{-2 x} \left (1-x+4 e^{2 x} x\right )} \left (e^{2 x} (2-4 x)+3 x-2 x^2\right ) \, dx=x\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{-2\,x}}{2}}\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^{-2\,x}}{2}} \] Input:
int(-exp(-2*x)*exp(-2*exp(-2*x)*(x*exp(2*x) - x/4 + 1/4))*((exp(2*x)*(4*x - 2))/2 - (3*x)/2 + x^2),x)
Output:
x*exp(-exp(-2*x)/2)*exp(-2*x)*exp((x*exp(-2*x))/2)
\[ \int \frac {1}{2} e^{-2 x-\frac {1}{2} e^{-2 x} \left (1-x+4 e^{2 x} x\right )} \left (e^{2 x} (2-4 x)+3 x-2 x^2\right ) \, dx=\int \frac {e^{\frac {x}{2 e^{2 x}}}}{e^{\frac {4 e^{2 x} x +1}{2 e^{2 x}}}}d x -\left (\int \frac {e^{\frac {x}{2 e^{2 x}}} x^{2}}{e^{\frac {8 e^{2 x} x +1}{2 e^{2 x}}}}d x \right )+\frac {3 \left (\int \frac {e^{\frac {x}{2 e^{2 x}}} x}{e^{\frac {8 e^{2 x} x +1}{2 e^{2 x}}}}d x \right )}{2}-2 \left (\int \frac {e^{\frac {x}{2 e^{2 x}}} x}{e^{\frac {4 e^{2 x} x +1}{2 e^{2 x}}}}d x \right ) \] Input:
int(1/2*((-4*x+2)*exp(x)^2-2*x^2+3*x)/exp(x)^2/exp(1/4*(4*x*exp(x)^2-x+1)/ exp(x)^2)^2,x)
Output:
(2*int(e**(x/(2*e**(2*x)))/e**((4*e**(2*x)*x + 1)/(2*e**(2*x))),x) - 2*int ((e**(x/(2*e**(2*x)))*x**2)/e**((8*e**(2*x)*x + 1)/(2*e**(2*x))),x) + 3*in t((e**(x/(2*e**(2*x)))*x)/e**((8*e**(2*x)*x + 1)/(2*e**(2*x))),x) - 4*int( (e**(x/(2*e**(2*x)))*x)/e**((4*e**(2*x)*x + 1)/(2*e**(2*x))),x))/2