Integrand size = 76, antiderivative size = 29 \[ \int \left (-1+e^{e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2} \left (-4+e^{6 x} (-2-12 x)+8 x+e^x (2+2 x)\right )\right ) \, dx=2 e^{e^{x^2+x \left (-2+e^x-e^{6 x}+x\right )}}-x \] Output:
2*exp(exp(x^2+(-2+exp(x)-exp(6*x)+x)*x))-x
Time = 5.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \left (-1+e^{e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2} \left (-4+e^{6 x} (-2-12 x)+8 x+e^x (2+2 x)\right )\right ) \, dx=2 e^{e^{-2 x+e^x x-e^{6 x} x+2 x^2}}-x \] Input:
Integrate[-1 + E^(E^(-2*x + E^x*x - E^(6*x)*x + 2*x^2) - 2*x + E^x*x - E^( 6*x)*x + 2*x^2)*(-4 + E^(6*x)*(-2 - 12*x) + 8*x + E^x*(2 + 2*x)),x]
Output:
2*E^E^(-2*x + E^x*x - E^(6*x)*x + 2*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.
\(\displaystyle \int \left (\left (e^{6 x} (-12 x-2)+8 x+e^x (2 x+2)-4\right ) \exp \left (2 x^2+e^{2 x^2+e^x x-e^{6 x} x-2 x}+e^x x-e^{6 x} x-2 x\right )-1\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 \int \exp \left (2 x^2+e^x x-e^{6 x} x-2 x+e^{2 x^2+e^x x-e^{6 x} x-2 x}\right )dx+2 \int \exp \left (2 x^2+e^x x-e^{6 x} x-x+e^{2 x^2+e^x x-e^{6 x} x-2 x}\right )dx-2 \int \exp \left (2 x^2+e^x x-e^{6 x} x+4 x+e^{2 x^2+e^x x-e^{6 x} x-2 x}\right )dx+8 \int \exp \left (2 x^2+e^x x-e^{6 x} x-2 x+e^{2 x^2+e^x x-e^{6 x} x-2 x}\right ) xdx+2 \int \exp \left (2 x^2+e^x x-e^{6 x} x-x+e^{2 x^2+e^x x-e^{6 x} x-2 x}\right ) xdx-12 \int \exp \left (2 x^2+e^x x-e^{6 x} x+4 x+e^{2 x^2+e^x x-e^{6 x} x-2 x}\right ) xdx-x\) |
Input:
Int[-1 + E^(E^(-2*x + E^x*x - E^(6*x)*x + 2*x^2) - 2*x + E^x*x - E^(6*x)*x + 2*x^2)*(-4 + E^(6*x)*(-2 - 12*x) + 8*x + E^x*(2 + 2*x)),x]
Output:
$Aborted
Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83
method | result | size |
risch | \(2 \,{\mathrm e}^{{\mathrm e}^{x \left ({\mathrm e}^{x}-{\mathrm e}^{6 x}+2 x -2\right )}}-x\) | \(24\) |
parallelrisch | \(2 \,{\mathrm e}^{{\mathrm e}^{x \left ({\mathrm e}^{x}-{\mathrm e}^{6 x}+2 x -2\right )}}-x\) | \(24\) |
Input:
int(((-12*x-2)*exp(6*x)+(2+2*x)*exp(x)+8*x-4)*exp(-x*exp(6*x)+exp(x)*x+2*x ^2-2*x)*exp(exp(-x*exp(6*x)+exp(x)*x+2*x^2-2*x))-1,x,method=_RETURNVERBOSE )
Output:
2*exp(exp(x*(exp(x)-exp(6*x)+2*x-2)))-x
Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (25) = 50\).
Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.14 \[ \int \left (-1+e^{e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2} \left (-4+e^{6 x} (-2-12 x)+8 x+e^x (2+2 x)\right )\right ) \, dx=-{\left (x e^{\left (2 \, x^{2} - x e^{\left (6 \, x\right )} + x e^{x} - 2 \, x\right )} - 2 \, e^{\left (2 \, x^{2} - x e^{\left (6 \, x\right )} + x e^{x} - 2 \, x + e^{\left (2 \, x^{2} - x e^{\left (6 \, x\right )} + x e^{x} - 2 \, x\right )}\right )}\right )} e^{\left (-2 \, x^{2} + x e^{\left (6 \, x\right )} - x e^{x} + 2 \, x\right )} \] Input:
integrate(((-12*x-2)*exp(6*x)+(2+2*x)*exp(x)+8*x-4)*exp(-x*exp(6*x)+exp(x) *x+2*x^2-2*x)*exp(exp(-x*exp(6*x)+exp(x)*x+2*x^2-2*x))-1,x, algorithm="fri cas")
Output:
-(x*e^(2*x^2 - x*e^(6*x) + x*e^x - 2*x) - 2*e^(2*x^2 - x*e^(6*x) + x*e^x - 2*x + e^(2*x^2 - x*e^(6*x) + x*e^x - 2*x)))*e^(-2*x^2 + x*e^(6*x) - x*e^x + 2*x)
Time = 0.63 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \left (-1+e^{e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2} \left (-4+e^{6 x} (-2-12 x)+8 x+e^x (2+2 x)\right )\right ) \, dx=- x + 2 e^{e^{2 x^{2} - x e^{6 x} + x e^{x} - 2 x}} \] Input:
integrate(((-12*x-2)*exp(6*x)+(2+2*x)*exp(x)+8*x-4)*exp(-x*exp(6*x)+exp(x) *x+2*x**2-2*x)*exp(exp(-x*exp(6*x)+exp(x)*x+2*x**2-2*x))-1,x)
Output:
-x + 2*exp(exp(2*x**2 - x*exp(6*x) + x*exp(x) - 2*x))
Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \left (-1+e^{e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2} \left (-4+e^{6 x} (-2-12 x)+8 x+e^x (2+2 x)\right )\right ) \, dx=-x + 2 \, e^{\left (e^{\left (2 \, x^{2} - x e^{\left (6 \, x\right )} + x e^{x} - 2 \, x\right )}\right )} \] Input:
integrate(((-12*x-2)*exp(6*x)+(2+2*x)*exp(x)+8*x-4)*exp(-x*exp(6*x)+exp(x) *x+2*x^2-2*x)*exp(exp(-x*exp(6*x)+exp(x)*x+2*x^2-2*x))-1,x, algorithm="max ima")
Output:
-x + 2*e^(e^(2*x^2 - x*e^(6*x) + x*e^x - 2*x))
\[ \int \left (-1+e^{e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2} \left (-4+e^{6 x} (-2-12 x)+8 x+e^x (2+2 x)\right )\right ) \, dx=\int { -2 \, {\left ({\left (6 \, x + 1\right )} e^{\left (6 \, x\right )} - {\left (x + 1\right )} e^{x} - 4 \, x + 2\right )} e^{\left (2 \, x^{2} - x e^{\left (6 \, x\right )} + x e^{x} - 2 \, x + e^{\left (2 \, x^{2} - x e^{\left (6 \, x\right )} + x e^{x} - 2 \, x\right )}\right )} - 1 \,d x } \] Input:
integrate(((-12*x-2)*exp(6*x)+(2+2*x)*exp(x)+8*x-4)*exp(-x*exp(6*x)+exp(x) *x+2*x^2-2*x)*exp(exp(-x*exp(6*x)+exp(x)*x+2*x^2-2*x))-1,x, algorithm="gia c")
Output:
integrate(-2*((6*x + 1)*e^(6*x) - (x + 1)*e^x - 4*x + 2)*e^(2*x^2 - x*e^(6 *x) + x*e^x - 2*x + e^(2*x^2 - x*e^(6*x) + x*e^x - 2*x)) - 1, x)
Time = 3.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \left (-1+e^{e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2} \left (-4+e^{6 x} (-2-12 x)+8 x+e^x (2+2 x)\right )\right ) \, dx=2\,{\mathrm {e}}^{{\mathrm {e}}^{x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{6\,x}}\,{\mathrm {e}}^{2\,x^2}}-x \] Input:
int(exp(exp(x*exp(x) - x*exp(6*x) - 2*x + 2*x^2))*exp(x*exp(x) - x*exp(6*x ) - 2*x + 2*x^2)*(8*x + exp(x)*(2*x + 2) - exp(6*x)*(12*x + 2) - 4) - 1,x)
Output:
2*exp(exp(x*exp(x))*exp(-2*x)*exp(-x*exp(6*x))*exp(2*x^2)) - x
\[ \int \left (-1+e^{e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2} \left (-4+e^{6 x} (-2-12 x)+8 x+e^x (2+2 x)\right )\right ) \, dx=-2 \left (\int \frac {e^{\frac {e^{e^{x} x +2 x^{2}}+e^{e^{6 x} x +3 x} x +2 e^{e^{6 x} x +2 x} x^{2}+4 e^{e^{6 x} x +2 x} x}{e^{e^{6 x} x +2 x}}}}{e^{e^{6 x} x}}d x \right )-4 \left (\int \frac {e^{\frac {e^{e^{x} x +2 x^{2}}+e^{e^{6 x} x +3 x} x +2 e^{e^{6 x} x +2 x} x^{2}}{e^{e^{6 x} x +2 x}}}}{e^{e^{6 x} x +2 x}}d x \right )+2 \left (\int \frac {e^{\frac {e^{e^{x} x +2 x^{2}}+e^{e^{6 x} x +3 x} x +2 e^{e^{6 x} x +2 x} x^{2}}{e^{e^{6 x} x +2 x}}}}{e^{e^{6 x} x +x}}d x \right )-12 \left (\int \frac {e^{\frac {e^{e^{x} x +2 x^{2}}+e^{e^{6 x} x +3 x} x +2 e^{e^{6 x} x +2 x} x^{2}+4 e^{e^{6 x} x +2 x} x}{e^{e^{6 x} x +2 x}}} x}{e^{e^{6 x} x}}d x \right )+8 \left (\int \frac {e^{\frac {e^{e^{x} x +2 x^{2}}+e^{e^{6 x} x +3 x} x +2 e^{e^{6 x} x +2 x} x^{2}}{e^{e^{6 x} x +2 x}}} x}{e^{e^{6 x} x +2 x}}d x \right )+2 \left (\int \frac {e^{\frac {e^{e^{x} x +2 x^{2}}+e^{e^{6 x} x +3 x} x +2 e^{e^{6 x} x +2 x} x^{2}}{e^{e^{6 x} x +2 x}}} x}{e^{e^{6 x} x +x}}d x \right )-x \] Input:
int(((-12*x-2)*exp(6*x)+(2+2*x)*exp(x)+8*x-4)*exp(-x*exp(6*x)+exp(x)*x+2*x ^2-2*x)*exp(exp(-x*exp(6*x)+exp(x)*x+2*x^2-2*x))-1,x)
Output:
- 2*int(e**((e**(e**x*x + 2*x**2) + e**(e**(6*x)*x + 3*x)*x + 2*e**(e**(6 *x)*x + 2*x)*x**2 + 4*e**(e**(6*x)*x + 2*x)*x)/e**(e**(6*x)*x + 2*x))/e**( e**(6*x)*x),x) - 4*int(e**((e**(e**x*x + 2*x**2) + e**(e**(6*x)*x + 3*x)*x + 2*e**(e**(6*x)*x + 2*x)*x**2)/e**(e**(6*x)*x + 2*x))/e**(e**(6*x)*x + 2 *x),x) + 2*int(e**((e**(e**x*x + 2*x**2) + e**(e**(6*x)*x + 3*x)*x + 2*e** (e**(6*x)*x + 2*x)*x**2)/e**(e**(6*x)*x + 2*x))/e**(e**(6*x)*x + x),x) - 1 2*int((e**((e**(e**x*x + 2*x**2) + e**(e**(6*x)*x + 3*x)*x + 2*e**(e**(6*x )*x + 2*x)*x**2 + 4*e**(e**(6*x)*x + 2*x)*x)/e**(e**(6*x)*x + 2*x))*x)/e** (e**(6*x)*x),x) + 8*int((e**((e**(e**x*x + 2*x**2) + e**(e**(6*x)*x + 3*x) *x + 2*e**(e**(6*x)*x + 2*x)*x**2)/e**(e**(6*x)*x + 2*x))*x)/e**(e**(6*x)* x + 2*x),x) + 2*int((e**((e**(e**x*x + 2*x**2) + e**(e**(6*x)*x + 3*x)*x + 2*e**(e**(6*x)*x + 2*x)*x**2)/e**(e**(6*x)*x + 2*x))*x)/e**(e**(6*x)*x + x),x) - x