Integrand size = 61, antiderivative size = 21 \[ \int \frac {1}{2} e^{e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )} \left (-1+4 e^{x^2} x\right ) \, dx=e^{e^{7-e^4+e^{x^2}-\frac {x}{2}}} \] Output:
exp(exp(exp(x^2)-exp(4)-1/2*x+7))
Time = 0.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2} e^{e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )} \left (-1+4 e^{x^2} x\right ) \, dx=e^{e^{7-e^4+e^{x^2}-\frac {x}{2}}} \] Input:
Integrate[(E^(E^((14 - 2*E^4 + 2*E^x^2 - x)/2) + (14 - 2*E^4 + 2*E^x^2 - x )/2)*(-1 + 4*E^x^2*x))/2,x]
Output:
E^E^(7 - E^4 + E^x^2 - 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.
\(\displaystyle \int \frac {1}{2} \left (4 e^{x^2} x-1\right ) \exp \left (\frac {1}{2} \left (2 e^{x^2}-x-2 e^4+14\right )+e^{\frac {1}{2} \left (2 e^{x^2}-x-2 e^4+14\right )}\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int -\exp \left (\frac {1}{2} \left (-x+2 e^{x^2}+2 \left (7-e^4\right )\right )+e^{\frac {1}{2} \left (-x+2 e^{x^2}+2 \left (7-e^4\right )\right )}\right ) \left (1-4 e^{x^2} x\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{2} \int \exp \left (\frac {1}{2} \left (-x+2 e^{x^2}+2 \left (7-e^4\right )\right )+e^{\frac {1}{2} \left (-x+2 e^{x^2}+2 \left (7-e^4\right )\right )}\right ) \left (1-4 e^{x^2} x\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{2} \int \left (\exp \left (\frac {1}{2} \left (-x+2 e^{x^2}+2 \left (7-e^4\right )\right )+e^{\frac {1}{2} \left (-x+2 e^{x^2}+2 \left (7-e^4\right )\right )}\right )-4 \exp \left (x^2+e^{\frac {1}{2} \left (-x+2 e^{x^2}+2 \left (7-e^4\right )\right )}+\frac {1}{2} \left (-x+2 e^{x^2}+2 \left (7-e^4\right )\right )\right ) x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (4 \int \exp \left (x^2+e^{\frac {1}{2} \left (-x+2 e^{x^2}+2 \left (7-e^4\right )\right )}+\frac {1}{2} \left (-x+2 e^{x^2}+2 \left (7-e^4\right )\right )\right ) xdx-\int \exp \left (\frac {1}{2} \left (-x+2 e^{x^2}+2 \left (7-e^4\right )\right )+e^{\frac {1}{2} \left (-x+2 e^{x^2}+2 \left (7-e^4\right )\right )}\right )dx\right )\) |
Input:
Int[(E^(E^((14 - 2*E^4 + 2*E^x^2 - x)/2) + (14 - 2*E^4 + 2*E^x^2 - x)/2)*( -1 + 4*E^x^2*x))/2,x]
Output:
$Aborted
Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76
method | result | size |
norman | \({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}}-{\mathrm e}^{4}-\frac {x}{2}+7}}\) | \(16\) |
risch | \({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}}-{\mathrm e}^{4}-\frac {x}{2}+7}}\) | \(16\) |
parallelrisch | \({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}}-{\mathrm e}^{4}-\frac {x}{2}+7}}\) | \(16\) |
Input:
int(1/2*(4*exp(x^2)*x-1)*exp(exp(x^2)-exp(4)-1/2*x+7)*exp(exp(exp(x^2)-exp (4)-1/2*x+7)),x,method=_RETURNVERBOSE)
Output:
exp(exp(exp(x^2)-exp(4)-1/2*x+7))
Time = 0.09 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {1}{2} e^{e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )} \left (-1+4 e^{x^2} x\right ) \, dx=e^{\left (e^{\left (-\frac {1}{2} \, x - e^{4} + e^{\left (x^{2}\right )} + 7\right )}\right )} \] Input:
integrate(1/2*(4*exp(x^2)*x-1)*exp(exp(x^2)-exp(4)-1/2*x+7)*exp(exp(exp(x^ 2)-exp(4)-1/2*x+7)),x, algorithm="fricas")
Output:
e^(e^(-1/2*x - e^4 + e^(x^2) + 7))
Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {1}{2} e^{e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )} \left (-1+4 e^{x^2} x\right ) \, dx=e^{e^{- \frac {x}{2} + e^{x^{2}} - e^{4} + 7}} \] Input:
integrate(1/2*(4*exp(x**2)*x-1)*exp(exp(x**2)-exp(4)-1/2*x+7)*exp(exp(exp( x**2)-exp(4)-1/2*x+7)),x)
Output:
exp(exp(-x/2 + exp(x**2) - exp(4) + 7))
Time = 0.46 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {1}{2} e^{e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )} \left (-1+4 e^{x^2} x\right ) \, dx=e^{\left (e^{\left (-\frac {1}{2} \, x - e^{4} + e^{\left (x^{2}\right )} + 7\right )}\right )} \] Input:
integrate(1/2*(4*exp(x^2)*x-1)*exp(exp(x^2)-exp(4)-1/2*x+7)*exp(exp(exp(x^ 2)-exp(4)-1/2*x+7)),x, algorithm="maxima")
Output:
e^(e^(-1/2*x - e^4 + e^(x^2) + 7))
\[ \int \frac {1}{2} e^{e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )} \left (-1+4 e^{x^2} x\right ) \, dx=\int { \frac {1}{2} \, {\left (4 \, x e^{\left (x^{2}\right )} - 1\right )} e^{\left (-\frac {1}{2} \, x - e^{4} + e^{\left (x^{2}\right )} + e^{\left (-\frac {1}{2} \, x - e^{4} + e^{\left (x^{2}\right )} + 7\right )} + 7\right )} \,d x } \] Input:
integrate(1/2*(4*exp(x^2)*x-1)*exp(exp(x^2)-exp(4)-1/2*x+7)*exp(exp(exp(x^ 2)-exp(4)-1/2*x+7)),x, algorithm="giac")
Output:
integrate(1/2*(4*x*e^(x^2) - 1)*e^(-1/2*x - e^4 + e^(x^2) + e^(-1/2*x - e^ 4 + e^(x^2) + 7) + 7), x)
Time = 2.67 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {1}{2} e^{e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )} \left (-1+4 e^{x^2} x\right ) \, dx={\mathrm {e}}^{{\mathrm {e}}^{-{\mathrm {e}}^4}\,{\mathrm {e}}^{-\frac {x}{2}}\,{\mathrm {e}}^7\,{\mathrm {e}}^{{\mathrm {e}}^{x^2}}} \] Input:
int((exp(exp(exp(x^2) - x/2 - exp(4) + 7))*exp(exp(x^2) - x/2 - exp(4) + 7 )*(4*x*exp(x^2) - 1))/2,x)
Output:
exp(exp(-exp(4))*exp(-x/2)*exp(7)*exp(exp(x^2)))
Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {1}{2} e^{e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )} \left (-1+4 e^{x^2} x\right ) \, dx=e^{\frac {e^{e^{x^{2}}} e^{7}}{e^{e^{4}+\frac {x}{2}}}} \] Input:
int(1/2*(4*exp(x^2)*x-1)*exp(exp(x^2)-exp(4)-1/2*x+7)*exp(exp(exp(x^2)-exp (4)-1/2*x+7)),x)
Output:
e**((e**(e**(x**2))*e**7)/e**((2*e**4 + x)/2))