Integrand size = 137, antiderivative size = 36 \[ \int \frac {1}{2} \left (4+e^{-e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x}+2 x} \left (-4+e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x} \left (2+e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )} \left (4 x+3 x^2+(2+2 x) \log (3)\right )\right )\right )\right ) \, dx=-e^{-e^{-e^{\frac {1}{2} (-2-x) x (x+\log (3))}+x}+2 x}+2 x \] Output:
2*x-exp(2*x-exp(x-exp(x*(ln(3)+x)*(-1-1/2*x))))
\[ \int \frac {1}{2} \left (4+e^{-e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x}+2 x} \left (-4+e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x} \left (2+e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )} \left (4 x+3 x^2+(2+2 x) \log (3)\right )\right )\right )\right ) \, dx=\int \frac {1}{2} \left (4+e^{-e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x}+2 x} \left (-4+e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x} \left (2+e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )} \left (4 x+3 x^2+(2+2 x) \log (3)\right )\right )\right )\right ) \, dx \] Input:
Integrate[(4 + E^(-E^(-E^((-2*x^2 - x^3 + (-2*x - x^2)*Log[3])/2) + x) + 2 *x)*(-4 + E^(-E^((-2*x^2 - x^3 + (-2*x - x^2)*Log[3])/2) + x)*(2 + E^((-2* x^2 - x^3 + (-2*x - x^2)*Log[3])/2)*(4*x + 3*x^2 + (2 + 2*x)*Log[3]))))/2, x]
Output:
Integrate[4 + E^(-E^(-E^((-2*x^2 - x^3 + (-2*x - x^2)*Log[3])/2) + x) + 2* x)*(-4 + E^(-E^((-2*x^2 - x^3 + (-2*x - x^2)*Log[3])/2) + x)*(2 + E^((-2*x ^2 - x^3 + (-2*x - x^2)*Log[3])/2)*(4*x + 3*x^2 + (2 + 2*x)*Log[3]))), 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.
| \(\displaystyle \int \frac {1}{2} \left (\exp \left (2 x-\exp \left (x-\exp \left (\frac {1}{2} \left (-x^3-2 x^2+\left (-x^2-2 x\right ) \log (3)\right )\right )\right )\right ) \left (\exp \left (x-\exp \left (\frac {1}{2} \left (-x^3-2 x^2+\left (-x^2-2 x\right ) \log (3)\right )\right )\right ) \left (\left (3 x^2+4 x+(2 x+2) \log (3)\right ) \exp \left (\frac {1}{2} \left (-x^3-2 x^2+\left (-x^2-2 x\right ) \log (3)\right )\right )+2\right )-4\right )+4\right ) \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \left (4-\exp \left (2 x-\exp \left (x-3^{\frac {1}{2} \left (-x^2-2 x\right )} e^{\frac {1}{2} \left (-x^3-2 x^2\right )}\right )\right ) \left (4-\exp \left (x-3^{\frac {1}{2} \left (-x^2-2 x\right )} e^{\frac {1}{2} \left (-x^3-2 x^2\right )}\right ) \left (3^{\frac {1}{2} \left (-x^2-2 x\right )} e^{\frac {1}{2} \left (-x^3-2 x^2\right )} \left (3 x^2+4 x+2 (x+1) \log (3)\right )+2\right )\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (3^{-\frac {1}{2} x (x+2)} \exp \left (-\frac {1}{2} (x+2) x^2+2 x-3^{-\frac {1}{2} x (x+2)} e^{-\frac {1}{2} x^2 (x+2)}-\exp \left (x-3^{-\frac {1}{2} x (x+2)} e^{-\frac {1}{2} x^2 (x+2)}\right )\right ) \left (3 e^x x^2+4 e^x \left (1+\frac {\log (3)}{2}\right ) x-4\ 3^{\frac {1}{2} x (x+2)} \exp \left (\frac {1}{2} (x+2) x^2+3^{-\frac {1}{2} x (x+2)} e^{-\frac {1}{2} x^2 (x+2)}\right )+2\ 3^{\frac {1}{2} x (x+2)} e^{\frac {1}{2} (x+2) x^2+x}+e^x \log (9)\right )+4\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (3^{-\frac {1}{2} x (x+2)} \exp \left (-\frac {1}{2} x \left (x^2+2 x-4\right )-3^{-\frac {1}{2} x (x+2)} e^{-\frac {1}{2} x^2 (x+2)}-\exp \left (x-3^{-\frac {1}{2} x (x+2)} e^{-\frac {1}{2} x^2 (x+2)}\right )\right ) \left (3 e^x x^2+4 e^x \left (1+\frac {\log (3)}{2}\right ) x+2\ 3^{\frac {1}{2} x (x+2)} e^{\frac {x^3}{2}+x^2+x}-4\ 3^{\frac {1}{2} x (x+2)} \exp \left (\frac {1}{2} (x+2) x^2+3^{-\frac {1}{2} x (x+2)} e^{-\frac {1}{2} x^2 (x+2)}\right )+e^x \log (9)\right )+4\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {1}{2} \int \left (3^{-\frac {1}{2} x (x+2)} \exp \left (-\frac {1}{2} x \left (x^2+2 x-4\right )-3^{-\frac {1}{2} x (x+2)} e^{-\frac {1}{2} x^2 (x+2)}-\exp \left (x-3^{-\frac {1}{2} x (x+2)} e^{-\frac {1}{2} x^2 (x+2)}\right )\right ) \left (3 e^x x^2+4 e^x \left (1+\frac {\log (3)}{2}\right ) x+2\ 3^{\frac {1}{2} x (x+2)} e^{\frac {x^3}{2}+x^2+x}-4\ 3^{\frac {1}{2} x (x+2)} \exp \left (\frac {1}{2} (x+2) x^2+3^{-\frac {1}{2} x (x+2)} e^{-\frac {1}{2} x^2 (x+2)}\right )+e^x \log (9)\right )+4\right )dx\) |
Input:
Int[(4 + E^(-E^(-E^((-2*x^2 - x^3 + (-2*x - x^2)*Log[3])/2) + x) + 2*x)*(- 4 + E^(-E^((-2*x^2 - x^3 + (-2*x - x^2)*Log[3])/2) + x)*(2 + E^((-2*x^2 - x^3 + (-2*x - x^2)*Log[3])/2)*(4*x + 3*x^2 + (2 + 2*x)*Log[3]))))/2,x]
Output:
$Aborted
Time = 2.57 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00
| method | result | size |
| risch | \(2 x -{\mathrm e}^{-{\mathrm e}^{-3^{-\frac {x \left (2+x \right )}{2}} {\mathrm e}^{-\frac {x^{2} \left (2+x \right )}{2}}+x}+2 x}\) | \(36\) |
| default | \(2 x -{\mathrm e}^{-{\mathrm e}^{-{\mathrm e}^{\frac {\left (-x^{2}-2 x \right ) \ln \left (3\right )}{2}-\frac {x^{3}}{2}-x^{2}}+x}+2 x}\) | \(44\) |
| norman | \(2 x -{\mathrm e}^{-{\mathrm e}^{-{\mathrm e}^{\frac {\left (-x^{2}-2 x \right ) \ln \left (3\right )}{2}-\frac {x^{3}}{2}-x^{2}}+x}+2 x}\) | \(44\) |
| parallelrisch | \(2 x -{\mathrm e}^{-{\mathrm e}^{-{\mathrm e}^{\frac {\left (-x^{2}-2 x \right ) \ln \left (3\right )}{2}-\frac {x^{3}}{2}-x^{2}}+x}+2 x}\) | \(44\) |
| parts | \(2 x -{\mathrm e}^{-{\mathrm e}^{-{\mathrm e}^{\frac {\left (-x^{2}-2 x \right ) \ln \left (3\right )}{2}-\frac {x^{3}}{2}-x^{2}}+x}+2 x}\) | \(44\) |
Input:
int(1/2*((((2+2*x)*ln(3)+3*x^2+4*x)*exp(1/2*(-x^2-2*x)*ln(3)-1/2*x^3-x^2)+ 2)*exp(-exp(1/2*(-x^2-2*x)*ln(3)-1/2*x^3-x^2)+x)-4)*exp(-exp(-exp(1/2*(-x^ 2-2*x)*ln(3)-1/2*x^3-x^2)+x)+2*x)+2,x,method=_RETURNVERBOSE)
Output:
2*x-exp(-exp(-3^(-1/2*x*(2+x))*exp(-1/2*x^2*(2+x))+x)+2*x)
Time = 0.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.14 \[ \int \frac {1}{2} \left (4+e^{-e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x}+2 x} \left (-4+e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x} \left (2+e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )} \left (4 x+3 x^2+(2+2 x) \log (3)\right )\right )\right )\right ) \, dx=2 \, x - e^{\left (2 \, x - e^{\left (x - e^{\left (-\frac {1}{2} \, x^{3} - x^{2} - \frac {1}{2} \, {\left (x^{2} + 2 \, x\right )} \log \left (3\right )\right )}\right )}\right )} \] Input:
integrate(1/2*((((2+2*x)*log(3)+3*x^2+4*x)*exp(1/2*(-x^2-2*x)*log(3)-1/2*x ^3-x^2)+2)*exp(-exp(1/2*(-x^2-2*x)*log(3)-1/2*x^3-x^2)+x)-4)*exp(-exp(-exp (1/2*(-x^2-2*x)*log(3)-1/2*x^3-x^2)+x)+2*x)+2,x, algorithm="fricas")
Output:
2*x - e^(2*x - e^(x - e^(-1/2*x^3 - x^2 - 1/2*(x^2 + 2*x)*log(3))))
Time = 1.35 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \[ \int \frac {1}{2} \left (4+e^{-e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x}+2 x} \left (-4+e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x} \left (2+e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )} \left (4 x+3 x^2+(2+2 x) \log (3)\right )\right )\right )\right ) \, dx=2 x - e^{2 x - e^{x - e^{- \frac {x^{3}}{2} - x^{2} + \left (- \frac {x^{2}}{2} - x\right ) \log {\left (3 \right )}}}} \] Input:
integrate(1/2*((((2+2*x)*ln(3)+3*x**2+4*x)*exp(1/2*(-x**2-2*x)*ln(3)-1/2*x **3-x**2)+2)*exp(-exp(1/2*(-x**2-2*x)*ln(3)-1/2*x**3-x**2)+x)-4)*exp(-exp( -exp(1/2*(-x**2-2*x)*ln(3)-1/2*x**3-x**2)+x)+2*x)+2,x)
Output:
2*x - exp(2*x - exp(x - exp(-x**3/2 - x**2 + (-x**2/2 - x)*log(3))))
Time = 0.33 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17 \[ \int \frac {1}{2} \left (4+e^{-e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x}+2 x} \left (-4+e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x} \left (2+e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )} \left (4 x+3 x^2+(2+2 x) \log (3)\right )\right )\right )\right ) \, dx=2 \, x - e^{\left (2 \, x - e^{\left (x - e^{\left (-\frac {1}{2} \, x^{3} - \frac {1}{2} \, x^{2} \log \left (3\right ) - x^{2} - x \log \left (3\right )\right )}\right )}\right )} \] Input:
integrate(1/2*((((2+2*x)*log(3)+3*x^2+4*x)*exp(1/2*(-x^2-2*x)*log(3)-1/2*x ^3-x^2)+2)*exp(-exp(1/2*(-x^2-2*x)*log(3)-1/2*x^3-x^2)+x)-4)*exp(-exp(-exp (1/2*(-x^2-2*x)*log(3)-1/2*x^3-x^2)+x)+2*x)+2,x, algorithm="maxima")
Output:
2*x - e^(2*x - e^(x - e^(-1/2*x^3 - 1/2*x^2*log(3) - x^2 - x*log(3))))
Time = 0.54 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17 \[ \int \frac {1}{2} \left (4+e^{-e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x}+2 x} \left (-4+e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x} \left (2+e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )} \left (4 x+3 x^2+(2+2 x) \log (3)\right )\right )\right )\right ) \, dx=2 \, x - e^{\left (2 \, x - e^{\left (x - e^{\left (-\frac {1}{2} \, x^{3} - \frac {1}{2} \, x^{2} \log \left (3\right ) - x^{2} - x \log \left (3\right )\right )}\right )}\right )} \] Input:
integrate(1/2*((((2+2*x)*log(3)+3*x^2+4*x)*exp(1/2*(-x^2-2*x)*log(3)-1/2*x ^3-x^2)+2)*exp(-exp(1/2*(-x^2-2*x)*log(3)-1/2*x^3-x^2)+x)-4)*exp(-exp(-exp (1/2*(-x^2-2*x)*log(3)-1/2*x^3-x^2)+x)+2*x)+2,x, algorithm="giac")
Output:
2*x - e^(2*x - e^(x - e^(-1/2*x^3 - 1/2*x^2*log(3) - x^2 - x*log(3))))
Time = 7.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17 \[ \int \frac {1}{2} \left (4+e^{-e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x}+2 x} \left (-4+e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x} \left (2+e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )} \left (4 x+3 x^2+(2+2 x) \log (3)\right )\right )\right )\right ) \, dx=2\,x-{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-{\mathrm {e}}^{-\frac {{\mathrm {e}}^{-x^2}}{3^x\,\sqrt {{\mathrm {e}}^{x^3}}\,\sqrt {3^{x^2}}}}\,{\mathrm {e}}^x} \] Input:
int((exp(2*x - exp(x - exp(- x^2 - x^3/2 - (log(3)*(2*x + x^2))/2)))*(exp( x - exp(- x^2 - x^3/2 - (log(3)*(2*x + x^2))/2))*(exp(- x^2 - x^3/2 - (log (3)*(2*x + x^2))/2)*(4*x + log(3)*(2*x + 2) + 3*x^2) + 2) - 4))/2 + 2,x)
Output:
2*x - exp(2*x)*exp(-exp(-exp(-x^2)/(3^x*exp(x^3)^(1/2)*(3^(x^2))^(1/2)))*e xp(x))
\[ \int \frac {1}{2} \left (4+e^{-e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x}+2 x} \left (-4+e^{-e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )}+x} \left (2+e^{\frac {1}{2} \left (-2 x^2-x^3+\left (-2 x-x^2\right ) \log (3)\right )} \left (4 x+3 x^2+(2+2 x) \log (3)\right )\right )\right )\right ) \, dx=\text {too large to display} \] Input:
int(1/2*((((2+2*x)*log(3)+3*x^2+4*x)*exp(1/2*(-x^2-2*x)*log(3)-1/2*x^3-x^2 )+2)*exp(-exp(1/2*(-x^2-2*x)*log(3)-1/2*x^3-x^2)+x)-4)*exp(-exp(-exp(1/2*( -x^2-2*x)*log(3)-1/2*x^3-x^2)+x)+2*x)+2,x)
Output:
(2*int(e**(3*x)/e**((e**((x**3 + 2*x**2 + 2*x)/2)*3**x*3**(x**2/2) + e**(1 /(e**((x**3 + 2*x**2)/2)*3**x*3**(x**2/2))))/(e**((e**((x**3 + 2*x**2)/2)* 3**x*3**(x**2/2)*x**3 + 2*e**((x**3 + 2*x**2)/2)*3**x*3**(x**2/2)*x**2 + 2 )/(2*e**((x**3 + 2*x**2)/2)*3**x*3**(x**2/2)))*3**x*3**(x**2/2))),x) + 2*i nt(e**(3*x)/(e**((e**((e**((x**3 + 2*x**2)/2)*3**x*3**(x**2/2)*x**3 + 2*e* *((x**3 + 2*x**2)/2)*3**x*3**(x**2/2)*x**2 + 2)/(2*e**((x**3 + 2*x**2)/2)* 3**x*3**(x**2/2)))*3**x*3**(x**2/2)*x**3 + 2*e**((e**((x**3 + 2*x**2)/2)*3 **x*3**(x**2/2)*x**3 + 2*e**((x**3 + 2*x**2)/2)*3**x*3**(x**2/2)*x**2 + 2) /(2*e**((x**3 + 2*x**2)/2)*3**x*3**(x**2/2)))*3**x*3**(x**2/2)*x**2 + 2*e* *((x**3 + 2*x**2 + 2*x)/2)*3**x*3**(x**2/2) + 2*e**(1/(e**((x**3 + 2*x**2) /2)*3**x*3**(x**2/2))))/(2*e**((e**((x**3 + 2*x**2)/2)*3**x*3**(x**2/2)*x* *3 + 2*e**((x**3 + 2*x**2)/2)*3**x*3**(x**2/2)*x**2 + 2)/(2*e**((x**3 + 2* x**2)/2)*3**x*3**(x**2/2)))*3**x*3**(x**2/2)))*3**x*3**(x**2/2)),x)*log(3) - 4*int(e**(2*x)/e**(e**x/e**(1/(e**((x**3 + 2*x**2)/2)*3**x*3**(x**2/2)) )),x) + 3*int((e**(3*x)*x**2)/(e**((e**((e**((x**3 + 2*x**2)/2)*3**x*3**(x **2/2)*x**3 + 2*e**((x**3 + 2*x**2)/2)*3**x*3**(x**2/2)*x**2 + 2)/(2*e**(( x**3 + 2*x**2)/2)*3**x*3**(x**2/2)))*3**x*3**(x**2/2)*x**3 + 2*e**((e**((x **3 + 2*x**2)/2)*3**x*3**(x**2/2)*x**3 + 2*e**((x**3 + 2*x**2)/2)*3**x*3** (x**2/2)*x**2 + 2)/(2*e**((x**3 + 2*x**2)/2)*3**x*3**(x**2/2)))*3**x*3**(x **2/2)*x**2 + 2*e**((x**3 + 2*x**2 + 2*x)/2)*3**x*3**(x**2/2) + 2*e**(1...