3.11.84 \(\int \frac {1}{2} (4+e^{-e^{-e^{\frac {1}{2} (-2 x^2-x^3+(-2 x-x^2) \log (3))}+x}+2 x} (-4+e^{-e^{\frac {1}{2} (-2 x^2-x^3+(-2 x-x^2) \log (3))}+x} (2+e^{\frac {1}{2} (-2 x^2-x^3+(-2 x-x^2) \log (3))} (4 x+3 x^2+(2+2 x) \log (3))))) \, dx\) [1084]

3.11.84.1 Optimal result

Integrand size = 137, antiderivative size = 36 \begin {dmath*} \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 \end {dmath*}

output

2*x-exp(2*x-exp(x-exp(x*(ln(3)+x)*(-1-1/2*x))))
 

3.11.84.2 Mathematica [F]

\begin {dmath*} \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 \end {dmath*}

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
 

3.11.84.3 Rubi [F]

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
 

3.11.84.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

3.11.84.4 Maple [A] (verified)

Time = 1.85 (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)
 

3.11.84.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.14 \begin {dmath*} \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 )} \end {dmath*}

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=\
 

output

2*x - e^(2*x - e^(x - e^(-1/2*x^3 - x^2 - 1/2*(x^2 + 2*x)*log(3))))
 

3.11.84.6 Sympy [A] (verification not implemented)

Time = 1.45 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \begin {dmath*} \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 )}}}} \end {dmath*}

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))))
 

3.11.84.7 Maxima [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17 \begin {dmath*} \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 )} \end {dmath*}

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=\
 

output

2*x - e^(2*x - e^(x - e^(-1/2*x^3 - 1/2*x^2*log(3) - x^2 - x*log(3))))
 

3.11.84.8 Giac [A] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17 \begin {dmath*} \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 )} \end {dmath*}

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=\
 

output

2*x - e^(2*x - e^(x - e^(-1/2*x^3 - 1/2*x^2*log(3) - x^2 - x*log(3))))
 

3.11.84.9 Mupad [B] (verification not implemented)

Time = 16.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17 \begin {dmath*} \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} \end {dmath*}

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))