Integrand size = 183, antiderivative size = 33 \[ \int \frac {e^{-x^2} x \left (2 x^2+e^{2+2 e^{e^3}} \left (-2+2 x^2\right )+e^2 \left (-8+6 x^2+2 x^4\right )+e^{e^{e^3}} \left (-2 x+e^2 \left (4 x-4 x^3\right )\right )\right )+e^{-x^2} x \left (8-6 x^2-2 x^4+e^{2 e^{e^3}} \left (2-2 x^2\right )+e^{e^{e^3}} \left (-4 x+4 x^3\right )\right ) \log \left (4+e^{2 e^{e^3}}-2 e^{e^{e^3}} x+x^2\right )}{4+e^{2 e^{e^3}}-2 e^{e^{e^3}} x+x^2} \, dx=e^{-x^2} x^2 \left (-e^2+\log \left (4+\left (e^{e^{e^3}}-x\right )^2\right )\right ) \] Output:
exp(ln(x)-x^2)*(ln((exp(exp(exp(3)))-x)^2+4)-exp(2))*x
Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30 \[ \int \frac {e^{-x^2} x \left (2 x^2+e^{2+2 e^{e^3}} \left (-2+2 x^2\right )+e^2 \left (-8+6 x^2+2 x^4\right )+e^{e^{e^3}} \left (-2 x+e^2 \left (4 x-4 x^3\right )\right )\right )+e^{-x^2} x \left (8-6 x^2-2 x^4+e^{2 e^{e^3}} \left (2-2 x^2\right )+e^{e^{e^3}} \left (-4 x+4 x^3\right )\right ) \log \left (4+e^{2 e^{e^3}}-2 e^{e^{e^3}} x+x^2\right )}{4+e^{2 e^{e^3}}-2 e^{e^{e^3}} x+x^2} \, dx=-e^{-x^2} x^2 \left (e^2-\log \left (4+e^{2 e^{e^3}}-2 e^{e^{e^3}} x+x^2\right )\right ) \] Input:
Integrate[((x*(2*x^2 + E^(2 + 2*E^E^3)*(-2 + 2*x^2) + E^2*(-8 + 6*x^2 + 2* x^4) + E^E^E^3*(-2*x + E^2*(4*x - 4*x^3))))/E^x^2 + (x*(8 - 6*x^2 - 2*x^4 + E^(2*E^E^3)*(2 - 2*x^2) + E^E^E^3*(-4*x + 4*x^3))*Log[4 + E^(2*E^E^3) - 2*E^E^E^3*x + x^2])/E^x^2)/(4 + E^(2*E^E^3) - 2*E^E^E^3*x + x^2),x]
Output:
-((x^2*(E^2 - Log[4 + E^(2*E^E^3) - 2*E^E^E^3*x + x^2]))/E^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.
| \(\displaystyle \int \frac {e^{-x^2} x \left (e^{e^{e^3}} \left (e^2 \left (4 x-4 x^3\right )-2 x\right )+2 x^2+e^{2+2 e^{e^3}} \left (2 x^2-2\right )+e^2 \left (2 x^4+6 x^2-8\right )\right )+e^{-x^2} x \left (-2 x^4+e^{e^{e^3}} \left (4 x^3-4 x\right )-6 x^2+e^{2 e^{e^3}} \left (2-2 x^2\right )+8\right ) \log \left (x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4\right )}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (-\frac {2 \left (1-2 e^2\right ) e^{e^{e^3}-x^2} x^2}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}-\frac {8 \left (1+\frac {1}{4} e^{2 e^{e^3}}\right ) e^{2-x^2} x}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}-\frac {4 e^{e^{e^3}-x^2} x^2 \log \left (x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4\right )}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}+\frac {8 \left (1+\frac {1}{4} e^{2 e^{e^3}}\right ) e^{-x^2} x \log \left (x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4\right )}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}+\frac {2 e^{2-x^2} x^5}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}-\frac {2 e^{-x^2} x^5 \log \left (x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4\right )}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}-\frac {4 e^{-x^2+e^{e^3}+2} x^4}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}+\frac {4 e^{e^{e^3}-x^2} x^4 \log \left (x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4\right )}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}+\frac {2 \left (1+e^2 \left (3+e^{2 e^{e^3}}\right )\right ) e^{-x^2} x^3}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}-\frac {6 \left (1+\frac {1}{3} e^{2 e^{e^3}}\right ) e^{-x^2} x^3 \log \left (x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4\right )}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 e^{-x^2} x \left (x^2-2 e^{2+e^{e^3}} \left (x^2-1\right ) x+e^{2+2 e^{e^3}} \left (x^2-1\right )-\left (x^2-1\right ) \left (x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4\right ) \log \left (x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4\right )+e^2 \left (x^4+3 x^2-4\right )-e^{e^{e^3}} x\right )}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {e^{-x^2} x \left (-x^2-2 e^{2+e^{e^3}} \left (1-x^2\right ) x+e^{e^{e^3}} x+e^{2+2 e^{e^3}} \left (1-x^2\right )+e^2 \left (-x^4-3 x^2+4\right )-\left (1-x^2\right ) \left (x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4\right ) \log \left (x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4\right )\right )}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {e^{-x^2} x \left (-x^2-2 e^{2+e^{e^3}} \left (1-x^2\right ) x+e^{e^{e^3}} x+e^{2+2 e^{e^3}} \left (1-x^2\right )+e^2 \left (-x^4-3 x^2+4\right )-\left (1-x^2\right ) \left (x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4\right ) \log \left (x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4\right )\right )}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -2 \int \left (-\frac {e^{-x^2} x^3}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}+\frac {e^{e^{e^3}-x^2} x^2}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}+\frac {2 e^{-x^2+e^{e^3}+2} (x-1) (x+1) x^2}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}+e^{-x^2} (x-1) (x+1) \log \left (x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4\right ) x-\frac {e^{-x^2+2 e^{e^3}+2} (x-1) (x+1) x}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}-\frac {e^{2-x^2} (x-1) (x+1) \left (x^2+4\right ) x}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {e^{-x^2} x \left (-x^2+2 e^{2+e^{e^3}} \left (x^2-1\right ) x+e^{e^{e^3}} x-e^{2+2 e^{e^3}} \left (x^2-1\right )-e^2 \left (x^4+3 x^2-4\right )+\left (x^2-1\right ) \left (x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4\right ) \log \left (x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4\right )\right )}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -2 \int \left (-\frac {e^{-x^2} x^3}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}+\frac {e^{e^{e^3}-x^2} x^2}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}+\frac {2 e^{-x^2+e^{e^3}+2} (x-1) (x+1) x^2}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}+e^{-x^2} (x-1) (x+1) \log \left (x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4\right ) x-\frac {e^{-x^2+2 e^{e^3}+2} (x-1) (x+1) x}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}-\frac {e^{2-x^2} (x-1) (x+1) \left (x^2+4\right ) x}{x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (\frac {1}{2} e^{2-x^2} x^2-\frac {1}{2} e^{-x^2} \log \left (x^2-2 e^{e^{e^3}} x+e^{2 e^{e^3}}+4\right ) x^2+\frac {e^{2-x^2}}{2}-\frac {3}{2} e^{2 \left (1+e^{e^3}\right )-x^2}+e^{2+e^{e^3}} \left (5-2 e^{2 e^{e^3}}\right ) \sqrt {\pi } \text {erf}(x)-e^{2+e^{e^3}} \left (5-3 e^{2 e^{e^3}}\right ) \sqrt {\pi } \text {erf}(x)-e^{2+3 e^{e^3}} \sqrt {\pi } \text {erf}(x)+\left (2 i+e^{e^{e^3}}\right )^2 \int \frac {e^{-x^2}}{2 x-2 e^{e^{e^3}}-4 i}dx+\frac {1}{2} \left (2-i e^{e^{e^3}}\right )^3 \int \frac {e^{-x^2}}{2 x-2 e^{e^{e^3}}-4 i}dx-\frac {1}{2} i \left (2 i+e^{e^{e^3}}\right )^2 \int \frac {e^{e^{e^3}-x^2}}{2 x-2 e^{e^{e^3}}-4 i}dx+\left (2 i+e^{e^{e^3}}\right )^2 \left (5 i+4 e^{e^{e^3}}-i e^{2 e^{e^3}}\right ) \int \frac {e^{-x^2+e^{e^3}+2}}{2 x-2 e^{e^{e^3}}-4 i}dx-\frac {1}{2} \left (4-i e^{e^{e^3}}\right ) \left ((-1+2 i)+e^{e^{e^3}}\right ) \left (2 i+e^{e^{e^3}}\right ) \left ((1+2 i)+e^{e^{e^3}}\right ) \int \frac {e^{-x^2+e^{e^3}+2}}{2 x-2 e^{e^{e^3}}-4 i}dx-\frac {1}{2} \left (2-i e^{e^{e^3}}\right ) \left ((-1+2 i)+e^{e^{e^3}}\right ) \left ((1+2 i)+e^{e^{e^3}}\right ) \int \frac {e^{-x^2+2 e^{e^3}+2}}{2 x-2 e^{e^{e^3}}-4 i}dx+\frac {1}{2} \left (2+i e^{e^{e^3}}\right )^3 \int \frac {e^{-x^2}}{2 x-2 e^{e^{e^3}}+4 i}dx+\left (2 i-e^{e^{e^3}}\right )^2 \int \frac {e^{-x^2}}{2 x-2 e^{e^{e^3}}+4 i}dx+\frac {1}{2} i \left (2 i-e^{e^{e^3}}\right )^2 \int \frac {e^{e^{e^3}-x^2}}{2 x-2 e^{e^{e^3}}+4 i}dx+\left (2 i-e^{e^{e^3}}\right )^2 \left (4 e^{e^{e^3}}-i \left (5-e^{2 e^{e^3}}\right )\right ) \int \frac {e^{-x^2+e^{e^3}+2}}{2 x-2 e^{e^{e^3}}+4 i}dx+\frac {1}{2} \left (4 i-e^{e^{e^3}}\right ) \left (2+i e^{e^{e^3}}\right ) \left ((-1-2 i)+e^{e^{e^3}}\right ) \left ((1-2 i)+e^{e^{e^3}}\right ) \int \frac {e^{-x^2+e^{e^3}+2}}{2 x-2 e^{e^{e^3}}+4 i}dx-\frac {1}{2} \left (2+i e^{e^{e^3}}\right ) \left ((-1-2 i)+e^{e^{e^3}}\right ) \left ((1-2 i)+e^{e^{e^3}}\right ) \int \frac {e^{-x^2+2 e^{e^3}+2}}{2 x-2 e^{e^{e^3}}+4 i}dx-\frac {1}{2} e^{2-x^2} \left (1-3 e^{2 e^{e^3}}\right )\right )\) |
Input:
Int[((x*(2*x^2 + E^(2 + 2*E^E^3)*(-2 + 2*x^2) + E^2*(-8 + 6*x^2 + 2*x^4) + E^E^E^3*(-2*x + E^2*(4*x - 4*x^3))))/E^x^2 + (x*(8 - 6*x^2 - 2*x^4 + E^(2 *E^E^3)*(2 - 2*x^2) + E^E^E^3*(-4*x + 4*x^3))*Log[4 + E^(2*E^E^3) - 2*E^E^ E^3*x + x^2])/E^x^2)/(4 + E^(2*E^E^3) - 2*E^E^E^3*x + x^2),x]
Output:
$Aborted
Time = 33.97 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09
| method | result | size |
| risch | \(\left (-{\mathrm e}^{2} x +x \ln \left ({\mathrm e}^{2 \,{\mathrm e}^{{\mathrm e}^{3}}}-2 x \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{3}}}+x^{2}+4\right )\right ) x \,{\mathrm e}^{-x^{2}}\) | \(36\) |
| parallelrisch | \(-{\mathrm e}^{2} x \,{\mathrm e}^{\ln \left (x \right )-x^{2}}+x \,{\mathrm e}^{\ln \left (x \right )-x^{2}} \ln \left ({\mathrm e}^{2 \,{\mathrm e}^{{\mathrm e}^{3}}}-2 x \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{3}}}+x^{2}+4\right )\) | \(46\) |
Input:
int((((-2*x^2+2)*exp(exp(exp(3)))^2+(4*x^3-4*x)*exp(exp(exp(3)))-2*x^4-6*x ^2+8)*exp(ln(x)-x^2)*ln(exp(exp(exp(3)))^2-2*x*exp(exp(exp(3)))+x^2+4)+((2 *x^2-2)*exp(2)*exp(exp(exp(3)))^2+((-4*x^3+4*x)*exp(2)-2*x)*exp(exp(exp(3) ))+(2*x^4+6*x^2-8)*exp(2)+2*x^2)*exp(ln(x)-x^2))/(exp(exp(exp(3)))^2-2*x*e xp(exp(exp(3)))+x^2+4),x,method=_RETURNVERBOSE)
Output:
(-exp(2)*x+x*ln(exp(2*exp(exp(3)))-2*x*exp(exp(exp(3)))+x^2+4))*x*exp(-x^2 )
Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {e^{-x^2} x \left (2 x^2+e^{2+2 e^{e^3}} \left (-2+2 x^2\right )+e^2 \left (-8+6 x^2+2 x^4\right )+e^{e^{e^3}} \left (-2 x+e^2 \left (4 x-4 x^3\right )\right )\right )+e^{-x^2} x \left (8-6 x^2-2 x^4+e^{2 e^{e^3}} \left (2-2 x^2\right )+e^{e^{e^3}} \left (-4 x+4 x^3\right )\right ) \log \left (4+e^{2 e^{e^3}}-2 e^{e^{e^3}} x+x^2\right )}{4+e^{2 e^{e^3}}-2 e^{e^{e^3}} x+x^2} \, dx=x e^{\left (-x^{2} + \log \left (x\right )\right )} \log \left (x^{2} - 2 \, x e^{\left (e^{\left (e^{3}\right )}\right )} + e^{\left (2 \, e^{\left (e^{3}\right )}\right )} + 4\right ) - x e^{\left (-x^{2} + \log \left (x\right ) + 2\right )} \] Input:
integrate((((-2*x^2+2)*exp(exp(exp(3)))^2+(4*x^3-4*x)*exp(exp(exp(3)))-2*x ^4-6*x^2+8)*exp(log(x)-x^2)*log(exp(exp(exp(3)))^2-2*x*exp(exp(exp(3)))+x^ 2+4)+((2*x^2-2)*exp(2)*exp(exp(exp(3)))^2+((-4*x^3+4*x)*exp(2)-2*x)*exp(ex p(exp(3)))+(2*x^4+6*x^2-8)*exp(2)+2*x^2)*exp(log(x)-x^2))/(exp(exp(exp(3)) )^2-2*x*exp(exp(exp(3)))+x^2+4),x, algorithm="fricas")
Output:
x*e^(-x^2 + log(x))*log(x^2 - 2*x*e^(e^(e^3)) + e^(2*e^(e^3)) + 4) - x*e^( -x^2 + log(x) + 2)
Time = 1.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {e^{-x^2} x \left (2 x^2+e^{2+2 e^{e^3}} \left (-2+2 x^2\right )+e^2 \left (-8+6 x^2+2 x^4\right )+e^{e^{e^3}} \left (-2 x+e^2 \left (4 x-4 x^3\right )\right )\right )+e^{-x^2} x \left (8-6 x^2-2 x^4+e^{2 e^{e^3}} \left (2-2 x^2\right )+e^{e^{e^3}} \left (-4 x+4 x^3\right )\right ) \log \left (4+e^{2 e^{e^3}}-2 e^{e^{e^3}} x+x^2\right )}{4+e^{2 e^{e^3}}-2 e^{e^{e^3}} x+x^2} \, dx=\left (x^{2} \log {\left (x^{2} - 2 x e^{e^{e^{3}}} + 4 + e^{2 e^{e^{3}}} \right )} - x^{2} e^{2}\right ) e^{- x^{2}} \] Input:
integrate((((-2*x**2+2)*exp(exp(exp(3)))**2+(4*x**3-4*x)*exp(exp(exp(3)))- 2*x**4-6*x**2+8)*exp(ln(x)-x**2)*ln(exp(exp(exp(3)))**2-2*x*exp(exp(exp(3) ))+x**2+4)+((2*x**2-2)*exp(2)*exp(exp(exp(3)))**2+((-4*x**3+4*x)*exp(2)-2* x)*exp(exp(exp(3)))+(2*x**4+6*x**2-8)*exp(2)+2*x**2)*exp(ln(x)-x**2))/(exp (exp(exp(3)))**2-2*x*exp(exp(exp(3)))+x**2+4),x)
Output:
(x**2*log(x**2 - 2*x*exp(exp(exp(3))) + 4 + exp(2*exp(exp(3)))) - x**2*exp (2))*exp(-x**2)
Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30 \[ \int \frac {e^{-x^2} x \left (2 x^2+e^{2+2 e^{e^3}} \left (-2+2 x^2\right )+e^2 \left (-8+6 x^2+2 x^4\right )+e^{e^{e^3}} \left (-2 x+e^2 \left (4 x-4 x^3\right )\right )\right )+e^{-x^2} x \left (8-6 x^2-2 x^4+e^{2 e^{e^3}} \left (2-2 x^2\right )+e^{e^{e^3}} \left (-4 x+4 x^3\right )\right ) \log \left (4+e^{2 e^{e^3}}-2 e^{e^{e^3}} x+x^2\right )}{4+e^{2 e^{e^3}}-2 e^{e^{e^3}} x+x^2} \, dx=x^{2} e^{\left (-x^{2}\right )} \log \left (x^{2} - 2 \, x e^{\left (e^{\left (e^{3}\right )}\right )} + e^{\left (2 \, e^{\left (e^{3}\right )}\right )} + 4\right ) - x^{2} e^{\left (-x^{2} + 2\right )} \] Input:
integrate((((-2*x^2+2)*exp(exp(exp(3)))^2+(4*x^3-4*x)*exp(exp(exp(3)))-2*x ^4-6*x^2+8)*exp(log(x)-x^2)*log(exp(exp(exp(3)))^2-2*x*exp(exp(exp(3)))+x^ 2+4)+((2*x^2-2)*exp(2)*exp(exp(exp(3)))^2+((-4*x^3+4*x)*exp(2)-2*x)*exp(ex p(exp(3)))+(2*x^4+6*x^2-8)*exp(2)+2*x^2)*exp(log(x)-x^2))/(exp(exp(exp(3)) )^2-2*x*exp(exp(exp(3)))+x^2+4),x, algorithm="maxima")
Output:
x^2*e^(-x^2)*log(x^2 - 2*x*e^(e^(e^3)) + e^(2*e^(e^3)) + 4) - x^2*e^(-x^2 + 2)
Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {e^{-x^2} x \left (2 x^2+e^{2+2 e^{e^3}} \left (-2+2 x^2\right )+e^2 \left (-8+6 x^2+2 x^4\right )+e^{e^{e^3}} \left (-2 x+e^2 \left (4 x-4 x^3\right )\right )\right )+e^{-x^2} x \left (8-6 x^2-2 x^4+e^{2 e^{e^3}} \left (2-2 x^2\right )+e^{e^{e^3}} \left (-4 x+4 x^3\right )\right ) \log \left (4+e^{2 e^{e^3}}-2 e^{e^{e^3}} x+x^2\right )}{4+e^{2 e^{e^3}}-2 e^{e^{e^3}} x+x^2} \, dx=-{\left (x^{2} e^{2} - x^{2} \log \left (x^{2} - 2 \, x e^{\left (e^{\left (e^{3}\right )}\right )} + e^{\left (2 \, e^{\left (e^{3}\right )}\right )} + 4\right )\right )} e^{\left (-x^{2}\right )} \] Input:
integrate((((-2*x^2+2)*exp(exp(exp(3)))^2+(4*x^3-4*x)*exp(exp(exp(3)))-2*x ^4-6*x^2+8)*exp(log(x)-x^2)*log(exp(exp(exp(3)))^2-2*x*exp(exp(exp(3)))+x^ 2+4)+((2*x^2-2)*exp(2)*exp(exp(exp(3)))^2+((-4*x^3+4*x)*exp(2)-2*x)*exp(ex p(exp(3)))+(2*x^4+6*x^2-8)*exp(2)+2*x^2)*exp(log(x)-x^2))/(exp(exp(exp(3)) )^2-2*x*exp(exp(exp(3)))+x^2+4),x, algorithm="giac")
Output:
-(x^2*e^2 - x^2*log(x^2 - 2*x*e^(e^(e^3)) + e^(2*e^(e^3)) + 4))*e^(-x^2)
Time = 2.92 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-x^2} x \left (2 x^2+e^{2+2 e^{e^3}} \left (-2+2 x^2\right )+e^2 \left (-8+6 x^2+2 x^4\right )+e^{e^{e^3}} \left (-2 x+e^2 \left (4 x-4 x^3\right )\right )\right )+e^{-x^2} x \left (8-6 x^2-2 x^4+e^{2 e^{e^3}} \left (2-2 x^2\right )+e^{e^{e^3}} \left (-4 x+4 x^3\right )\right ) \log \left (4+e^{2 e^{e^3}}-2 e^{e^{e^3}} x+x^2\right )}{4+e^{2 e^{e^3}}-2 e^{e^{e^3}} x+x^2} \, dx=x^2\,{\mathrm {e}}^{-x^2}\,\left (\ln \left (x^2-2\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^3}}\,x+{\mathrm {e}}^{2\,{\mathrm {e}}^{{\mathrm {e}}^3}}+4\right )-{\mathrm {e}}^2\right ) \] Input:
int((exp(log(x) - x^2)*(exp(2)*(6*x^2 + 2*x^4 - 8) + 2*x^2 - exp(exp(exp(3 )))*(2*x - exp(2)*(4*x - 4*x^3)) + exp(2)*exp(2*exp(exp(3)))*(2*x^2 - 2)) - log(exp(2*exp(exp(3))) - 2*x*exp(exp(exp(3))) + x^2 + 4)*exp(log(x) - x^ 2)*(exp(2*exp(exp(3)))*(2*x^2 - 2) + 6*x^2 + 2*x^4 + exp(exp(exp(3)))*(4*x - 4*x^3) - 8))/(exp(2*exp(exp(3))) - 2*x*exp(exp(exp(3))) + x^2 + 4),x)
Output:
x^2*exp(-x^2)*(log(exp(2*exp(exp(3))) - 2*x*exp(exp(exp(3))) + x^2 + 4) - exp(2))
Time = 1.66 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {e^{-x^2} x \left (2 x^2+e^{2+2 e^{e^3}} \left (-2+2 x^2\right )+e^2 \left (-8+6 x^2+2 x^4\right )+e^{e^{e^3}} \left (-2 x+e^2 \left (4 x-4 x^3\right )\right )\right )+e^{-x^2} x \left (8-6 x^2-2 x^4+e^{2 e^{e^3}} \left (2-2 x^2\right )+e^{e^{e^3}} \left (-4 x+4 x^3\right )\right ) \log \left (4+e^{2 e^{e^3}}-2 e^{e^{e^3}} x+x^2\right )}{4+e^{2 e^{e^3}}-2 e^{e^{e^3}} x+x^2} \, dx=\frac {x^{2} \left (\mathrm {log}\left (e^{2 e^{e^{3}}}-2 e^{e^{e^{3}}} x +x^{2}+4\right )-e^{2}\right )}{e^{x^{2}}} \] Input:
int((((-2*x^2+2)*exp(exp(exp(3)))^2+(4*x^3-4*x)*exp(exp(exp(3)))-2*x^4-6*x ^2+8)*exp(log(x)-x^2)*log(exp(exp(exp(3)))^2-2*x*exp(exp(exp(3)))+x^2+4)+( (2*x^2-2)*exp(2)*exp(exp(exp(3)))^2+((-4*x^3+4*x)*exp(2)-2*x)*exp(exp(exp( 3)))+(2*x^4+6*x^2-8)*exp(2)+2*x^2)*exp(log(x)-x^2))/(exp(exp(exp(3)))^2-2* x*exp(exp(exp(3)))+x^2+4),x)
Output:
(x**2*(log(e**(2*e**(e**3)) - 2*e**(e**(e**3))*x + x**2 + 4) - e**2))/e**( x**2)