Integrand size = 107, antiderivative size = 32 \[ \int \frac {-6-2 x+e^2 \left (9 x+20 x^2+5 x^3\right )+\left (2+e^2 \left (-6 x-6 x^2\right )\right ) \log \left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )+e^2 x \log ^2\left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )}{e^2 x} \, dx=x \left (x+\left (-3-x+\log \left (4 e^{-2-x+\frac {-\frac {1}{e^2}+x}{x}}\right )\right )^2\right ) \] Output:
(x+(ln(4*exp((x-exp(-2))/x-2-x))-x-3)^2)*x
Leaf count is larger than twice the leaf count of optimal. \(80\) vs. \(2(32)=64\).
Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.50 \[ \int \frac {-6-2 x+e^2 \left (9 x+20 x^2+5 x^3\right )+\left (2+e^2 \left (-6 x-6 x^2\right )\right ) \log \left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )+e^2 x \log ^2\left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )}{e^2 x} \, dx=\frac {2 (-3+x)}{e^2}+\frac {2}{e^4 x}+x \left (9+7 x+x^2\right )+\left (\frac {2}{e^2}-2 x (3+x)\right ) \log \left (4 e^{-1-\frac {1}{e^2 x}-x}\right )+x \log ^2\left (4 e^{-1-\frac {1}{e^2 x}-x}\right ) \] Input:
Integrate[(-6 - 2*x + E^2*(9*x + 20*x^2 + 5*x^3) + (2 + E^2*(-6*x - 6*x^2) )*Log[4*E^((-1 + E^2*(-x - x^2))/(E^2*x))] + E^2*x*Log[4*E^((-1 + E^2*(-x - x^2))/(E^2*x))]^2)/(E^2*x),x]
Output:
(2*(-3 + x))/E^2 + 2/(E^4*x) + x*(9 + 7*x + x^2) + (2/E^2 - 2*x*(3 + x))*L og[4*E^(-1 - 1/(E^2*x) - x)] + x*Log[4*E^(-1 - 1/(E^2*x) - 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.
| \(\displaystyle \int \frac {e^2 x \log ^2\left (4 e^{\frac {e^2 \left (-x^2-x\right )-1}{e^2 x}}\right )+\left (e^2 \left (-6 x^2-6 x\right )+2\right ) \log \left (4 e^{\frac {e^2 \left (-x^2-x\right )-1}{e^2 x}}\right )+e^2 \left (5 x^3+20 x^2+9 x\right )-2 x-6}{e^2 x} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {-e^2 x \log ^2\left (4 e^{-\frac {e^2 \left (x^2+x\right )+1}{e^2 x}}\right )-2 \left (1-3 e^2 \left (x^2+x\right )\right ) \log \left (4 e^{-\frac {e^2 \left (x^2+x\right )+1}{e^2 x}}\right )+2 x-e^2 \left (5 x^3+20 x^2+9 x\right )+6}{x}dx}{e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {-e^2 x \log ^2\left (4 e^{-\frac {e^2 \left (x^2+x\right )+1}{e^2 x}}\right )-2 \left (1-3 e^2 \left (x^2+x\right )\right ) \log \left (4 e^{-\frac {e^2 \left (x^2+x\right )+1}{e^2 x}}\right )+2 x-e^2 \left (5 x^3+20 x^2+9 x\right )+6}{x}dx}{e^2}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle -\frac {\int \left (-e^2 \log ^2\left (4 e^{-x-1-\frac {1}{e^2 x}}\right )+\frac {2 \left (3 e^2 x^2+3 e^2 x-1\right ) \log \left (4 e^{-x-1-\frac {1}{e^2 x}}\right )}{x}+\frac {-5 e^2 x^3-20 e^2 x^2+\left (2-9 e^2\right ) x+6}{x}\right )dx}{e^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-e^2 \int \log ^2\left (4 e^{-x-1-\frac {1}{e^2 x}}\right )dx-\frac {2}{3} e^2 x^3-7 e^2 x^2+3 e^2 x^2 \log \left (4 e^{-x-\frac {1}{e^2 x}-1}\right )+\left (2-9 e^2\right ) x-x-\frac {2}{e^2 x}+6 e^2 x \log \left (4 e^{-x-\frac {1}{e^2 x}-1}\right )-2 x \log (x)-2 \log \left (4 e^{-x-\frac {1}{e^2 x}-1}\right ) \log (x)-\frac {2 \log (x)}{e^2 x}}{e^2}\) |
Input:
Int[(-6 - 2*x + E^2*(9*x + 20*x^2 + 5*x^3) + (2 + E^2*(-6*x - 6*x^2))*Log[ 4*E^((-1 + E^2*(-x - x^2))/(E^2*x))] + E^2*x*Log[4*E^((-1 + E^2*(-x - x^2) )/(E^2*x))]^2)/(E^2*x),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(89\) vs. \(2(30)=60\).
Time = 0.47 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.81
| method | result | size |
| risch | \(9 x -4 x^{2} \ln \left (2\right )-12 x \ln \left (2\right )+4 x \ln \left (2\right )^{2}+7 x^{2}+x^{3}+x \ln \left ({\mathrm e}^{-\frac {\left (x^{2} {\mathrm e}^{2}+{\mathrm e}^{2} x +1\right ) {\mathrm e}^{-2}}{x}}\right )^{2}-x \left (6-4 \ln \left (2\right )+2 x \right ) \ln \left ({\mathrm e}^{-\frac {\left (x^{2} {\mathrm e}^{2}+{\mathrm e}^{2} x +1\right ) {\mathrm e}^{-2}}{x}}\right )\) | \(90\) |
| parallelrisch | \(\frac {{\mathrm e}^{-4} \left ({\mathrm e}^{4} x^{6}-2 \,{\mathrm e}^{4} x^{5} \ln \left (4 \,{\mathrm e}^{\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}}\right )+{\mathrm e}^{4} x^{4} \ln \left (4 \,{\mathrm e}^{\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}}\right )^{2}+7 x^{5} {\mathrm e}^{4}-6 \,{\mathrm e}^{4} \ln \left (4 \,{\mathrm e}^{\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}}\right ) x^{4}-9 \,{\mathrm e}^{4} x^{3} \ln \left (4 \,{\mathrm e}^{\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}}\right )-9 x^{2} {\mathrm e}^{2}\right )}{x^{3}}\) | \(175\) |
| default | \({\mathrm e}^{-2} \left (x^{3} {\mathrm e}^{2}+7 x^{2} {\mathrm e}^{2}+9 \,{\mathrm e}^{2} x +2 x +x \,{\mathrm e}^{2} \ln \left (4 \,{\mathrm e}^{\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}}\right )^{2}-2 \,{\mathrm e}^{2} x^{2} \ln \left (4 \,{\mathrm e}^{\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}}\right )+2 \ln \left (4 \,{\mathrm e}^{\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}}\right )-\frac {2 \,{\mathrm e}^{-2} \ln \left (x \right )}{x}-6 \ln \left (4 \,{\mathrm e}^{\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}}\right ) {\mathrm e}^{2} x -2 \,{\mathrm e}^{-2} \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )\right )\) | \(182\) |
| parts | \(x^{3}-\left (\ln \left (4 \,{\mathrm e}^{\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}}\right )-\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}\right ) x^{2}+x {\left (\ln \left (4 \,{\mathrm e}^{\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}}\right )-\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}\right )}^{2}+8 x^{2}-2 x \left (\ln \left (4 \,{\mathrm e}^{\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}}\right )-\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}\right )+10 x -2 \,{\mathrm e}^{-2} \ln \left (x \right ) \left (\ln \left (4 \,{\mathrm e}^{\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}}\right )-\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}\right )+2 \,{\mathrm e}^{-2} \ln \left (x \right )+\frac {{\mathrm e}^{-4}}{x}-3 x^{2} \ln \left (4 \,{\mathrm e}^{\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}}\right )-6 \ln \left (4 \,{\mathrm e}^{\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}}\right ) x +2 \,{\mathrm e}^{-2} \ln \left (x \right ) \ln \left (4 \,{\mathrm e}^{\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}}\right )+x \,{\mathrm e}^{-2}+2 \ln \left (x \right ) {\mathrm e}^{-2} x +\frac {2 \,{\mathrm e}^{-4} \ln \left (x \right )}{x}\) | \(369\) |
Input:
int((x*exp(2)*ln(4*exp(((-x^2-x)*exp(2)-1)/exp(2)/x))^2+((-6*x^2-6*x)*exp( 2)+2)*ln(4*exp(((-x^2-x)*exp(2)-1)/exp(2)/x))+(5*x^3+20*x^2+9*x)*exp(2)-2* x-6)/exp(2)/x,x,method=_RETURNVERBOSE)
Output:
9*x-4*x^2*ln(2)-12*x*ln(2)+4*x*ln(2)^2+7*x^2+x^3+x*ln(exp(-(x^2*exp(2)+exp (2)*x+1)*exp(-2)/x))^2-x*(6-4*ln(2)+2*x)*ln(exp(-(x^2*exp(2)+exp(2)*x+1)*e xp(-2)/x))
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.88 \[ \int \frac {-6-2 x+e^2 \left (9 x+20 x^2+5 x^3\right )+\left (2+e^2 \left (-6 x-6 x^2\right )\right ) \log \left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )+e^2 x \log ^2\left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )}{e^2 x} \, dx=\frac {{\left (4 \, x^{2} e^{4} \log \left (2\right )^{2} + 4 \, x^{2} e^{2} - 8 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{4} \log \left (2\right ) + {\left (4 \, x^{4} + 17 \, x^{3} + 16 \, x^{2}\right )} e^{4} + 1\right )} e^{\left (-4\right )}}{x} \] Input:
integrate((x*exp(2)*log(4*exp(((-x^2-x)*exp(2)-1)/exp(2)/x))^2+((-6*x^2-6* x)*exp(2)+2)*log(4*exp(((-x^2-x)*exp(2)-1)/exp(2)/x))+(5*x^3+20*x^2+9*x)*e xp(2)-2*x-6)/exp(2)/x,x, algorithm="fricas")
Output:
(4*x^2*e^4*log(2)^2 + 4*x^2*e^2 - 8*(x^3 + 2*x^2)*e^4*log(2) + (4*x^4 + 17 *x^3 + 16*x^2)*e^4 + 1)*e^(-4)/x
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (24) = 48\).
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.91 \[ \int \frac {-6-2 x+e^2 \left (9 x+20 x^2+5 x^3\right )+\left (2+e^2 \left (-6 x-6 x^2\right )\right ) \log \left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )+e^2 x \log ^2\left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )}{e^2 x} \, dx=\frac {4 x^{3} e^{4} + x^{2} \left (- 8 e^{4} \log {\left (2 \right )} + 17 e^{4}\right ) + x \left (- 16 e^{4} \log {\left (2 \right )} + 4 e^{2} + 4 e^{4} \log {\left (2 \right )}^{2} + 16 e^{4}\right ) + \frac {1}{x}}{e^{4}} \] Input:
integrate((x*exp(2)*ln(4*exp(((-x**2-x)*exp(2)-1)/exp(2)/x))**2+((-6*x**2- 6*x)*exp(2)+2)*ln(4*exp(((-x**2-x)*exp(2)-1)/exp(2)/x))+(5*x**3+20*x**2+9* x)*exp(2)-2*x-6)/exp(2)/x,x)
Output:
(4*x**3*exp(4) + x**2*(-8*exp(4)*log(2) + 17*exp(4)) + x*(-16*exp(4)*log(2 ) + 4*exp(2) + 4*exp(4)*log(2)**2 + 16*exp(4)) + 1/x)*exp(-4)
Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (30) = 60\).
Time = 0.05 (sec) , antiderivative size = 220, normalized size of antiderivative = 6.88 \[ \int \frac {-6-2 x+e^2 \left (9 x+20 x^2+5 x^3\right )+\left (2+e^2 \left (-6 x-6 x^2\right )\right ) \log \left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )+e^2 x \log ^2\left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )}{e^2 x} \, dx=\frac {1}{3} \, {\left (2 \, x^{3} e^{2} - 9 \, x^{2} e^{2} \log \left (4 \, e^{\left (-x - \frac {e^{\left (-2\right )}}{x} - 1\right )}\right ) + 3 \, x e^{2} \log \left (4 \, e^{\left (-x - \frac {e^{\left (-2\right )}}{x} - 1\right )}\right )^{2} + 30 \, x^{2} e^{2} - 18 \, x e^{2} \log \left (4 \, e^{\left (-x - \frac {e^{\left (-2\right )}}{x} - 1\right )}\right ) - 9 \, {\left (x^{2} - e^{\left (-2\right )} \log \left (x^{2}\right )\right )} e^{2} + {\left (3 \, {\left (x^{2} - e^{\left (-2\right )} \log \left (x^{2}\right )\right )} \log \left (4 \, e^{\left (-x - \frac {e^{\left (-2\right )}}{x} - 1\right )}\right ) + \frac {{\left (x^{4} e^{4} + 3 \, x^{2} e^{2} - 6 \, {\left (x^{2} e^{2} + 1\right )} \log \left (x\right ) - 6\right )} e^{\left (-4\right )}}{x}\right )} e^{2} + 27 \, x e^{2} + 6 \, {\left (x + \frac {e^{\left (-2\right )}}{x}\right )} \log \left (x\right ) + 6 \, \log \left (x\right ) \log \left (4 \, e^{\left (-x - \frac {e^{\left (-2\right )}}{x} - 1\right )}\right ) - 3 \, x + \frac {6 \, e^{\left (-2\right )}}{x} - 18 \, \log \left (x\right )\right )} e^{\left (-2\right )} \] Input:
integrate((x*exp(2)*log(4*exp(((-x^2-x)*exp(2)-1)/exp(2)/x))^2+((-6*x^2-6* x)*exp(2)+2)*log(4*exp(((-x^2-x)*exp(2)-1)/exp(2)/x))+(5*x^3+20*x^2+9*x)*e xp(2)-2*x-6)/exp(2)/x,x, algorithm="maxima")
Output:
1/3*(2*x^3*e^2 - 9*x^2*e^2*log(4*e^(-x - e^(-2)/x - 1)) + 3*x*e^2*log(4*e^ (-x - e^(-2)/x - 1))^2 + 30*x^2*e^2 - 18*x*e^2*log(4*e^(-x - e^(-2)/x - 1) ) - 9*(x^2 - e^(-2)*log(x^2))*e^2 + (3*(x^2 - e^(-2)*log(x^2))*log(4*e^(-x - e^(-2)/x - 1)) + (x^4*e^4 + 3*x^2*e^2 - 6*(x^2*e^2 + 1)*log(x) - 6)*e^( -4)/x)*e^2 + 27*x*e^2 + 6*(x + e^(-2)/x)*log(x) + 6*log(x)*log(4*e^(-x - e ^(-2)/x - 1)) - 3*x + 6*e^(-2)/x - 18*log(x))*e^(-2)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (30) = 60\).
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.97 \[ \int \frac {-6-2 x+e^2 \left (9 x+20 x^2+5 x^3\right )+\left (2+e^2 \left (-6 x-6 x^2\right )\right ) \log \left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )+e^2 x \log ^2\left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )}{e^2 x} \, dx={\left ({\left (4 \, x^{3} e^{14} - 8 \, x^{2} e^{14} \log \left (2\right ) + 4 \, x e^{14} \log \left (2\right )^{2} + 17 \, x^{2} e^{14} - 16 \, x e^{14} \log \left (2\right ) + 16 \, x e^{14} + 4 \, x e^{12}\right )} e^{\left (-12\right )} + \frac {e^{\left (-2\right )}}{x}\right )} e^{\left (-2\right )} \] Input:
integrate((x*exp(2)*log(4*exp(((-x^2-x)*exp(2)-1)/exp(2)/x))^2+((-6*x^2-6* x)*exp(2)+2)*log(4*exp(((-x^2-x)*exp(2)-1)/exp(2)/x))+(5*x^3+20*x^2+9*x)*e xp(2)-2*x-6)/exp(2)/x,x, algorithm="giac")
Output:
((4*x^3*e^14 - 8*x^2*e^14*log(2) + 4*x*e^14*log(2)^2 + 17*x^2*e^14 - 16*x* e^14*log(2) + 16*x*e^14 + 4*x*e^12)*e^(-12) + e^(-2)/x)*e^(-2)
Time = 3.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {-6-2 x+e^2 \left (9 x+20 x^2+5 x^3\right )+\left (2+e^2 \left (-6 x-6 x^2\right )\right ) \log \left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )+e^2 x \log ^2\left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )}{e^2 x} \, dx=\frac {{\mathrm {e}}^{-4}}{x}-x^2\,\left (8\,\ln \left (2\right )-17\right )+4\,x^3+x\,\left (4\,{\mathrm {e}}^{-2}-16\,\ln \left (2\right )+4\,{\ln \left (2\right )}^2+16\right ) \] Input:
int(-(exp(-2)*(2*x - exp(2)*(9*x + 20*x^2 + 5*x^3) + log(4*exp(-(exp(-2)*( exp(2)*(x + x^2) + 1))/x))*(exp(2)*(6*x + 6*x^2) - 2) - x*exp(2)*log(4*exp (-(exp(-2)*(exp(2)*(x + x^2) + 1))/x))^2 + 6))/x,x)
Output:
exp(-4)/x - x^2*(8*log(2) - 17) + 4*x^3 + x*(4*exp(-2) - 16*log(2) + 4*log (2)^2 + 16)
\[ \int \frac {-6-2 x+e^2 \left (9 x+20 x^2+5 x^3\right )+\left (2+e^2 \left (-6 x-6 x^2\right )\right ) \log \left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )+e^2 x \log ^2\left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )}{e^2 x} \, dx=\frac {3 \left (\int \mathrm {log}\left (\frac {4}{e^{\frac {e^{2} x^{2}+1}{e^{2} x}} e}\right )^{2}d x \right ) e^{2}-18 \left (\int \mathrm {log}\left (\frac {4}{e^{\frac {e^{2} x^{2}+1}{e^{2} x}} e}\right )d x \right ) e^{2}+6 \left (\int \frac {\mathrm {log}\left (\frac {4}{e^{\frac {e^{2} x^{2}+1}{e^{2} x}} e}\right )}{x}d x \right )-18 \left (\int \mathrm {log}\left (\frac {4}{e^{\frac {e^{2} x^{2}+1}{e^{2} x}} e}\right ) x d x \right ) e^{2}-18 \,\mathrm {log}\left (x \right )+5 e^{2} x^{3}+30 e^{2} x^{2}+27 e^{2} x -6 x}{3 e^{2}} \] Input:
int((x*exp(2)*log(4*exp(((-x^2-x)*exp(2)-1)/exp(2)/x))^2+((-6*x^2-6*x)*exp (2)+2)*log(4*exp(((-x^2-x)*exp(2)-1)/exp(2)/x))+(5*x^3+20*x^2+9*x)*exp(2)- 2*x-6)/exp(2)/x,x)
Output:
(3*int(log(4/(e**((e**2*x**2 + 1)/(e**2*x))*e))**2,x)*e**2 - 18*int(log(4/ (e**((e**2*x**2 + 1)/(e**2*x))*e)),x)*e**2 + 6*int(log(4/(e**((e**2*x**2 + 1)/(e**2*x))*e))/x,x) - 18*int(log(4/(e**((e**2*x**2 + 1)/(e**2*x))*e))*x ,x)*e**2 - 18*log(x) + 5*e**2*x**3 + 30*e**2*x**2 + 27*e**2*x - 6*x)/(3*e* *2)