Integrand size = 132, antiderivative size = 24 \[ \int e^{-2 x} \left (6 x^5-2 x^6+e^x \left (-6 x^2-6 x^3+2 x^4+10 e^{-2+x} x^4+12 x^5-2 x^6\right )+e^{2 x} \left (2+2 x-6 x^2-8 x^3+6 x^5+e^{-4+2 x} \left (4 x^3+2 x^4\right )+e^{-2+x} \left (-4 x-8 x^2-2 x^3+10 x^4+2 x^5\right )\right )\right ) \, dx=\left (1+x-x^2 \left (e^{-2+x}+x+e^{-x} x\right )\right )^2 \] Output:
(1+x-(exp(-2+x)+x/exp(x)+x)*x^2)^2
Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(24)=48\).
Time = 0.47 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.50 \[ \int e^{-2 x} \left (6 x^5-2 x^6+e^x \left (-6 x^2-6 x^3+2 x^4+10 e^{-2+x} x^4+12 x^5-2 x^6\right )+e^{2 x} \left (2+2 x-6 x^2-8 x^3+6 x^5+e^{-4+2 x} \left (4 x^3+2 x^4\right )+e^{-2+x} \left (-4 x-8 x^2-2 x^3+10 x^4+2 x^5\right )\right )\right ) \, dx=2 \left (x+\frac {x^2}{2}-x^3-x^4+\frac {1}{2} e^{-4+2 x} x^4+\frac {x^5}{e^2}+\frac {x^6}{2}+\frac {1}{2} e^{-2 x} x^6+e^x \left (-\frac {x^2}{e^2}-\frac {x^3}{e^2}+\frac {x^5}{e^2}\right )-e^{-x} \left (x^3+x^4-x^6\right )\right ) \] Input:
Integrate[(6*x^5 - 2*x^6 + E^x*(-6*x^2 - 6*x^3 + 2*x^4 + 10*E^(-2 + x)*x^4 + 12*x^5 - 2*x^6) + E^(2*x)*(2 + 2*x - 6*x^2 - 8*x^3 + 6*x^5 + E^(-4 + 2* x)*(4*x^3 + 2*x^4) + E^(-2 + x)*(-4*x - 8*x^2 - 2*x^3 + 10*x^4 + 2*x^5)))/ E^(2*x),x]
Output:
2*(x + x^2/2 - x^3 - x^4 + (E^(-4 + 2*x)*x^4)/2 + x^5/E^2 + x^6/2 + x^6/(2 *E^(2*x)) + E^x*(-(x^2/E^2) - x^3/E^2 + x^5/E^2) - (x^3 + x^4 - x^6)/E^x)
Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(24)=48\).
Time = 1.97 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.50, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {7293, 2009}
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.
| \(\displaystyle \int e^{-2 x} \left (-2 x^6+6 x^5+e^{2 x} \left (6 x^5-8 x^3-6 x^2+e^{2 x-4} \left (2 x^4+4 x^3\right )+e^{x-2} \left (2 x^5+10 x^4-2 x^3-8 x^2-4 x\right )+2 x+2\right )+e^x \left (-2 x^6+12 x^5+10 e^{x-2} x^4+2 x^4-6 x^3-6 x^2\right )\right ) \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-2 e^{-2 x} x^6+6 e^{-2 x} x^5+2 e^{-x-2} \left (-e^2 x^4+6 e^2 x^3+5 e^x x^2+e^2 x^2-3 e^2 x-3 e^2\right ) x^2+\frac {2 \left (e^{x+2} x^5+3 e^4 x^5+e^{2 x} x^4+5 e^{x+2} x^4+2 e^{2 x} x^3-e^{x+2} x^3-4 e^4 x^3-4 e^{x+2} x^2-3 e^4 x^2-2 e^{x+2} x+e^4 x+e^4\right )}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle e^{-2 x} x^6+2 e^{-x} x^6+x^6+2 e^{x-2} x^5+\frac {2 x^5}{e^2}-2 e^{-x} x^4+e^{2 x-4} x^4-2 x^4-2 e^{x-2} x^3-2 e^{-x} x^3-2 x^3-2 e^{x-2} x^2+x^2+2 x\) |
Input:
Int[(6*x^5 - 2*x^6 + E^x*(-6*x^2 - 6*x^3 + 2*x^4 + 10*E^(-2 + x)*x^4 + 12* x^5 - 2*x^6) + E^(2*x)*(2 + 2*x - 6*x^2 - 8*x^3 + 6*x^5 + E^(-4 + 2*x)*(4* x^3 + 2*x^4) + E^(-2 + x)*(-4*x - 8*x^2 - 2*x^3 + 10*x^4 + 2*x^5)))/E^(2*x ),x]
Output:
2*x + x^2 - 2*E^(-2 + x)*x^2 - 2*x^3 - 2*E^(-2 + x)*x^3 - (2*x^3)/E^x - 2* x^4 - (2*x^4)/E^x + E^(-4 + 2*x)*x^4 + (2*x^5)/E^2 + 2*E^(-2 + x)*x^5 + x^ 6 + x^6/E^(2*x) + (2*x^6)/E^x
Leaf count of result is larger than twice the leaf count of optimal. \(93\) vs. \(2(22)=44\).
Time = 12.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.92
| method | result | size |
| risch | \({\mathrm e}^{-4} {\mathrm e}^{2 x} x^{4}+2 \,{\mathrm e}^{-2} x^{5}+x^{6}-2 x^{4}-2 x^{3}+x^{2}+2 x +\left (2 \,{\mathrm e}^{-2} x^{5}-2 \,{\mathrm e}^{-2} x^{3}-2 \,{\mathrm e}^{-2} x^{2}\right ) {\mathrm e}^{x}+\left (2 x^{6}-2 x^{4}-2 x^{3}\right ) {\mathrm e}^{-x}+{\mathrm e}^{-2 x} x^{6}\) | \(94\) |
| parallelrisch | \(-\left (-{\mathrm e}^{2 x} x^{6}-2 \,{\mathrm e}^{2 x} {\mathrm e}^{-2+x} x^{5}-{\mathrm e}^{2 x} {\mathrm e}^{2 x -4} x^{4}-2 x^{6} {\mathrm e}^{x}-2 \,{\mathrm e}^{-2+x} {\mathrm e}^{x} x^{5}-x^{6}+2 \,{\mathrm e}^{2 x} x^{4}+2 \,{\mathrm e}^{2 x} {\mathrm e}^{-2+x} x^{3}+2 \,{\mathrm e}^{x} x^{4}+2 \,{\mathrm e}^{2 x} x^{3}+2 \,{\mathrm e}^{2 x} {\mathrm e}^{-2+x} x^{2}+2 \,{\mathrm e}^{x} x^{3}-{\mathrm e}^{2 x} x^{2}-2 \ln \left ({\mathrm e}^{x}\right ) {\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x}\) | \(144\) |
| parts | \(2 x -2 \,{\mathrm e}^{-x} x^{4}-2 \,{\mathrm e}^{-x} x^{3}+2 \,{\mathrm e}^{-2} x^{5}+2 \,{\mathrm e}^{-x} x^{6}+\left (-2+x \right )^{4} {\mathrm e}^{2 x -4}+8 \,{\mathrm e}^{2 x -4} \left (-2+x \right )^{3}+24 \left (-2+x \right )^{2} {\mathrm e}^{2 x -4}+32 \,{\mathrm e}^{2 x -4} \left (-2+x \right )+16 \,{\mathrm e}^{2 x -4}+{\mathrm e}^{-2 x} x^{6}+2 \left (-2+x \right )^{5} {\mathrm e}^{-2+x}+20 \left (-2+x \right )^{4} {\mathrm e}^{-2+x}+78 \,{\mathrm e}^{-2+x} \left (-2+x \right )^{3}+146 \,{\mathrm e}^{-2+x} \left (-2+x \right )^{2}+128 \,{\mathrm e}^{-2+x} \left (-2+x \right )+40 \,{\mathrm e}^{-2+x}+x^{2}-2 x^{3}-2 x^{4}+x^{6}\) | \(179\) |
| default | \(x^{6}-2 x^{4}-2 x^{3}+x^{2}+2 x +2 \,{\mathrm e}^{-2} x^{5}+{\mathrm e}^{-2 x} x^{6}-2 \,{\mathrm e}^{-x} x^{3}-2 \,{\mathrm e}^{-x} x^{4}+2 \,{\mathrm e}^{-x} x^{6}-4 \,{\mathrm e}^{-2} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )-8 \,{\mathrm e}^{-2} \left ({\mathrm e}^{x} x^{2}-2 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}\right )-2 \,{\mathrm e}^{-2} \left ({\mathrm e}^{x} x^{3}-3 \,{\mathrm e}^{x} x^{2}+6 \,{\mathrm e}^{x} x -6 \,{\mathrm e}^{x}\right )+10 \,{\mathrm e}^{-2} \left ({\mathrm e}^{x} x^{4}-4 \,{\mathrm e}^{x} x^{3}+12 \,{\mathrm e}^{x} x^{2}-24 \,{\mathrm e}^{x} x +24 \,{\mathrm e}^{x}\right )+2 \,{\mathrm e}^{-2} \left (x^{5} {\mathrm e}^{x}-5 \,{\mathrm e}^{x} x^{4}+20 \,{\mathrm e}^{x} x^{3}-60 \,{\mathrm e}^{x} x^{2}+120 \,{\mathrm e}^{x} x -120 \,{\mathrm e}^{x}\right )+4 \,{\mathrm e}^{-4} \left (\frac {{\mathrm e}^{2 x} x^{3}}{2}-\frac {3 \,{\mathrm e}^{2 x} x^{2}}{4}+\frac {3 x \,{\mathrm e}^{2 x}}{4}-\frac {3 \,{\mathrm e}^{2 x}}{8}\right )+2 \,{\mathrm e}^{-4} \left (\frac {{\mathrm e}^{2 x} x^{4}}{2}-{\mathrm e}^{2 x} x^{3}+\frac {3 \,{\mathrm e}^{2 x} x^{2}}{2}-\frac {3 x \,{\mathrm e}^{2 x}}{2}+\frac {3 \,{\mathrm e}^{2 x}}{4}\right )\) | \(283\) |
Input:
int((((2*x^4+4*x^3)*exp(-2+x)^2+(2*x^5+10*x^4-2*x^3-8*x^2-4*x)*exp(-2+x)+6 *x^5-8*x^3-6*x^2+2*x+2)*exp(x)^2+(10*x^4*exp(-2+x)-2*x^6+12*x^5+2*x^4-6*x^ 3-6*x^2)*exp(x)-2*x^6+6*x^5)/exp(x)^2,x,method=_RETURNVERBOSE)
Output:
exp(-2)^2*exp(x)^2*x^4+2*exp(-2)*x^5+x^6-2*x^4-2*x^3+x^2+2*x+(2*exp(-2)*x^ 5-2*exp(-2)*x^3-2*exp(-2)*x^2)*exp(x)+(2*x^6-2*x^4-2*x^3)/exp(x)+1/exp(x)^ 2*x^6
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (23) = 46\).
Time = 0.08 (sec) , antiderivative size = 100, normalized size of antiderivative = 4.17 \[ \int e^{-2 x} \left (6 x^5-2 x^6+e^x \left (-6 x^2-6 x^3+2 x^4+10 e^{-2+x} x^4+12 x^5-2 x^6\right )+e^{2 x} \left (2+2 x-6 x^2-8 x^3+6 x^5+e^{-4+2 x} \left (4 x^3+2 x^4\right )+e^{-2+x} \left (-4 x-8 x^2-2 x^3+10 x^4+2 x^5\right )\right )\right ) \, dx={\left (x^{6} e^{4} + x^{4} e^{\left (4 \, x\right )} + {\left (2 \, x^{5} e^{2} + {\left (x^{6} - 2 \, x^{4} - 2 \, x^{3} + x^{2} + 2 \, x\right )} e^{4}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{5} - x^{3} - x^{2}\right )} e^{\left (3 \, x + 2\right )} + 2 \, {\left (x^{6} - x^{4} - x^{3}\right )} e^{\left (x + 4\right )}\right )} e^{\left (-2 \, x - 4\right )} \] Input:
integrate((((2*x^4+4*x^3)*exp(-2+x)^2+(2*x^5+10*x^4-2*x^3-8*x^2-4*x)*exp(- 2+x)+6*x^5-8*x^3-6*x^2+2*x+2)*exp(x)^2+(10*x^4*exp(-2+x)-2*x^6+12*x^5+2*x^ 4-6*x^3-6*x^2)*exp(x)-2*x^6+6*x^5)/exp(x)^2,x, algorithm="fricas")
Output:
(x^6*e^4 + x^4*e^(4*x) + (2*x^5*e^2 + (x^6 - 2*x^4 - 2*x^3 + x^2 + 2*x)*e^ 4)*e^(2*x) + 2*(x^5 - x^3 - x^2)*e^(3*x + 2) + 2*(x^6 - x^4 - x^3)*e^(x + 4))*e^(-2*x - 4)
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (19) = 38\).
Time = 0.19 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.67 \[ \int e^{-2 x} \left (6 x^5-2 x^6+e^x \left (-6 x^2-6 x^3+2 x^4+10 e^{-2+x} x^4+12 x^5-2 x^6\right )+e^{2 x} \left (2+2 x-6 x^2-8 x^3+6 x^5+e^{-4+2 x} \left (4 x^3+2 x^4\right )+e^{-2+x} \left (-4 x-8 x^2-2 x^3+10 x^4+2 x^5\right )\right )\right ) \, dx=x^{6} + \frac {2 x^{5}}{e^{2}} - 2 x^{4} - 2 x^{3} + x^{2} + 2 x + \frac {x^{6} e^{6} e^{- 2 x} + x^{4} e^{2} e^{2 x} + \left (2 x^{5} e^{4} - 2 x^{3} e^{4} - 2 x^{2} e^{4}\right ) e^{x} + \left (2 x^{6} e^{6} - 2 x^{4} e^{6} - 2 x^{3} e^{6}\right ) e^{- x}}{e^{6}} \] Input:
integrate((((2*x**4+4*x**3)*exp(-2+x)**2+(2*x**5+10*x**4-2*x**3-8*x**2-4*x )*exp(-2+x)+6*x**5-8*x**3-6*x**2+2*x+2)*exp(x)**2+(10*x**4*exp(-2+x)-2*x** 6+12*x**5+2*x**4-6*x**3-6*x**2)*exp(x)-2*x**6+6*x**5)/exp(x)**2,x)
Output:
x**6 + 2*x**5*exp(-2) - 2*x**4 - 2*x**3 + x**2 + 2*x + (x**6*exp(6)*exp(-2 *x) + x**4*exp(2)*exp(2*x) + (2*x**5*exp(4) - 2*x**3*exp(4) - 2*x**2*exp(4 ))*exp(x) + (2*x**6*exp(6) - 2*x**4*exp(6) - 2*x**3*exp(6))*exp(-x))*exp(- 6)
Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (23) = 46\).
Time = 0.05 (sec) , antiderivative size = 360, normalized size of antiderivative = 15.00 \[ \int e^{-2 x} \left (6 x^5-2 x^6+e^x \left (-6 x^2-6 x^3+2 x^4+10 e^{-2+x} x^4+12 x^5-2 x^6\right )+e^{2 x} \left (2+2 x-6 x^2-8 x^3+6 x^5+e^{-4+2 x} \left (4 x^3+2 x^4\right )+e^{-2+x} \left (-4 x-8 x^2-2 x^3+10 x^4+2 x^5\right )\right )\right ) \, dx=x^{6} + 2 \, x^{5} e^{\left (-2\right )} - 2 \, x^{4} - 2 \, x^{3} + x^{2} + 2 \, {\left (x^{6} + 6 \, x^{5} + 30 \, x^{4} + 120 \, x^{3} + 360 \, x^{2} + 720 \, x + 720\right )} e^{\left (-x\right )} - 12 \, {\left (x^{5} + 5 \, x^{4} + 20 \, x^{3} + 60 \, x^{2} + 120 \, x + 120\right )} e^{\left (-x\right )} - 2 \, {\left (x^{4} + 4 \, x^{3} + 12 \, x^{2} + 24 \, x + 24\right )} e^{\left (-x\right )} + 6 \, {\left (x^{3} + 3 \, x^{2} + 6 \, x + 6\right )} e^{\left (-x\right )} + 6 \, {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} + \frac {1}{4} \, {\left (4 \, x^{6} + 12 \, x^{5} + 30 \, x^{4} + 60 \, x^{3} + 90 \, x^{2} + 90 \, x + 45\right )} e^{\left (-2 \, x\right )} - \frac {3}{4} \, {\left (4 \, x^{5} + 10 \, x^{4} + 20 \, x^{3} + 30 \, x^{2} + 30 \, x + 15\right )} e^{\left (-2 \, x\right )} + \frac {1}{2} \, {\left (2 \, x^{4} - 4 \, x^{3} + 6 \, x^{2} - 6 \, x + 3\right )} e^{\left (2 \, x - 4\right )} + \frac {1}{2} \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x - 4\right )} + 2 \, {\left (x^{5} - 5 \, x^{4} + 20 \, x^{3} - 60 \, x^{2} + 120 \, x - 120\right )} e^{\left (x - 2\right )} + 10 \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{\left (x - 2\right )} - 2 \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{\left (x - 2\right )} - 8 \, {\left (x^{2} - 2 \, x + 2\right )} e^{\left (x - 2\right )} - 4 \, {\left (x - 1\right )} e^{\left (x - 2\right )} + 2 \, x \] Input:
integrate((((2*x^4+4*x^3)*exp(-2+x)^2+(2*x^5+10*x^4-2*x^3-8*x^2-4*x)*exp(- 2+x)+6*x^5-8*x^3-6*x^2+2*x+2)*exp(x)^2+(10*x^4*exp(-2+x)-2*x^6+12*x^5+2*x^ 4-6*x^3-6*x^2)*exp(x)-2*x^6+6*x^5)/exp(x)^2,x, algorithm="maxima")
Output:
x^6 + 2*x^5*e^(-2) - 2*x^4 - 2*x^3 + x^2 + 2*(x^6 + 6*x^5 + 30*x^4 + 120*x ^3 + 360*x^2 + 720*x + 720)*e^(-x) - 12*(x^5 + 5*x^4 + 20*x^3 + 60*x^2 + 1 20*x + 120)*e^(-x) - 2*(x^4 + 4*x^3 + 12*x^2 + 24*x + 24)*e^(-x) + 6*(x^3 + 3*x^2 + 6*x + 6)*e^(-x) + 6*(x^2 + 2*x + 2)*e^(-x) + 1/4*(4*x^6 + 12*x^5 + 30*x^4 + 60*x^3 + 90*x^2 + 90*x + 45)*e^(-2*x) - 3/4*(4*x^5 + 10*x^4 + 20*x^3 + 30*x^2 + 30*x + 15)*e^(-2*x) + 1/2*(2*x^4 - 4*x^3 + 6*x^2 - 6*x + 3)*e^(2*x - 4) + 1/2*(4*x^3 - 6*x^2 + 6*x - 3)*e^(2*x - 4) + 2*(x^5 - 5*x ^4 + 20*x^3 - 60*x^2 + 120*x - 120)*e^(x - 2) + 10*(x^4 - 4*x^3 + 12*x^2 - 24*x + 24)*e^(x - 2) - 2*(x^3 - 3*x^2 + 6*x - 6)*e^(x - 2) - 8*(x^2 - 2*x + 2)*e^(x - 2) - 4*(x - 1)*e^(x - 2) + 2*x
Leaf count of result is larger than twice the leaf count of optimal. 371 vs. \(2 (23) = 46\).
Time = 0.12 (sec) , antiderivative size = 371, normalized size of antiderivative = 15.46 \[ \int e^{-2 x} \left (6 x^5-2 x^6+e^x \left (-6 x^2-6 x^3+2 x^4+10 e^{-2+x} x^4+12 x^5-2 x^6\right )+e^{2 x} \left (2+2 x-6 x^2-8 x^3+6 x^5+e^{-4+2 x} \left (4 x^3+2 x^4\right )+e^{-2+x} \left (-4 x-8 x^2-2 x^3+10 x^4+2 x^5\right )\right )\right ) \, dx={\left ({\left (x - 2\right )}^{6} e^{4} + 2 \, {\left (x - 2\right )}^{6} e^{\left (-x + 4\right )} + {\left (x - 2\right )}^{6} e^{\left (-2 \, x + 4\right )} + 12 \, {\left (x - 2\right )}^{5} e^{4} + 2 \, {\left (x - 2\right )}^{5} e^{2} + 2 \, {\left (x - 2\right )}^{5} e^{\left (x + 2\right )} + 24 \, {\left (x - 2\right )}^{5} e^{\left (-x + 4\right )} + 12 \, {\left (x - 2\right )}^{5} e^{\left (-2 \, x + 4\right )} + 58 \, {\left (x - 2\right )}^{4} e^{4} + 20 \, {\left (x - 2\right )}^{4} e^{2} + {\left (x - 2\right )}^{4} e^{\left (2 \, x\right )} + 20 \, {\left (x - 2\right )}^{4} e^{\left (x + 2\right )} + 118 \, {\left (x - 2\right )}^{4} e^{\left (-x + 4\right )} + 60 \, {\left (x - 2\right )}^{4} e^{\left (-2 \, x + 4\right )} + 142 \, {\left (x - 2\right )}^{3} e^{4} + 80 \, {\left (x - 2\right )}^{3} e^{2} + 8 \, {\left (x - 2\right )}^{3} e^{\left (2 \, x\right )} + 78 \, {\left (x - 2\right )}^{3} e^{\left (x + 2\right )} + 302 \, {\left (x - 2\right )}^{3} e^{\left (-x + 4\right )} + 160 \, {\left (x - 2\right )}^{3} e^{\left (-2 \, x + 4\right )} + 181 \, {\left (x - 2\right )}^{2} e^{4} + 160 \, {\left (x - 2\right )}^{2} e^{2} + 24 \, {\left (x - 2\right )}^{2} e^{\left (2 \, x\right )} + 146 \, {\left (x - 2\right )}^{2} e^{\left (x + 2\right )} + 420 \, {\left (x - 2\right )}^{2} e^{\left (-x + 4\right )} + 240 \, {\left (x - 2\right )}^{2} e^{\left (-2 \, x + 4\right )} + 110 \, {\left (x - 2\right )} e^{4} + 160 \, {\left (x - 2\right )} e^{2} + 32 \, {\left (x - 2\right )} e^{\left (2 \, x\right )} + 128 \, {\left (x - 2\right )} e^{\left (x + 2\right )} + 296 \, {\left (x - 2\right )} e^{\left (-x + 4\right )} + 192 \, {\left (x - 2\right )} e^{\left (-2 \, x + 4\right )} + 16 \, e^{\left (2 \, x\right )} + 40 \, e^{\left (x + 2\right )} + 80 \, e^{\left (-x + 4\right )} + 64 \, e^{\left (-2 \, x + 4\right )}\right )} e^{\left (-4\right )} \] Input:
integrate((((2*x^4+4*x^3)*exp(-2+x)^2+(2*x^5+10*x^4-2*x^3-8*x^2-4*x)*exp(- 2+x)+6*x^5-8*x^3-6*x^2+2*x+2)*exp(x)^2+(10*x^4*exp(-2+x)-2*x^6+12*x^5+2*x^ 4-6*x^3-6*x^2)*exp(x)-2*x^6+6*x^5)/exp(x)^2,x, algorithm="giac")
Output:
((x - 2)^6*e^4 + 2*(x - 2)^6*e^(-x + 4) + (x - 2)^6*e^(-2*x + 4) + 12*(x - 2)^5*e^4 + 2*(x - 2)^5*e^2 + 2*(x - 2)^5*e^(x + 2) + 24*(x - 2)^5*e^(-x + 4) + 12*(x - 2)^5*e^(-2*x + 4) + 58*(x - 2)^4*e^4 + 20*(x - 2)^4*e^2 + (x - 2)^4*e^(2*x) + 20*(x - 2)^4*e^(x + 2) + 118*(x - 2)^4*e^(-x + 4) + 60*( x - 2)^4*e^(-2*x + 4) + 142*(x - 2)^3*e^4 + 80*(x - 2)^3*e^2 + 8*(x - 2)^3 *e^(2*x) + 78*(x - 2)^3*e^(x + 2) + 302*(x - 2)^3*e^(-x + 4) + 160*(x - 2) ^3*e^(-2*x + 4) + 181*(x - 2)^2*e^4 + 160*(x - 2)^2*e^2 + 24*(x - 2)^2*e^( 2*x) + 146*(x - 2)^2*e^(x + 2) + 420*(x - 2)^2*e^(-x + 4) + 240*(x - 2)^2* e^(-2*x + 4) + 110*(x - 2)*e^4 + 160*(x - 2)*e^2 + 32*(x - 2)*e^(2*x) + 12 8*(x - 2)*e^(x + 2) + 296*(x - 2)*e^(-x + 4) + 192*(x - 2)*e^(-2*x + 4) + 16*e^(2*x) + 40*e^(x + 2) + 80*e^(-x + 4) + 64*e^(-2*x + 4))*e^(-4)
Time = 0.41 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.88 \[ \int e^{-2 x} \left (6 x^5-2 x^6+e^x \left (-6 x^2-6 x^3+2 x^4+10 e^{-2+x} x^4+12 x^5-2 x^6\right )+e^{2 x} \left (2+2 x-6 x^2-8 x^3+6 x^5+e^{-4+2 x} \left (4 x^3+2 x^4\right )+e^{-2+x} \left (-4 x-8 x^2-2 x^3+10 x^4+2 x^5\right )\right )\right ) \, dx=2\,x-{\mathrm {e}}^x\,\left (-2\,{\mathrm {e}}^{-2}\,x^5+2\,{\mathrm {e}}^{-2}\,x^3+2\,{\mathrm {e}}^{-2}\,x^2\right )+x^6\,{\mathrm {e}}^{-2\,x}+2\,x^5\,{\mathrm {e}}^{-2}-{\mathrm {e}}^{-x}\,\left (-2\,x^6+2\,x^4+2\,x^3\right )+x^4\,{\mathrm {e}}^{2\,x-4}+x^2-2\,x^3-2\,x^4+x^6 \] Input:
int(exp(-2*x)*(exp(x)*(10*x^4*exp(x - 2) - 6*x^2 - 6*x^3 + 2*x^4 + 12*x^5 - 2*x^6) + 6*x^5 - 2*x^6 + exp(2*x)*(2*x - exp(x - 2)*(4*x + 8*x^2 + 2*x^3 - 10*x^4 - 2*x^5) + exp(2*x - 4)*(4*x^3 + 2*x^4) - 6*x^2 - 8*x^3 + 6*x^5 + 2)),x)
Output:
2*x - exp(x)*(2*x^2*exp(-2) + 2*x^3*exp(-2) - 2*x^5*exp(-2)) + x^6*exp(-2* x) + 2*x^5*exp(-2) - exp(-x)*(2*x^3 + 2*x^4 - 2*x^6) + x^4*exp(2*x - 4) + x^2 - 2*x^3 - 2*x^4 + x^6
Time = 0.19 (sec) , antiderivative size = 170, normalized size of antiderivative = 7.08 \[ \int e^{-2 x} \left (6 x^5-2 x^6+e^x \left (-6 x^2-6 x^3+2 x^4+10 e^{-2+x} x^4+12 x^5-2 x^6\right )+e^{2 x} \left (2+2 x-6 x^2-8 x^3+6 x^5+e^{-4+2 x} \left (4 x^3+2 x^4\right )+e^{-2+x} \left (-4 x-8 x^2-2 x^3+10 x^4+2 x^5\right )\right )\right ) \, dx=\frac {x \left (e^{4 x} x^{3}+2 e^{3 x} e^{2} x^{4}-2 e^{3 x} e^{2} x^{2}-2 e^{3 x} e^{2} x +e^{2 x} e^{4} x^{5}-2 e^{2 x} e^{4} x^{3}-2 e^{2 x} e^{4} x^{2}+e^{2 x} e^{4} x +2 e^{2 x} e^{4}+2 e^{2 x} e^{2} x^{4}+2 e^{x} e^{4} x^{5}-2 e^{x} e^{4} x^{3}-2 e^{x} e^{4} x^{2}+e^{4} x^{5}\right )}{e^{2 x} e^{4}} \] Input:
int((((2*x^4+4*x^3)*exp(-2+x)^2+(2*x^5+10*x^4-2*x^3-8*x^2-4*x)*exp(-2+x)+6 *x^5-8*x^3-6*x^2+2*x+2)*exp(x)^2+(10*x^4*exp(-2+x)-2*x^6+12*x^5+2*x^4-6*x^ 3-6*x^2)*exp(x)-2*x^6+6*x^5)/exp(x)^2,x)
Output:
(x*(e**(4*x)*x**3 + 2*e**(3*x)*e**2*x**4 - 2*e**(3*x)*e**2*x**2 - 2*e**(3* x)*e**2*x + e**(2*x)*e**4*x**5 - 2*e**(2*x)*e**4*x**3 - 2*e**(2*x)*e**4*x* *2 + e**(2*x)*e**4*x + 2*e**(2*x)*e**4 + 2*e**(2*x)*e**2*x**4 + 2*e**x*e** 4*x**5 - 2*e**x*e**4*x**3 - 2*e**x*e**4*x**2 + e**4*x**5))/(e**(2*x)*e**4)