Integrand size = 75, antiderivative size = 27 \[ \int \frac {e^{-x} \left (8-1016 x-257 x^2+112 x^3-8 x^4+e^{1+x} \left (16-2048 x+512 x^2-32 x^3\right )\right )}{2 x^2-256 x^3+8224 x^4-2048 x^5+128 x^6} \, dx=\frac {e+\frac {e^{-x}}{2}}{\frac {x}{-8+x}+8 x^2} \] Output:
(1/2/exp(x)+exp(1))/(8*x^2+x/(-8+x))
Time = 6.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {e^{-x} \left (8-1016 x-257 x^2+112 x^3-8 x^4+e^{1+x} \left (16-2048 x+512 x^2-32 x^3\right )\right )}{2 x^2-256 x^3+8224 x^4-2048 x^5+128 x^6} \, dx=\frac {e^{-x} \left (1+2 e^{1+x}\right ) (-8+x)}{2 x \left (1-64 x+8 x^2\right )} \] Input:
Integrate[(8 - 1016*x - 257*x^2 + 112*x^3 - 8*x^4 + E^(1 + x)*(16 - 2048*x + 512*x^2 - 32*x^3))/(E^x*(2*x^2 - 256*x^3 + 8224*x^4 - 2048*x^5 + 128*x^ 6)),x]
Output:
((1 + 2*E^(1 + x))*(-8 + x))/(2*E^x*x*(1 - 64*x + 8*x^2))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 4.04 (sec) , antiderivative size = 625, normalized size of antiderivative = 23.15, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2026, 2463, 27, 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.
| \(\displaystyle \int \frac {e^{-x} \left (-8 x^4+112 x^3-257 x^2+e^{x+1} \left (-32 x^3+512 x^2-2048 x+16\right )-1016 x+8\right )}{128 x^6-2048 x^5+8224 x^4-256 x^3+2 x^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {e^{-x} \left (-8 x^4+112 x^3-257 x^2+e^{x+1} \left (-32 x^3+512 x^2-2048 x+16\right )-1016 x+8\right )}{x^2 \left (128 x^4-2048 x^3+8224 x^2-256 x+2\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \frac {e^{-x} \left (-8 x^4+112 x^3-257 x^2+e^{x+1} \left (-32 x^3+512 x^2-2048 x+16\right )-1016 x+8\right )}{2 x^2 \left (8 x^2-64 x+1\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {e^{-x} \left (-8 x^4+112 x^3-257 x^2-1016 x+16 e^{x+1} \left (-2 x^3+32 x^2-128 x+1\right )+8\right )}{x^2 \left (8 x^2-64 x+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (-\frac {8 e^{-x} x^2}{\left (8 x^2-64 x+1\right )^2}+\frac {112 e^{-x} x}{\left (8 x^2-64 x+1\right )^2}-\frac {257 e^{-x}}{\left (8 x^2-64 x+1\right )^2}-\frac {1016 e^{-x}}{\left (8 x^2-64 x+1\right )^2 x}-\frac {16 e \left (2 x^3-32 x^2+128 x-1\right )}{\left (8 x^2-64 x+1\right )^2 x^2}+\frac {8 e^{-x}}{\left (8 x^2-64 x+1\right )^2 x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {8}{127} \left (8128+511 \sqrt {254}\right ) e^{\frac {1}{4} \left (\sqrt {254}-16\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (-4 x-\sqrt {254}+16\right )\right )+4 \left (127+8 \sqrt {254}\right ) e^{\frac {1}{4} \left (\sqrt {254}-16\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (-4 x-\sqrt {254}+16\right )\right )-\frac {255 \left (16-\sqrt {254}\right ) e^{\frac {1}{4} \left (\sqrt {254}-16\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (-4 x-\sqrt {254}+16\right )\right )}{1016}-\frac {e^{\frac {1}{4} \left (\sqrt {254}-16\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (-4 x-\sqrt {254}+16\right )\right )}{508 \sqrt {254}}-\frac {1000}{127} \sqrt {\frac {2}{127}} e^{\frac {1}{4} \left (\sqrt {254}-16\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (-4 x-\sqrt {254}+16\right )\right )+\frac {1018}{127} e^{\frac {1}{4} \left (\sqrt {254}-16\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (-4 x-\sqrt {254}+16\right )\right )-\frac {255 \left (16+\sqrt {254}\right ) e^{-4-\frac {\sqrt {\frac {127}{2}}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (-4 x+\sqrt {254}+16\right )\right )}{1016}+4 \left (127-8 \sqrt {254}\right ) e^{-4-\frac {\sqrt {\frac {127}{2}}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (-4 x+\sqrt {254}+16\right )\right )-\frac {8}{127} \left (8128-511 \sqrt {254}\right ) e^{-4-\frac {\sqrt {\frac {127}{2}}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (-4 x+\sqrt {254}+16\right )\right )+\frac {e^{-4-\frac {\sqrt {\frac {127}{2}}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (-4 x+\sqrt {254}+16\right )\right )}{508 \sqrt {254}}+\frac {1000}{127} \sqrt {\frac {2}{127}} e^{-4-\frac {\sqrt {\frac {127}{2}}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (-4 x+\sqrt {254}+16\right )\right )+\frac {1018}{127} e^{-4-\frac {\sqrt {\frac {127}{2}}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (-4 x+\sqrt {254}+16\right )\right )-\frac {2 e (511-64 x)}{8 x^2-64 x+1}+\frac {255 \left (16-\sqrt {254}\right ) e^{-x}}{254 \left (-4 x-\sqrt {254}+16\right )}-\frac {4072 e^{-x}}{127 \left (-4 x-\sqrt {254}+16\right )}+\frac {255 \left (16+\sqrt {254}\right ) e^{-x}}{254 \left (-4 x+\sqrt {254}+16\right )}-\frac {4072 e^{-x}}{127 \left (-4 x+\sqrt {254}+16\right )}-\frac {8 e^{-x}}{x}-\frac {16 e}{x}\right )\) |
Input:
Int[(8 - 1016*x - 257*x^2 + 112*x^3 - 8*x^4 + E^(1 + x)*(16 - 2048*x + 512 *x^2 - 32*x^3))/(E^x*(2*x^2 - 256*x^3 + 8224*x^4 - 2048*x^5 + 128*x^6)),x]
Output:
(-4072/(127*E^x*(16 - Sqrt[254] - 4*x)) + (255*(16 - Sqrt[254]))/(254*E^x* (16 - Sqrt[254] - 4*x)) - 4072/(127*E^x*(16 + Sqrt[254] - 4*x)) + (255*(16 + Sqrt[254]))/(254*E^x*(16 + Sqrt[254] - 4*x)) - (16*E)/x - 8/(E^x*x) - ( 2*E*(511 - 64*x))/(1 - 64*x + 8*x^2) + (1018*E^((-16 + Sqrt[254])/4)*ExpIn tegralEi[(16 - Sqrt[254] - 4*x)/4])/127 - (1000*Sqrt[2/127]*E^((-16 + Sqrt [254])/4)*ExpIntegralEi[(16 - Sqrt[254] - 4*x)/4])/127 - (E^((-16 + Sqrt[2 54])/4)*ExpIntegralEi[(16 - Sqrt[254] - 4*x)/4])/(508*Sqrt[254]) - (255*(1 6 - Sqrt[254])*E^((-16 + Sqrt[254])/4)*ExpIntegralEi[(16 - Sqrt[254] - 4*x )/4])/1016 + 4*(127 + 8*Sqrt[254])*E^((-16 + Sqrt[254])/4)*ExpIntegralEi[( 16 - Sqrt[254] - 4*x)/4] - (8*(8128 + 511*Sqrt[254])*E^((-16 + Sqrt[254])/ 4)*ExpIntegralEi[(16 - Sqrt[254] - 4*x)/4])/127 + (1018*E^(-4 - Sqrt[127/2 ]/2)*ExpIntegralEi[(16 + Sqrt[254] - 4*x)/4])/127 + (1000*Sqrt[2/127]*E^(- 4 - Sqrt[127/2]/2)*ExpIntegralEi[(16 + Sqrt[254] - 4*x)/4])/127 + (E^(-4 - Sqrt[127/2]/2)*ExpIntegralEi[(16 + Sqrt[254] - 4*x)/4])/(508*Sqrt[254]) - (8*(8128 - 511*Sqrt[254])*E^(-4 - Sqrt[127/2]/2)*ExpIntegralEi[(16 + Sqrt [254] - 4*x)/4])/127 + 4*(127 - 8*Sqrt[254])*E^(-4 - Sqrt[127/2]/2)*ExpInt egralEi[(16 + Sqrt[254] - 4*x)/4] - (255*(16 + Sqrt[254])*E^(-4 - Sqrt[127 /2]/2)*ExpIntegralEi[(16 + Sqrt[254] - 4*x)/4])/1016)/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 1.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81
| method | result | size |
| parallelrisch | \(\frac {\left (-8-16 \,{\mathrm e}^{x} {\mathrm e} x^{3}+128 \,{\mathrm e}^{x} {\mathrm e} x^{2}-16 \,{\mathrm e} \,{\mathrm e}^{x}+x \right ) {\mathrm e}^{-x}}{2 x \left (8 x^{2}-64 x +1\right )}\) | \(49\) |
| norman | \(\frac {\left (-4+64 \,{\mathrm e}^{x} {\mathrm e} x^{2}-8 \,{\mathrm e}^{x} {\mathrm e} x^{3}+\frac {x}{2}-8 \,{\mathrm e} \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{x \left (8 x^{2}-64 x +1\right )}\) | \(50\) |
| risch | \(\frac {x \,{\mathrm e}-8 \,{\mathrm e}}{\left (8 x^{2}-64 x +1\right ) x}+\frac {\left (-8+x \right ) {\mathrm e}^{-x}}{2 \left (8 x^{2}-64 x +1\right ) x}\) | \(53\) |
| parts | \(\frac {257 \,{\mathrm e}^{-x} \left (x -4\right )}{508 \left (8 x^{2}-64 x +1\right )}-\frac {4 \,{\mathrm e}^{-x} \left (2036 x^{2}-16272 x +127\right )}{127 \left (8 x^{2}-64 x +1\right ) x}+\frac {2 \,{\mathrm e}^{-x} \left (32 x -255\right )}{8 x^{2}-64 x +1}-\frac {7 \,{\mathrm e}^{-x} \left (32 x -1\right )}{254 \left (8 x^{2}-64 x +1\right )}+\frac {{\mathrm e}^{-x} \left (255 x -4\right )}{4064 x^{2}-32512 x +508}-8 \,{\mathrm e} \left (\frac {-x +\frac {511}{64}}{x^{2}-8 x +\frac {1}{8}}+\frac {1}{x}\right )\) | \(148\) |
| default | \(8 \,{\mathrm e} \left (-\frac {16 \left (\frac {255 x}{4064}-\frac {509}{1016}\right )}{x^{2}-8 x +\frac {1}{8}}-64 \ln \left (8 x^{2}-64 x +1\right )-\frac {129539 \sqrt {254}\, \operatorname {arctanh}\left (\frac {\left (16 x -64\right ) \sqrt {254}}{1016}\right )}{16129}-\frac {1}{x}+128 \ln \left (x \right )\right )-\frac {4 \,{\mathrm e}^{-x} \left (2036 x^{2}-16272 x +127\right )}{127 \left (8 x^{2}-64 x +1\right ) x}-1024 \,{\mathrm e} \left (-\frac {16 \left (\frac {x}{1016}-\frac {255}{32512}\right )}{x^{2}-8 x +\frac {1}{8}}-\frac {\ln \left (8 x^{2}-64 x +1\right )}{2}-\frac {1012 \sqrt {254}\, \operatorname {arctanh}\left (\frac {\left (16 x -64\right ) \sqrt {254}}{1016}\right )}{16129}+\ln \left (x \right )\right )+256 \,{\mathrm e} \left (-\frac {16 x -64}{4064 \left (8 x^{2}-64 x +1\right )}+\frac {\sqrt {254}\, \operatorname {arctanh}\left (\frac {\left (16 x -64\right ) \sqrt {254}}{1016}\right )}{129032}\right )-16 \,{\mathrm e} \left (-\frac {64 x -2}{4064 \left (8 x^{2}-64 x +1\right )}+\frac {\sqrt {254}\, \operatorname {arctanh}\left (\frac {\left (16 x -64\right ) \sqrt {254}}{1016}\right )}{32258}\right )+\frac {2 \,{\mathrm e}^{-x} \left (32 x -255\right )}{8 x^{2}-64 x +1}+\frac {257 \,{\mathrm e}^{-x} \left (x -4\right )}{508 \left (8 x^{2}-64 x +1\right )}-\frac {7 \,{\mathrm e}^{-x} \left (32 x -1\right )}{254 \left (8 x^{2}-64 x +1\right )}+\frac {{\mathrm e}^{-x} \left (255 x -4\right )}{4064 x^{2}-32512 x +508}\) | \(316\) |
| orering | \(-\frac {\left (8 x^{5}-160 x^{4}+769 x^{3}+2032 x^{2}-16336 x +128\right ) \left (\left (-32 x^{3}+512 x^{2}-2048 x +16\right ) {\mathrm e} \,{\mathrm e}^{x}-8 x^{4}+112 x^{3}-257 x^{2}-1016 x +8\right ) {\mathrm e}^{-x}}{8 \left (2 x^{4}-46 x^{3}+336 x^{2}-641 x -1014\right ) \left (128 x^{6}-2048 x^{5}+8224 x^{4}-256 x^{3}+2 x^{2}\right )}-\frac {\left (x^{2}-16 x +64\right ) x \left (8 x^{2}-64 x +1\right ) \left (\frac {\left (\left (-96 x^{2}+1024 x -2048\right ) {\mathrm e} \,{\mathrm e}^{x}+\left (-32 x^{3}+512 x^{2}-2048 x +16\right ) {\mathrm e} \,{\mathrm e}^{x}-32 x^{3}+336 x^{2}-514 x -1016\right ) {\mathrm e}^{-x}}{128 x^{6}-2048 x^{5}+8224 x^{4}-256 x^{3}+2 x^{2}}-\frac {\left (\left (-32 x^{3}+512 x^{2}-2048 x +16\right ) {\mathrm e} \,{\mathrm e}^{x}-8 x^{4}+112 x^{3}-257 x^{2}-1016 x +8\right ) {\mathrm e}^{-x} \left (768 x^{5}-10240 x^{4}+32896 x^{3}-768 x^{2}+4 x \right )}{\left (128 x^{6}-2048 x^{5}+8224 x^{4}-256 x^{3}+2 x^{2}\right )^{2}}-\frac {\left (\left (-32 x^{3}+512 x^{2}-2048 x +16\right ) {\mathrm e} \,{\mathrm e}^{x}-8 x^{4}+112 x^{3}-257 x^{2}-1016 x +8\right ) {\mathrm e}^{-x}}{128 x^{6}-2048 x^{5}+8224 x^{4}-256 x^{3}+2 x^{2}}\right )}{8 \left (2 x^{4}-46 x^{3}+336 x^{2}-641 x -1014\right )}\) | \(422\) |
Input:
int(((-32*x^3+512*x^2-2048*x+16)*exp(1)*exp(x)-8*x^4+112*x^3-257*x^2-1016* x+8)/(128*x^6-2048*x^5+8224*x^4-256*x^3+2*x^2)/exp(x),x,method=_RETURNVERB OSE)
Output:
1/2*(-8-16*exp(x)*exp(1)*x^3+128*exp(x)*exp(1)*x^2-16*exp(1)*exp(x)+x)/exp (x)/x/(8*x^2-64*x+1)
Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^{-x} \left (8-1016 x-257 x^2+112 x^3-8 x^4+e^{1+x} \left (16-2048 x+512 x^2-32 x^3\right )\right )}{2 x^2-256 x^3+8224 x^4-2048 x^5+128 x^6} \, dx=\frac {{\left ({\left (x - 8\right )} e + 2 \, {\left (x - 8\right )} e^{\left (x + 2\right )}\right )} e^{\left (-x - 1\right )}}{2 \, {\left (8 \, x^{3} - 64 \, x^{2} + x\right )}} \] Input:
integrate(((-32*x^3+512*x^2-2048*x+16)*exp(1)*exp(x)-8*x^4+112*x^3-257*x^2 -1016*x+8)/(128*x^6-2048*x^5+8224*x^4-256*x^3+2*x^2)/exp(x),x, algorithm=" fricas")
Output:
1/2*((x - 8)*e + 2*(x - 8)*e^(x + 2))*e^(-x - 1)/(8*x^3 - 64*x^2 + x)
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).
Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {e^{-x} \left (8-1016 x-257 x^2+112 x^3-8 x^4+e^{1+x} \left (16-2048 x+512 x^2-32 x^3\right )\right )}{2 x^2-256 x^3+8224 x^4-2048 x^5+128 x^6} \, dx=\frac {\left (x - 8\right ) e^{- x}}{16 x^{3} - 128 x^{2} + 2 x} - \frac {- e x + 8 e}{8 x^{3} - 64 x^{2} + x} \] Input:
integrate(((-32*x**3+512*x**2-2048*x+16)*exp(1)*exp(x)-8*x**4+112*x**3-257 *x**2-1016*x+8)/(128*x**6-2048*x**5+8224*x**4-256*x**3+2*x**2)/exp(x),x)
Output:
(x - 8)*exp(-x)/(16*x**3 - 128*x**2 + 2*x) - (-E*x + 8*E)/(8*x**3 - 64*x** 2 + x)
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {e^{-x} \left (8-1016 x-257 x^2+112 x^3-8 x^4+e^{1+x} \left (16-2048 x+512 x^2-32 x^3\right )\right )}{2 x^2-256 x^3+8224 x^4-2048 x^5+128 x^6} \, dx=\frac {2 \, x e + {\left (x - 8\right )} e^{\left (-x\right )} - 16 \, e}{2 \, {\left (8 \, x^{3} - 64 \, x^{2} + x\right )}} \] Input:
integrate(((-32*x^3+512*x^2-2048*x+16)*exp(1)*exp(x)-8*x^4+112*x^3-257*x^2 -1016*x+8)/(128*x^6-2048*x^5+8224*x^4-256*x^3+2*x^2)/exp(x),x, algorithm=" maxima")
Output:
1/2*(2*x*e + (x - 8)*e^(-x) - 16*e)/(8*x^3 - 64*x^2 + x)
Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {e^{-x} \left (8-1016 x-257 x^2+112 x^3-8 x^4+e^{1+x} \left (16-2048 x+512 x^2-32 x^3\right )\right )}{2 x^2-256 x^3+8224 x^4-2048 x^5+128 x^6} \, dx=\frac {2 \, {\left (x + 1\right )} e + {\left (x + 1\right )} e^{\left (-x\right )} - 18 \, e - 9 \, e^{\left (-x\right )}}{2 \, {\left (8 \, {\left (x + 1\right )}^{3} - 88 \, {\left (x + 1\right )}^{2} + 153 \, x + 80\right )}} \] Input:
integrate(((-32*x^3+512*x^2-2048*x+16)*exp(1)*exp(x)-8*x^4+112*x^3-257*x^2 -1016*x+8)/(128*x^6-2048*x^5+8224*x^4-256*x^3+2*x^2)/exp(x),x, algorithm=" giac")
Output:
1/2*(2*(x + 1)*e + (x + 1)*e^(-x) - 18*e - 9*e^(-x))/(8*(x + 1)^3 - 88*(x + 1)^2 + 153*x + 80)
Time = 0.15 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {e^{-x} \left (8-1016 x-257 x^2+112 x^3-8 x^4+e^{1+x} \left (16-2048 x+512 x^2-32 x^3\right )\right )}{2 x^2-256 x^3+8224 x^4-2048 x^5+128 x^6} \, dx=-\frac {4\,{\mathrm {e}}^{-x}+8\,\mathrm {e}-x\,\left (\frac {{\mathrm {e}}^{-x}}{2}+\mathrm {e}\right )}{x\,\left (8\,x^2-64\,x+1\right )} \] Input:
int(-(exp(-x)*(1016*x + 257*x^2 - 112*x^3 + 8*x^4 + exp(1)*exp(x)*(2048*x - 512*x^2 + 32*x^3 - 16) - 8))/(2*x^2 - 256*x^3 + 8224*x^4 - 2048*x^5 + 12 8*x^6),x)
Output:
-(4*exp(-x) + 8*exp(1) - x*(exp(-x)/2 + exp(1)))/(x*(8*x^2 - 64*x + 1))
Time = 0.17 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^{-x} \left (8-1016 x-257 x^2+112 x^3-8 x^4+e^{1+x} \left (16-2048 x+512 x^2-32 x^3\right )\right )}{2 x^2-256 x^3+8224 x^4-2048 x^5+128 x^6} \, dx=\frac {2 e^{x} e x -16 e^{x} e +x -8}{2 e^{x} x \left (8 x^{2}-64 x +1\right )} \] Input:
int(((-32*x^3+512*x^2-2048*x+16)*exp(1)*exp(x)-8*x^4+112*x^3-257*x^2-1016* x+8)/(128*x^6-2048*x^5+8224*x^4-256*x^3+2*x^2)/exp(x),x)
Output:
(2*e**x*e*x - 16*e**x*e + x - 8)/(2*e**x*x*(8*x**2 - 64*x + 1))