3.14.36 \(\int e^{-2 x} (6 x^5-2 x^6+e^x (-6 x^2-6 x^3+2 x^4+10 e^{-2+x} x^4+12 x^5-2 x^6)+e^{2 x} (2+2 x-6 x^2-8 x^3+6 x^5+e^{-4+2 x} (4 x^3+2 x^4)+e^{-2+x} (-4 x-8 x^2-2 x^3+10 x^4+2 x^5))) \, dx\)

Optimal. Leaf size=24 \[ \left (1+x-x^2 \left (e^{-2+x}+x+e^{-x} x\right )\right )^2 \]

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Rubi [B]  time = 2.64, antiderivative size = 108, normalized size of antiderivative = 4.50, number of steps used = 74, number of rules used = 5, integrand size = 132, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6742, 2176, 2194, 6688, 2196} \begin {gather*} 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 \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

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

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rule 6688

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (6 e^{-2 x} x^5-2 e^{-2 x} x^6+2 e^{-2-x} x^2 \left (-3 e^2-3 e^2 x+e^2 x^2+5 e^x x^2+6 e^2 x^3-e^2 x^4\right )+\frac {2 \left (e^4+e^4 x-2 e^{2+x} x-3 e^4 x^2-4 e^{2+x} x^2-4 e^4 x^3+2 e^{2 x} x^3-e^{2+x} x^3+e^{2 x} x^4+5 e^{2+x} x^4+3 e^4 x^5+e^{2+x} x^5\right )}{e^4}\right ) \, dx\\ &=-\left (2 \int e^{-2 x} x^6 \, dx\right )+2 \int e^{-2-x} x^2 \left (-3 e^2-3 e^2 x+e^2 x^2+5 e^x x^2+6 e^2 x^3-e^2 x^4\right ) \, dx+6 \int e^{-2 x} x^5 \, dx+\frac {2 \int \left (e^4+e^4 x-2 e^{2+x} x-3 e^4 x^2-4 e^{2+x} x^2-4 e^4 x^3+2 e^{2 x} x^3-e^{2+x} x^3+e^{2 x} x^4+5 e^{2+x} x^4+3 e^4 x^5+e^{2+x} x^5\right ) \, dx}{e^4}\\ &=2 x+x^2-2 x^3-2 x^4-3 e^{-2 x} x^5+x^6+e^{-2 x} x^6+2 \int \left (\frac {5 x^4}{e^2}+e^{-x} x^2 \left (-3-3 x+x^2+6 x^3-x^4\right )\right ) \, dx-6 \int e^{-2 x} x^5 \, dx+15 \int e^{-2 x} x^4 \, dx-\frac {2 \int e^{2+x} x^3 \, dx}{e^4}+\frac {2 \int e^{2 x} x^4 \, dx}{e^4}+\frac {2 \int e^{2+x} x^5 \, dx}{e^4}-\frac {4 \int e^{2+x} x \, dx}{e^4}+\frac {4 \int e^{2 x} x^3 \, dx}{e^4}-\frac {8 \int e^{2+x} x^2 \, dx}{e^4}+\frac {10 \int e^{2+x} x^4 \, dx}{e^4}\\ &=2 x-4 e^{-2+x} x+x^2-8 e^{-2+x} x^2-2 x^3-2 e^{-2+x} x^3+2 e^{-4+2 x} x^3-2 x^4+10 e^{-2+x} x^4-\frac {15}{2} e^{-2 x} x^4+e^{-4+2 x} x^4+\frac {2 x^5}{e^2}+2 e^{-2+x} x^5+x^6+e^{-2 x} x^6+2 \int e^{-x} x^2 \left (-3-3 x+x^2+6 x^3-x^4\right ) \, dx-15 \int e^{-2 x} x^4 \, dx+30 \int e^{-2 x} x^3 \, dx+\frac {4 \int e^{2+x} \, dx}{e^4}-\frac {4 \int e^{2 x} x^3 \, dx}{e^4}-\frac {6 \int e^{2 x} x^2 \, dx}{e^4}+\frac {6 \int e^{2+x} x^2 \, dx}{e^4}-\frac {10 \int e^{2+x} x^4 \, dx}{e^4}+\frac {16 \int e^{2+x} x \, dx}{e^4}-\frac {40 \int e^{2+x} x^3 \, dx}{e^4}\\ &=4 e^{-2+x}+2 x+12 e^{-2+x} x+x^2-2 e^{-2+x} x^2-3 e^{-4+2 x} x^2-2 x^3-42 e^{-2+x} x^3-15 e^{-2 x} x^3-2 x^4+e^{-4+2 x} x^4+\frac {2 x^5}{e^2}+2 e^{-2+x} x^5+x^6+e^{-2 x} x^6+2 \int \left (-3 e^{-x} x^2-3 e^{-x} x^3+e^{-x} x^4+6 e^{-x} x^5-e^{-x} x^6\right ) \, dx-30 \int e^{-2 x} x^3 \, dx+45 \int e^{-2 x} x^2 \, dx+\frac {6 \int e^{2 x} x \, dx}{e^4}+\frac {6 \int e^{2 x} x^2 \, dx}{e^4}-\frac {12 \int e^{2+x} x \, dx}{e^4}-\frac {16 \int e^{2+x} \, dx}{e^4}+\frac {40 \int e^{2+x} x^3 \, dx}{e^4}+\frac {120 \int e^{2+x} x^2 \, dx}{e^4}\\ &=-12 e^{-2+x}+2 x+3 e^{-4+2 x} x+x^2+118 e^{-2+x} x^2-\frac {45}{2} e^{-2 x} x^2-2 x^3-2 e^{-2+x} x^3-2 x^4+e^{-4+2 x} x^4+\frac {2 x^5}{e^2}+2 e^{-2+x} x^5+x^6+e^{-2 x} x^6+2 \int e^{-x} x^4 \, dx-2 \int e^{-x} x^6 \, dx-6 \int e^{-x} x^2 \, dx-6 \int e^{-x} x^3 \, dx+12 \int e^{-x} x^5 \, dx+45 \int e^{-2 x} x \, dx-45 \int e^{-2 x} x^2 \, dx-\frac {3 \int e^{2 x} \, dx}{e^4}-\frac {6 \int e^{2 x} x \, dx}{e^4}+\frac {12 \int e^{2+x} \, dx}{e^4}-\frac {120 \int e^{2+x} x^2 \, dx}{e^4}-\frac {240 \int e^{2+x} x \, dx}{e^4}\\ &=-\frac {3}{2} e^{-4+2 x}+2 x-240 e^{-2+x} x-\frac {45}{2} e^{-2 x} x+x^2-2 e^{-2+x} x^2+6 e^{-x} x^2-2 x^3-2 e^{-2+x} x^3+6 e^{-x} x^3-2 x^4-2 e^{-x} x^4+e^{-4+2 x} x^4+\frac {2 x^5}{e^2}+2 e^{-2+x} x^5-12 e^{-x} x^5+x^6+e^{-2 x} x^6+2 e^{-x} x^6+8 \int e^{-x} x^3 \, dx-12 \int e^{-x} x \, dx-12 \int e^{-x} x^5 \, dx-18 \int e^{-x} x^2 \, dx+\frac {45}{2} \int e^{-2 x} \, dx-45 \int e^{-2 x} x \, dx+60 \int e^{-x} x^4 \, dx+\frac {3 \int e^{2 x} \, dx}{e^4}+\frac {240 \int e^{2+x} \, dx}{e^4}+\frac {240 \int e^{2+x} x \, dx}{e^4}\\ &=240 e^{-2+x}-\frac {45 e^{-2 x}}{4}+2 x+12 e^{-x} x+x^2-2 e^{-2+x} x^2+24 e^{-x} x^2-2 x^3-2 e^{-2+x} x^3-2 e^{-x} x^3-2 x^4-62 e^{-x} x^4+e^{-4+2 x} x^4+\frac {2 x^5}{e^2}+2 e^{-2+x} x^5+x^6+e^{-2 x} x^6+2 e^{-x} x^6-12 \int e^{-x} \, dx-\frac {45}{2} \int e^{-2 x} \, dx+24 \int e^{-x} x^2 \, dx-36 \int e^{-x} x \, dx-60 \int e^{-x} x^4 \, dx+240 \int e^{-x} x^3 \, dx-\frac {240 \int e^{2+x} \, dx}{e^4}\\ &=12 e^{-x}+2 x+48 e^{-x} x+x^2-2 e^{-2+x} x^2-2 x^3-2 e^{-2+x} x^3-242 e^{-x} x^3-2 x^4-2 e^{-x} x^4+e^{-4+2 x} x^4+\frac {2 x^5}{e^2}+2 e^{-2+x} x^5+x^6+e^{-2 x} x^6+2 e^{-x} x^6-36 \int e^{-x} \, dx+48 \int e^{-x} x \, dx-240 \int e^{-x} x^3 \, dx+720 \int e^{-x} x^2 \, dx\\ &=48 e^{-x}+2 x+x^2-2 e^{-2+x} x^2-720 e^{-x} x^2-2 x^3-2 e^{-2+x} x^3-2 e^{-x} x^3-2 x^4-2 e^{-x} x^4+e^{-4+2 x} x^4+\frac {2 x^5}{e^2}+2 e^{-2+x} x^5+x^6+e^{-2 x} x^6+2 e^{-x} x^6+48 \int e^{-x} \, dx-720 \int e^{-x} x^2 \, dx+1440 \int e^{-x} x \, dx\\ &=2 x-1440 e^{-x} x+x^2-2 e^{-2+x} x^2-2 x^3-2 e^{-2+x} x^3-2 e^{-x} x^3-2 x^4-2 e^{-x} x^4+e^{-4+2 x} x^4+\frac {2 x^5}{e^2}+2 e^{-2+x} x^5+x^6+e^{-2 x} x^6+2 e^{-x} x^6+1440 \int e^{-x} \, dx-1440 \int e^{-x} x \, dx\\ &=-1440 e^{-x}+2 x+x^2-2 e^{-2+x} x^2-2 x^3-2 e^{-2+x} x^3-2 e^{-x} x^3-2 x^4-2 e^{-x} x^4+e^{-4+2 x} x^4+\frac {2 x^5}{e^2}+2 e^{-2+x} x^5+x^6+e^{-2 x} x^6+2 e^{-x} x^6-1440 \int e^{-x} \, dx\\ &=2 x+x^2-2 e^{-2+x} x^2-2 x^3-2 e^{-2+x} x^3-2 e^{-x} x^3-2 x^4-2 e^{-x} x^4+e^{-4+2 x} x^4+\frac {2 x^5}{e^2}+2 e^{-2+x} x^5+x^6+e^{-2 x} x^6+2 e^{-x} x^6\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.25, size = 108, normalized size = 4.50 \begin {gather*} 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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

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)

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fricas [B]  time = 1.48, size = 100, normalized size = 4.17 \begin {gather*} {\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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^4+4*x^3)*exp(x-2)^2+(2*x^5+10*x^4-2*x^3-8*x^2-4*x)*exp(x-2)+6*x^5-8*x^3-6*x^2+2*x+2)*exp(x)^2
+(10*x^4*exp(x-2)-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")

[Out]

(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)

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giac [B]  time = 0.40, size = 122, normalized size = 5.08 \begin {gather*} {\left (x^{6} e^{6} + 2 \, x^{6} e^{\left (-x + 6\right )} + x^{6} e^{\left (-2 \, x + 6\right )} + 2 \, x^{5} e^{4} + 2 \, x^{5} e^{\left (x + 4\right )} - 2 \, x^{4} e^{6} + x^{4} e^{\left (2 \, x + 2\right )} - 2 \, x^{4} e^{\left (-x + 6\right )} - 2 \, x^{3} e^{6} - 2 \, x^{3} e^{\left (x + 4\right )} - 2 \, x^{3} e^{\left (-x + 6\right )} + x^{2} e^{6} - 2 \, x^{2} e^{\left (x + 4\right )} + 2 \, x e^{6}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^4+4*x^3)*exp(x-2)^2+(2*x^5+10*x^4-2*x^3-8*x^2-4*x)*exp(x-2)+6*x^5-8*x^3-6*x^2+2*x+2)*exp(x)^2
+(10*x^4*exp(x-2)-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")

[Out]

(x^6*e^6 + 2*x^6*e^(-x + 6) + x^6*e^(-2*x + 6) + 2*x^5*e^4 + 2*x^5*e^(x + 4) - 2*x^4*e^6 + x^4*e^(2*x + 2) - 2
*x^4*e^(-x + 6) - 2*x^3*e^6 - 2*x^3*e^(x + 4) - 2*x^3*e^(-x + 6) + x^2*e^6 - 2*x^2*e^(x + 4) + 2*x*e^6)*e^(-6)

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maple [B]  time = 0.06, size = 92, normalized size = 3.83

method	result	size
risch	\(x^{4} {\mathrm e}^{2 x -4}+2 \,{\mathrm e}^{-2} x^{5}+x^{6}-2 x^{4}-2 x^{3}+x^{2}+2 x +\left (2 \,{\mathrm e}^{-2} x^{5}-2 \,{\mathrm e}^{-2} x^{3}-2 x^{2} {\mathrm e}^{-2}\right ) {\mathrm e}^{x}+\left (2 x^{6}-2 x^{4}-2 x^{3}\right ) {\mathrm e}^{-x}+{\mathrm e}^{-2 x} x^{6}\)	\(92\)
default	\(x^{6}-2 x^{4}-2 x^{3}+x^{2}+2 x -2 x^{3} {\mathrm e}^{-x}-2 x^{4} {\mathrm e}^{-x}+2 \,{\mathrm e}^{-2} x^{5}+{\mathrm e}^{-2 x} x^{6}+2 \,{\mathrm e}^{-x} x^{6}-4 \,{\mathrm e}^{-2} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )-8 \,{\mathrm e}^{-2} \left ({\mathrm e}^{x} x^{2}-2 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}\right )-2 \,{\mathrm e}^{-2} \left ({\mathrm e}^{x} x^{3}-3 \,{\mathrm e}^{x} x^{2}+6 \,{\mathrm e}^{x} x -6 \,{\mathrm e}^{x}\right )+10 \,{\mathrm e}^{-2} \left ({\mathrm e}^{x} x^{4}-4 \,{\mathrm e}^{x} x^{3}+12 \,{\mathrm e}^{x} x^{2}-24 \,{\mathrm e}^{x} x +24 \,{\mathrm e}^{x}\right )+2 \,{\mathrm e}^{-2} \left (x^{5} {\mathrm e}^{x}-5 \,{\mathrm e}^{x} x^{4}+20 \,{\mathrm e}^{x} x^{3}-60 \,{\mathrm e}^{x} x^{2}+120 \,{\mathrm e}^{x} x -120 \,{\mathrm e}^{x}\right )+4 \,{\mathrm e}^{-4} \left (\frac {{\mathrm e}^{2 x} x^{3}}{2}-\frac {3 \,{\mathrm e}^{2 x} x^{2}}{4}+\frac {3 x \,{\mathrm e}^{2 x}}{4}-\frac {3 \,{\mathrm e}^{2 x}}{8}\right )+2 \,{\mathrm e}^{-4} \left (\frac {{\mathrm e}^{2 x} x^{4}}{2}-{\mathrm e}^{2 x} x^{3}+\frac {3 \,{\mathrm e}^{2 x} x^{2}}{2}-\frac {3 x \,{\mathrm e}^{2 x}}{2}+\frac {3 \,{\mathrm e}^{2 x}}{4}\right )\)	\(283\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^4+4*x^3)*exp(x-2)^2+(2*x^5+10*x^4-2*x^3-8*x^2-4*x)*exp(x-2)+6*x^5-8*x^3-6*x^2+2*x+2)*exp(x)^2+(10*x
^4*exp(x-2)-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)

[Out]

x^4*exp(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*x^2*exp(-2))*exp(x)+(2*x^6
-2*x^4-2*x^3)*exp(-x)+exp(-2*x)*x^6

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maxima [B]  time = 0.74, size = 360, normalized size = 15.00 \begin {gather*} 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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^4+4*x^3)*exp(x-2)^2+(2*x^5+10*x^4-2*x^3-8*x^2-4*x)*exp(x-2)+6*x^5-8*x^3-6*x^2+2*x+2)*exp(x)^2
+(10*x^4*exp(x-2)-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")

[Out]

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 + 120*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 - 1
20)*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

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mupad [B]  time = 0.37, size = 93, normalized size = 3.88 \begin {gather*} 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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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

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sympy [B]  time = 0.33, size = 112, normalized size = 4.67 \begin {gather*} 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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**4+4*x**3)*exp(x-2)**2+(2*x**5+10*x**4-2*x**3-8*x**2-4*x)*exp(x-2)+6*x**5-8*x**3-6*x**2+2*x+2
)*exp(x)**2+(10*x**4*exp(x-2)-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)

[Out]

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)

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