3.90.25 \(\int e^{-2 e^4-2 x} (-2 x-2 x^2+e^{2 e^4} (2-2 e^2-8 e x-8 x^2)+e^{e^4} (-2+4 x+8 x^2+e (2+4 x))) \, dx\)

Optimal. Leaf size=26 \[ e^{-2 x} \left (1+e+2 x-e^{-e^4} (1+x)\right )^2 \]

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Rubi [B]  time = 0.28, antiderivative size = 192, normalized size of antiderivative = 7.38, number of steps used = 23, number of rules used = 3, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {2196, 2176, 2194} \begin {gather*} e^{-2 x-2 e^4} x^2-4 e^{-2 x-e^4} x^2+4 e^{-2 x} x^2+4 e^{1-2 x} x+2 e^{-2 x-2 e^4} x-4 e^{-2 x-e^4} x+4 e^{-2 x} x-2 (1+e) e^{-2 x-e^4} x+2 e^{1-2 x}+e^{-2 x-2 e^4}+(1-e) e^{-2 x-e^4}-2 e^{-2 x-e^4}+2 e^{-2 x}-\left (1-e^2\right ) e^{-2 x}-(1+e) e^{-2 x-e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-2*E^4 - 2*x)*(-2*x - 2*x^2 + E^(2*E^4)*(2 - 2*E^2 - 8*E*x - 8*x^2) + E^E^4*(-2 + 4*x + 8*x^2 + E*(2 +
4*x))),x]

[Out]

2*E^(1 - 2*x) + E^(-2*E^4 - 2*x) - 2*E^(-E^4 - 2*x) + (1 - E)*E^(-E^4 - 2*x) + 2/E^(2*x) - E^(-E^4 - 2*x)*(1 +
E) - (1 - E^2)/E^(2*x) + 4*E^(1 - 2*x)*x + 2*E^(-2*E^4 - 2*x)*x - 4*E^(-E^4 - 2*x)*x + (4*x)/E^(2*x) - 2*E^(-
E^4 - 2*x)*(1 + E)*x + E^(-2*E^4 - 2*x)*x^2 - 4*E^(-E^4 - 2*x)*x^2 + (4*x^2)/E^(2*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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-2 e^{-2 e^4-2 x} x-2 e^{-2 e^4-2 x} x^2-2 e^{-2 x} \left (-1+e^2+4 e x+4 x^2\right )+2 e^{-e^4-2 x} \left (-1+e+2 (1+e) x+4 x^2\right )\right ) \, dx\\ &=-\left (2 \int e^{-2 e^4-2 x} x \, dx\right )-2 \int e^{-2 e^4-2 x} x^2 \, dx-2 \int e^{-2 x} \left (-1+e^2+4 e x+4 x^2\right ) \, dx+2 \int e^{-e^4-2 x} \left (-1+e+2 (1+e) x+4 x^2\right ) \, dx\\ &=e^{-2 e^4-2 x} x+e^{-2 e^4-2 x} x^2-2 \int e^{-2 e^4-2 x} x \, dx+2 \int \left ((-1+e) e^{-e^4-2 x}+2 e^{-e^4-2 x} (1+e) x+4 e^{-e^4-2 x} x^2\right ) \, dx-2 \int \left (e^{-2 x} \left (-1+e^2\right )+4 e^{1-2 x} x+4 e^{-2 x} x^2\right ) \, dx-\int e^{-2 e^4-2 x} \, dx\\ &=\frac {1}{2} e^{-2 e^4-2 x}+2 e^{-2 e^4-2 x} x+e^{-2 e^4-2 x} x^2-8 \int e^{1-2 x} x \, dx+8 \int e^{-e^4-2 x} x^2 \, dx-8 \int e^{-2 x} x^2 \, dx-(2 (1-e)) \int e^{-e^4-2 x} \, dx+(4 (1+e)) \int e^{-e^4-2 x} x \, dx+\left (2 \left (1-e^2\right )\right ) \int e^{-2 x} \, dx-\int e^{-2 e^4-2 x} \, dx\\ &=e^{-2 e^4-2 x}+(1-e) e^{-e^4-2 x}-e^{-2 x} \left (1-e^2\right )+4 e^{1-2 x} x+2 e^{-2 e^4-2 x} x-2 e^{-e^4-2 x} (1+e) x+e^{-2 e^4-2 x} x^2-4 e^{-e^4-2 x} x^2+4 e^{-2 x} x^2-4 \int e^{1-2 x} \, dx+8 \int e^{-e^4-2 x} x \, dx-8 \int e^{-2 x} x \, dx+(2 (1+e)) \int e^{-e^4-2 x} \, dx\\ &=2 e^{1-2 x}+e^{-2 e^4-2 x}+(1-e) e^{-e^4-2 x}-e^{-e^4-2 x} (1+e)-e^{-2 x} \left (1-e^2\right )+4 e^{1-2 x} x+2 e^{-2 e^4-2 x} x-4 e^{-e^4-2 x} x+4 e^{-2 x} x-2 e^{-e^4-2 x} (1+e) x+e^{-2 e^4-2 x} x^2-4 e^{-e^4-2 x} x^2+4 e^{-2 x} x^2+4 \int e^{-e^4-2 x} \, dx-4 \int e^{-2 x} \, dx\\ &=2 e^{1-2 x}+e^{-2 e^4-2 x}-2 e^{-e^4-2 x}+(1-e) e^{-e^4-2 x}+2 e^{-2 x}-e^{-e^4-2 x} (1+e)-e^{-2 x} \left (1-e^2\right )+4 e^{1-2 x} x+2 e^{-2 e^4-2 x} x-4 e^{-e^4-2 x} x+4 e^{-2 x} x-2 e^{-e^4-2 x} (1+e) x+e^{-2 e^4-2 x} x^2-4 e^{-e^4-2 x} x^2+4 e^{-2 x} x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.48, size = 35, normalized size = 1.35 \begin {gather*} e^{-2 \left (e^4+x\right )} \left (-1+e^{1+e^4}-x+e^{e^4} (1+2 x)\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-2*E^4 - 2*x)*(-2*x - 2*x^2 + E^(2*E^4)*(2 - 2*E^2 - 8*E*x - 8*x^2) + E^E^4*(-2 + 4*x + 8*x^2 + E
*(2 + 4*x))),x]

[Out]

(-1 + E^(1 + E^4) - x + E^E^4*(1 + 2*x))^2/E^(2*(E^4 + x))

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fricas [B]  time = 0.47, size = 72, normalized size = 2.77 \begin {gather*} {\left (4 \, x^{2} + 2 \, {\left (2 \, x + 1\right )} e + 4 \, x + e^{2} + 1\right )} e^{\left (-2 \, x\right )} - 2 \, {\left (2 \, x^{2} + {\left (x + 1\right )} e + 3 \, x + 1\right )} e^{\left (-2 \, x - e^{4}\right )} + {\left (x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x - 2 \, e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(1)^2-8*x*exp(1)-8*x^2+2)*exp(exp(4))^2+((4*x+2)*exp(1)+8*x^2+4*x-2)*exp(exp(4))-2*x^2-2*x)/
exp(x)^2/exp(exp(4))^2,x, algorithm="fricas")

[Out]

(4*x^2 + 2*(2*x + 1)*e + 4*x + e^2 + 1)*e^(-2*x) - 2*(2*x^2 + (x + 1)*e + 3*x + 1)*e^(-2*x - e^4) + (x^2 + 2*x
+ 1)*e^(-2*x - 2*e^4)

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giac [B]  time = 0.17, size = 89, normalized size = 3.42 \begin {gather*} {\left (4 \, x^{2} + 4 \, x + 1\right )} e^{\left (-2 \, x\right )} - 2 \, {\left (x + 1\right )} e^{\left (-2 \, x - e^{4} + 1\right )} - 2 \, {\left (2 \, x^{2} + 3 \, x + 1\right )} e^{\left (-2 \, x - e^{4}\right )} + {\left (x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x - 2 \, e^{4}\right )} + 2 \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x + 1\right )} + e^{\left (-2 \, x + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(1)^2-8*x*exp(1)-8*x^2+2)*exp(exp(4))^2+((4*x+2)*exp(1)+8*x^2+4*x-2)*exp(exp(4))-2*x^2-2*x)/
exp(x)^2/exp(exp(4))^2,x, algorithm="giac")

[Out]

(4*x^2 + 4*x + 1)*e^(-2*x) - 2*(x + 1)*e^(-2*x - e^4 + 1) - 2*(2*x^2 + 3*x + 1)*e^(-2*x - e^4) + (x^2 + 2*x +
1)*e^(-2*x - 2*e^4) + 2*(2*x + 1)*e^(-2*x + 1) + e^(-2*x + 2)

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maple [A]  time = 0.07, size = 33, normalized size = 1.27

method	result	size
gosper	\(\left ({\mathrm e}^{{\mathrm e}^{4}} {\mathrm e}+2 x \,{\mathrm e}^{{\mathrm e}^{4}}+{\mathrm e}^{{\mathrm e}^{4}}-x -1\right )^{2} {\mathrm e}^{-2 x} {\mathrm e}^{-2 \,{\mathrm e}^{4}}\)	\(33\)
risch	\(\left ({\mathrm e}^{2 \,{\mathrm e}^{4}+2}+4 x \,{\mathrm e}^{2 \,{\mathrm e}^{4}+1}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}+2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}+1}-2 x \,{\mathrm e}^{{\mathrm e}^{4}+1}+4 x \,{\mathrm e}^{2 \,{\mathrm e}^{4}}-4 x^{2} {\mathrm e}^{{\mathrm e}^{4}}-2 \,{\mathrm e}^{{\mathrm e}^{4}+1}+{\mathrm e}^{2 \,{\mathrm e}^{4}}-6 x \,{\mathrm e}^{{\mathrm e}^{4}}+x^{2}-2 \,{\mathrm e}^{{\mathrm e}^{4}}+2 x +1\right ) {\mathrm e}^{-2 \,{\mathrm e}^{4}-2 x}\)	\(102\)
norman	\(\left (\left ({\mathrm e}^{2} {\mathrm e}^{2 \,{\mathrm e}^{4}}+{\mathrm e}^{2 \,{\mathrm e}^{4}}-2 \,{\mathrm e}^{{\mathrm e}^{4}} {\mathrm e}-2 \,{\mathrm e}^{{\mathrm e}^{4}}+1+2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}\right ) {\mathrm e}^{-{\mathrm e}^{4}}+\left (4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}}-4 \,{\mathrm e}^{{\mathrm e}^{4}}+1\right ) {\mathrm e}^{-{\mathrm e}^{4}} x^{2}+2 \left (2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}-{\mathrm e}^{{\mathrm e}^{4}} {\mathrm e}-3 \,{\mathrm e}^{{\mathrm e}^{4}}+1+2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}}\right ) {\mathrm e}^{-{\mathrm e}^{4}} x \right ) {\mathrm e}^{-2 x} {\mathrm e}^{-{\mathrm e}^{4}}\)	\(117\)
meijerg	\(-{\mathrm e}^{2} \left (1-{\mathrm e}^{-2 x}\right )+1-{\mathrm e}^{-2 x}+{\mathrm e}^{1-{\mathrm e}^{4}} \left (1-{\mathrm e}^{-2 x}\right )-{\mathrm e}^{-{\mathrm e}^{4}} \left (1-{\mathrm e}^{-2 x}\right )+\frac {{\mathrm e}^{-2 \,{\mathrm e}^{4}} \left (-8 \,{\mathrm e}^{2 \,{\mathrm e}^{4}+1}+4 \,{\mathrm e}^{{\mathrm e}^{4}+1}+4 \,{\mathrm e}^{{\mathrm e}^{4}}-2\right ) \left (1-\frac {\left (4 x +2\right ) {\mathrm e}^{-2 x}}{2}\right )}{4}+\frac {{\mathrm e}^{-2 \,{\mathrm e}^{4}} \left (-8 \,{\mathrm e}^{2 \,{\mathrm e}^{4}}+8 \,{\mathrm e}^{{\mathrm e}^{4}}-2\right ) \left (2-\frac {\left (12 x^{2}+12 x +6\right ) {\mathrm e}^{-2 x}}{3}\right )}{8}\)	\(134\)
default	\({\mathrm e}^{-2 \,{\mathrm e}^{4}} \left (2 x \,{\mathrm e}^{-2 x}+{\mathrm e}^{-2 x}+x^{2} {\mathrm e}^{-2 x}+{\mathrm e}^{-2 x} {\mathrm e}^{{\mathrm e}^{4}}-{\mathrm e}^{-2 x} {\mathrm e}^{2 \,{\mathrm e}^{4}}+4 \,{\mathrm e}^{{\mathrm e}^{4}} \left (-\frac {x \,{\mathrm e}^{-2 x}}{2}-\frac {{\mathrm e}^{-2 x}}{4}\right )+8 \,{\mathrm e}^{{\mathrm e}^{4}} \left (-\frac {x^{2} {\mathrm e}^{-2 x}}{2}-\frac {x \,{\mathrm e}^{-2 x}}{2}-\frac {{\mathrm e}^{-2 x}}{4}\right )-8 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} \left (-\frac {x^{2} {\mathrm e}^{-2 x}}{2}-\frac {x \,{\mathrm e}^{-2 x}}{2}-\frac {{\mathrm e}^{-2 x}}{4}\right )-{\mathrm e}^{-2 x} {\mathrm e}^{{\mathrm e}^{4}} {\mathrm e}+{\mathrm e}^{-2 x} {\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{2}+4 \,{\mathrm e}^{{\mathrm e}^{4}} {\mathrm e} \left (-\frac {x \,{\mathrm e}^{-2 x}}{2}-\frac {{\mathrm e}^{-2 x}}{4}\right )-8 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e} \left (-\frac {x \,{\mathrm e}^{-2 x}}{2}-\frac {{\mathrm e}^{-2 x}}{4}\right )\right )\)	\(192\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*exp(1)^2-8*x*exp(1)-8*x^2+2)*exp(exp(4))^2+((4*x+2)*exp(1)+8*x^2+4*x-2)*exp(exp(4))-2*x^2-2*x)/exp(x)
^2/exp(exp(4))^2,x,method=_RETURNVERBOSE)

[Out]

(exp(exp(4))*exp(1)+2*x*exp(exp(4))+exp(exp(4))-x-1)^2/exp(x)^2/exp(exp(4))^2

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maxima [B]  time = 0.36, size = 157, normalized size = 6.04 \begin {gather*} 2 \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} + 2 \, {\left (2 \, x e + e\right )} e^{\left (-2 \, x\right )} - 2 \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x - e^{4}\right )} - {\left (2 \, x e + e\right )} e^{\left (-2 \, x - e^{4}\right )} - {\left (2 \, x + 1\right )} e^{\left (-2 \, x - e^{4}\right )} + \frac {1}{2} \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x - 2 \, e^{4}\right )} + \frac {1}{2} \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x - 2 \, e^{4}\right )} - e^{\left (-2 \, x\right )} - e^{\left (-2 \, x - e^{4} + 1\right )} + e^{\left (-2 \, x - e^{4}\right )} + e^{\left (-2 \, x + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(1)^2-8*x*exp(1)-8*x^2+2)*exp(exp(4))^2+((4*x+2)*exp(1)+8*x^2+4*x-2)*exp(exp(4))-2*x^2-2*x)/
exp(x)^2/exp(exp(4))^2,x, algorithm="maxima")

[Out]

2*(2*x^2 + 2*x + 1)*e^(-2*x) + 2*(2*x*e + e)*e^(-2*x) - 2*(2*x^2 + 2*x + 1)*e^(-2*x - e^4) - (2*x*e + e)*e^(-2
*x - e^4) - (2*x + 1)*e^(-2*x - e^4) + 1/2*(2*x^2 + 2*x + 1)*e^(-2*x - 2*e^4) + 1/2*(2*x + 1)*e^(-2*x - 2*e^4)
- e^(-2*x) - e^(-2*x - e^4 + 1) + e^(-2*x - e^4) + e^(-2*x + 2)

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mupad [B]  time = 7.52, size = 109, normalized size = 4.19 \begin {gather*} {\mathrm {e}}^{-2\,x-2\,{\mathrm {e}}^4}\,\left ({\mathrm {e}}^{2\,{\mathrm {e}}^4}-2\,{\mathrm {e}}^{{\mathrm {e}}^4+1}+2\,{\mathrm {e}}^{2\,{\mathrm {e}}^4+1}+{\mathrm {e}}^{2\,{\mathrm {e}}^4+2}-2\,{\mathrm {e}}^{{\mathrm {e}}^4}+1\right )+x^2\,{\mathrm {e}}^{-2\,x-2\,{\mathrm {e}}^4}\,{\left (2\,{\mathrm {e}}^{{\mathrm {e}}^4}-1\right )}^2+x\,{\mathrm {e}}^{-2\,x-2\,{\mathrm {e}}^4}\,\left (4\,{\mathrm {e}}^{2\,{\mathrm {e}}^4}-2\,{\mathrm {e}}^{{\mathrm {e}}^4+1}+4\,{\mathrm {e}}^{2\,{\mathrm {e}}^4+1}-6\,{\mathrm {e}}^{{\mathrm {e}}^4}+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-2*exp(4))*exp(-2*x)*(2*x + exp(2*exp(4))*(2*exp(2) + 8*x*exp(1) + 8*x^2 - 2) - exp(exp(4))*(4*x + 8*
x^2 + exp(1)*(4*x + 2) - 2) + 2*x^2),x)

[Out]

exp(- 2*x - 2*exp(4))*(exp(2*exp(4)) - 2*exp(exp(4) + 1) + 2*exp(2*exp(4) + 1) + exp(2*exp(4) + 2) - 2*exp(exp
(4)) + 1) + x^2*exp(- 2*x - 2*exp(4))*(2*exp(exp(4)) - 1)^2 + x*exp(- 2*x - 2*exp(4))*(4*exp(2*exp(4)) - 2*exp
(exp(4) + 1) + 4*exp(2*exp(4) + 1) - 6*exp(exp(4)) + 2)

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sympy [B]  time = 0.24, size = 131, normalized size = 5.04 \begin {gather*} \frac {\left (- 4 x^{2} e^{e^{4}} + x^{2} + 4 x^{2} e^{2 e^{4}} - 6 x e^{e^{4}} - 2 e x e^{e^{4}} + 2 x + 4 x e^{2 e^{4}} + 4 e x e^{2 e^{4}} - 2 e e^{e^{4}} - 2 e^{e^{4}} + 1 + e^{2 e^{4}} + 2 e e^{2 e^{4}} + e^{2} e^{2 e^{4}}\right ) e^{- 2 x}}{e^{2 e^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(1)**2-8*x*exp(1)-8*x**2+2)*exp(exp(4))**2+((4*x+2)*exp(1)+8*x**2+4*x-2)*exp(exp(4))-2*x**2-
2*x)/exp(x)**2/exp(exp(4))**2,x)

[Out]

(-4*x**2*exp(exp(4)) + x**2 + 4*x**2*exp(2*exp(4)) - 6*x*exp(exp(4)) - 2*E*x*exp(exp(4)) + 2*x + 4*x*exp(2*exp
(4)) + 4*E*x*exp(2*exp(4)) - 2*E*exp(exp(4)) - 2*exp(exp(4)) + 1 + exp(2*exp(4)) + 2*E*exp(2*exp(4)) + exp(2)*
exp(2*exp(4)))*exp(-2*x)*exp(-2*exp(4))

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