3.41.93 \(\int e^{-8 e^4} (-4 e^{2 e^4}-12 x^2) (x^3+e^{2 e^4} (3+x))^3 \, dx\)

Optimal. Leaf size=22 \[ \frac {1}{4}-\left (3+x+e^{-2 e^4} x^3\right )^4 \]

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Rubi [A]  time = 0.04, antiderivative size = 26, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {12, 1588} \begin {gather*} -e^{-8 e^4} \left (x^3+e^{2 e^4} (x+3)\right )^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-4*E^(2*E^4) - 12*x^2)*(x^3 + E^(2*E^4)*(3 + x))^3)/E^(8*E^4),x]

[Out]

-((x^3 + E^(2*E^4)*(3 + x))^4/E^(8*E^4))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^{-8 e^4} \int \left (-4 e^{2 e^4}-12 x^2\right ) \left (x^3+e^{2 e^4} (3+x)\right )^3 \, dx\\ &=-e^{-8 e^4} \left (x^3+e^{2 e^4} (3+x)\right )^4\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.05, size = 84, normalized size = 3.82 \begin {gather*} -e^{-8 e^4} \left (x^{12}+4 e^{2 e^4} x^9 (3+x)+6 e^{4 e^4} x^6 (3+x)^2+4 e^{6 e^4} x^3 (3+x)^3+e^{8 e^4} x \left (108+54 x+12 x^2+x^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-4*E^(2*E^4) - 12*x^2)*(x^3 + E^(2*E^4)*(3 + x))^3)/E^(8*E^4),x]

[Out]

-((x^12 + 4*E^(2*E^4)*x^9*(3 + x) + 6*E^(4*E^4)*x^6*(3 + x)^2 + 4*E^(6*E^4)*x^3*(3 + x)^3 + E^(8*E^4)*x*(108 +
54*x + 12*x^2 + x^3))/E^(8*E^4))

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fricas [B]  time = 0.67, size = 97, normalized size = 4.41 \begin {gather*} -{\left (x^{12} + {\left (x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x\right )} e^{\left (8 \, e^{4}\right )} + 4 \, {\left (x^{6} + 9 \, x^{5} + 27 \, x^{4} + 27 \, x^{3}\right )} e^{\left (6 \, e^{4}\right )} + 6 \, {\left (x^{8} + 6 \, x^{7} + 9 \, x^{6}\right )} e^{\left (4 \, e^{4}\right )} + 4 \, {\left (x^{10} + 3 \, x^{9}\right )} e^{\left (2 \, e^{4}\right )}\right )} e^{\left (-8 \, e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(exp(4))^2-12*x^2)*((3+x)*exp(exp(4))^2+x^3)^3/exp(exp(4))^8,x, algorithm="fricas")

[Out]

-(x^12 + (x^4 + 12*x^3 + 54*x^2 + 108*x)*e^(8*e^4) + 4*(x^6 + 9*x^5 + 27*x^4 + 27*x^3)*e^(6*e^4) + 6*(x^8 + 6*
x^7 + 9*x^6)*e^(4*e^4) + 4*(x^10 + 3*x^9)*e^(2*e^4))*e^(-8*e^4)

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giac [B]  time = 0.19, size = 79, normalized size = 3.59 \begin {gather*} -{\left ({\left (x^{3} + x e^{\left (2 \, e^{4}\right )}\right )}^{4} + 12 \, {\left (x^{3} + x e^{\left (2 \, e^{4}\right )}\right )}^{3} e^{\left (2 \, e^{4}\right )} + 54 \, {\left (x^{3} + x e^{\left (2 \, e^{4}\right )}\right )}^{2} e^{\left (4 \, e^{4}\right )} + 108 \, {\left (x^{3} + x e^{\left (2 \, e^{4}\right )}\right )} e^{\left (6 \, e^{4}\right )}\right )} e^{\left (-8 \, e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(exp(4))^2-12*x^2)*((3+x)*exp(exp(4))^2+x^3)^3/exp(exp(4))^8,x, algorithm="giac")

[Out]

-((x^3 + x*e^(2*e^4))^4 + 12*(x^3 + x*e^(2*e^4))^3*e^(2*e^4) + 54*(x^3 + x*e^(2*e^4))^2*e^(4*e^4) + 108*(x^3 +
x*e^(2*e^4))*e^(6*e^4))*e^(-8*e^4)

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maple [B]  time = 0.07, size = 135, normalized size = 6.14

method	result	size
norman	\(\left (-{\mathrm e}^{-{\mathrm e}^{4}} x^{12}-12 \,{\mathrm e}^{{\mathrm e}^{4}} x^{9}-4 \,{\mathrm e}^{{\mathrm e}^{4}} x^{10}-36 \,{\mathrm e}^{3 \,{\mathrm e}^{4}} x^{7}-6 \,{\mathrm e}^{3 \,{\mathrm e}^{4}} x^{8}-36 \,{\mathrm e}^{5 \,{\mathrm e}^{4}} x^{5}-108 \,{\mathrm e}^{7 \,{\mathrm e}^{4}} x -54 \,{\mathrm e}^{7 \,{\mathrm e}^{4}} x^{2}-2 \,{\mathrm e}^{3 \,{\mathrm e}^{4}} \left (2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}}+27\right ) x^{6}-12 \,{\mathrm e}^{5 \,{\mathrm e}^{4}} \left ({\mathrm e}^{2 \,{\mathrm e}^{4}}+9\right ) x^{3}-{\mathrm e}^{5 \,{\mathrm e}^{4}} \left ({\mathrm e}^{2 \,{\mathrm e}^{4}}+108\right ) x^{4}\right ) {\mathrm e}^{-7 \,{\mathrm e}^{4}}\)	\(135\)
gosper	\(-x \left (x^{3} {\mathrm e}^{8 \,{\mathrm e}^{4}}+4 \,{\mathrm e}^{6 \,{\mathrm e}^{4}} x^{5}+6 \,{\mathrm e}^{4 \,{\mathrm e}^{4}} x^{7}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{9}+x^{11}+12 \,{\mathrm e}^{8 \,{\mathrm e}^{4}} x^{2}+36 \,{\mathrm e}^{6 \,{\mathrm e}^{4}} x^{4}+36 \,{\mathrm e}^{4 \,{\mathrm e}^{4}} x^{6}+12 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{8}+54 \,{\mathrm e}^{8 \,{\mathrm e}^{4}} x +108 x^{3} {\mathrm e}^{6 \,{\mathrm e}^{4}}+54 x^{5} {\mathrm e}^{4 \,{\mathrm e}^{4}}+108 \,{\mathrm e}^{8 \,{\mathrm e}^{4}}+108 x^{2} {\mathrm e}^{6 \,{\mathrm e}^{4}}\right ) {\mathrm e}^{-8 \,{\mathrm e}^{4}}\)	\(137\)
default	\({\mathrm e}^{-8 \,{\mathrm e}^{4}} \left (-x^{12}-4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{10}-12 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{9}-6 \,{\mathrm e}^{4 \,{\mathrm e}^{4}} x^{8}-36 \,{\mathrm e}^{4 \,{\mathrm e}^{4}} x^{7}+\frac {\left (-24 \,{\mathrm e}^{6 \,{\mathrm e}^{4}}-324 \,{\mathrm e}^{4 \,{\mathrm e}^{4}}\right ) x^{6}}{6}-36 \,{\mathrm e}^{6 \,{\mathrm e}^{4}} x^{5}+\frac {\left (-4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} \left ({\mathrm e}^{6 \,{\mathrm e}^{4}}+27 \,{\mathrm e}^{4 \,{\mathrm e}^{4}}\right )-324 \,{\mathrm e}^{6 \,{\mathrm e}^{4}}\right ) x^{4}}{4}+\frac {\left (-36 \,{\mathrm e}^{8 \,{\mathrm e}^{4}}-324 \,{\mathrm e}^{6 \,{\mathrm e}^{4}}\right ) x^{3}}{3}-54 \,{\mathrm e}^{8 \,{\mathrm e}^{4}} x^{2}-108 \,{\mathrm e}^{8 \,{\mathrm e}^{4}} x \right )\)	\(154\)
risch	\(-{\mathrm e}^{-8 \,{\mathrm e}^{4}} x^{12}-4 \,{\mathrm e}^{-6 \,{\mathrm e}^{4}} x^{10}-12 \,{\mathrm e}^{-6 \,{\mathrm e}^{4}} x^{9}-6 \,{\mathrm e}^{-4 \,{\mathrm e}^{4}} x^{8}-36 \,{\mathrm e}^{-4 \,{\mathrm e}^{4}} x^{7}-4 \,{\mathrm e}^{-8 \,{\mathrm e}^{4}} x^{6} {\mathrm e}^{6 \,{\mathrm e}^{4}}-54 \,{\mathrm e}^{-8 \,{\mathrm e}^{4}} x^{6} {\mathrm e}^{4 \,{\mathrm e}^{4}}-36 \,{\mathrm e}^{-2 \,{\mathrm e}^{4}} x^{5}-{\mathrm e}^{-8 \,{\mathrm e}^{4}} {\mathrm e}^{6 \,{\mathrm e}^{4}} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{4}-27 \,{\mathrm e}^{-8 \,{\mathrm e}^{4}} {\mathrm e}^{4 \,{\mathrm e}^{4}} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{4}-81 \,{\mathrm e}^{-8 \,{\mathrm e}^{4}} {\mathrm e}^{6 \,{\mathrm e}^{4}} x^{4}-12 \,{\mathrm e}^{-8 \,{\mathrm e}^{4}} x^{3} {\mathrm e}^{8 \,{\mathrm e}^{4}}-108 \,{\mathrm e}^{-8 \,{\mathrm e}^{4}} x^{3} {\mathrm e}^{6 \,{\mathrm e}^{4}}-54 x^{2}-108 x\)	\(185\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*exp(exp(4))^2-12*x^2)*((3+x)*exp(exp(4))^2+x^3)^3/exp(exp(4))^8,x,method=_RETURNVERBOSE)

[Out]

(-1/exp(exp(4))*x^12-12*exp(exp(4))*x^9-4*exp(exp(4))*x^10-36*exp(exp(4))^3*x^7-6*exp(exp(4))^3*x^8-36*exp(exp
(4))^5*x^5-108*exp(exp(4))^7*x-54*exp(exp(4))^7*x^2-2*exp(exp(4))^3*(2*exp(exp(4))^2+27)*x^6-12*exp(exp(4))^5*
(exp(exp(4))^2+9)*x^3-exp(exp(4))^5*(exp(exp(4))^2+108)*x^4)/exp(exp(4))^7

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maxima [A]  time = 0.36, size = 22, normalized size = 1.00 \begin {gather*} -{\left (x^{3} + {\left (x + 3\right )} e^{\left (2 \, e^{4}\right )}\right )}^{4} e^{\left (-8 \, e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(exp(4))^2-12*x^2)*((3+x)*exp(exp(4))^2+x^3)^3/exp(exp(4))^8,x, algorithm="maxima")

[Out]

-(x^3 + (x + 3)*e^(2*e^4))^4*e^(-8*e^4)

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mupad [B]  time = 3.09, size = 122, normalized size = 5.55 \begin {gather*} -{\mathrm {e}}^{-8\,{\mathrm {e}}^4}\,x^{12}-4\,{\mathrm {e}}^{-6\,{\mathrm {e}}^4}\,x^{10}-12\,{\mathrm {e}}^{-6\,{\mathrm {e}}^4}\,x^9-6\,{\mathrm {e}}^{-4\,{\mathrm {e}}^4}\,x^8-36\,{\mathrm {e}}^{-4\,{\mathrm {e}}^4}\,x^7-2\,{\mathrm {e}}^{-4\,{\mathrm {e}}^4}\,\left (2\,{\mathrm {e}}^{2\,{\mathrm {e}}^4}+27\right )\,x^6-36\,{\mathrm {e}}^{-2\,{\mathrm {e}}^4}\,x^5-{\mathrm {e}}^{-2\,{\mathrm {e}}^4}\,\left ({\mathrm {e}}^{2\,{\mathrm {e}}^4}+108\right )\,x^4-12\,{\mathrm {e}}^{-2\,{\mathrm {e}}^4}\,\left ({\mathrm {e}}^{2\,{\mathrm {e}}^4}+9\right )\,x^3-54\,x^2-108\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-8*exp(4))*(4*exp(2*exp(4)) + 12*x^2)*(exp(2*exp(4))*(x + 3) + x^3)^3,x)

[Out]

- 108*x - 36*x^5*exp(-2*exp(4)) - 36*x^7*exp(-4*exp(4)) - 6*x^8*exp(-4*exp(4)) - 12*x^9*exp(-6*exp(4)) - 4*x^1
0*exp(-6*exp(4)) - x^12*exp(-8*exp(4)) - 54*x^2 - 2*x^6*exp(-4*exp(4))*(2*exp(2*exp(4)) + 27) - 12*x^3*exp(-2*
exp(4))*(exp(2*exp(4)) + 9) - x^4*exp(-2*exp(4))*(exp(2*exp(4)) + 108)

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sympy [B]  time = 0.09, size = 141, normalized size = 6.41 \begin {gather*} - \frac {x^{12}}{e^{8 e^{4}}} - \frac {4 x^{10}}{e^{6 e^{4}}} - \frac {12 x^{9}}{e^{6 e^{4}}} - \frac {6 x^{8}}{e^{4 e^{4}}} - \frac {36 x^{7}}{e^{4 e^{4}}} + \frac {x^{6} \left (- 4 e^{2 e^{4}} - 54\right )}{e^{4 e^{4}}} - \frac {36 x^{5}}{e^{2 e^{4}}} + \frac {x^{4} \left (- e^{2 e^{4}} - 108\right )}{e^{2 e^{4}}} + \frac {x^{3} \left (- 12 e^{2 e^{4}} - 108\right )}{e^{2 e^{4}}} - 54 x^{2} - 108 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(exp(4))**2-12*x**2)*((3+x)*exp(exp(4))**2+x**3)**3/exp(exp(4))**8,x)

[Out]

-x**12*exp(-8*exp(4)) - 4*x**10*exp(-6*exp(4)) - 12*x**9*exp(-6*exp(4)) - 6*x**8*exp(-4*exp(4)) - 36*x**7*exp(
-4*exp(4)) + x**6*(-4*exp(2*exp(4)) - 54)*exp(-4*exp(4)) - 36*x**5*exp(-2*exp(4)) + x**4*(-exp(2*exp(4)) - 108
)*exp(-2*exp(4)) + x**3*(-12*exp(2*exp(4)) - 108)*exp(-2*exp(4)) - 54*x**2 - 108*x

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