3.28.20 \(\int \frac {1}{3} (3+e^{-6+2 e^3+\frac {1}{3} e^{-6+2 e^3} (18 x^2-2 x^4)} (36 x-8 x^3)) \, dx\)

Optimal. Leaf size=29 \[ 5+e^{\frac {2}{3} e^{-6+2 e^3} (3-x) x^2 (3+x)}+x \]

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Rubi [A]  time = 0.42, antiderivative size = 28, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 3, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {12, 1593, 6706} \begin {gather*} e^{\frac {2}{3} e^{2 e^3-6} \left (9 x^2-x^4\right )}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + E^(-6 + 2*E^3 + (E^(-6 + 2*E^3)*(18*x^2 - 2*x^4))/3)*(36*x - 8*x^3))/3,x]

[Out]

E^((2*E^(-6 + 2*E^3)*(9*x^2 - x^4))/3) + x

Rule 12

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

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \left (3+\exp \left (-6+2 e^3+\frac {1}{3} e^{-6+2 e^3} \left (18 x^2-2 x^4\right )\right ) \left (36 x-8 x^3\right )\right ) \, dx\\ &=x+\frac {1}{3} \int \exp \left (-6+2 e^3+\frac {1}{3} e^{-6+2 e^3} \left (18 x^2-2 x^4\right )\right ) \left (36 x-8 x^3\right ) \, dx\\ &=x+\frac {1}{3} \int \exp \left (-6+2 e^3+\frac {1}{3} e^{-6+2 e^3} \left (18 x^2-2 x^4\right )\right ) x \left (36-8 x^2\right ) \, dx\\ &=e^{\frac {2}{3} e^{-6+2 e^3} \left (9 x^2-x^4\right )}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.36, size = 35, normalized size = 1.21 \begin {gather*} e^{6 e^{-6+2 e^3} x^2-\frac {2}{3} e^{-6+2 e^3} x^4}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + E^(-6 + 2*E^3 + (E^(-6 + 2*E^3)*(18*x^2 - 2*x^4))/3)*(36*x - 8*x^3))/3,x]

[Out]

E^(6*E^(-6 + 2*E^3)*x^2 - (2*E^(-6 + 2*E^3)*x^4)/3) + x

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fricas [A]  time = 0.70, size = 43, normalized size = 1.48 \begin {gather*} {\left (x e^{\left (2 \, e^{3} - 6\right )} + e^{\left (-\frac {2}{3} \, {\left (x^{4} - 9 \, x^{2}\right )} e^{\left (2 \, e^{3} - 6\right )} + 2 \, e^{3} - 6\right )}\right )} e^{\left (-2 \, e^{3} + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-8*x^3+36*x)*exp(exp(3)-3)^2*exp(1/3*(-2*x^4+18*x^2)*exp(exp(3)-3)^2)+1,x, algorithm="fricas")

[Out]

(x*e^(2*e^3 - 6) + e^(-2/3*(x^4 - 9*x^2)*e^(2*e^3 - 6) + 2*e^3 - 6))*e^(-2*e^3 + 6)

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giac [A]  time = 0.20, size = 28, normalized size = 0.97 \begin {gather*} x + e^{\left (-\frac {2}{3} \, x^{4} e^{\left (2 \, e^{3} - 6\right )} + 6 \, x^{2} e^{\left (2 \, e^{3} - 6\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-8*x^3+36*x)*exp(exp(3)-3)^2*exp(1/3*(-2*x^4+18*x^2)*exp(exp(3)-3)^2)+1,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.10, size = 22, normalized size = 0.76

method	result	size
risch	\({\mathrm e}^{-\frac {2 x^{2} \left (x -3\right ) \left (3+x \right ) {\mathrm e}^{2 \,{\mathrm e}^{3}-6}}{3}}+x\)	\(22\)
default	\(x +{\mathrm e}^{\frac {\left (-2 x^{4}+18 x^{2}\right ) {\mathrm e}^{2 \,{\mathrm e}^{3}-6}}{3}}\)	\(24\)
norman	\(x +{\mathrm e}^{\frac {\left (-2 x^{4}+18 x^{2}\right ) {\mathrm e}^{2 \,{\mathrm e}^{3}-6}}{3}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(-8*x^3+36*x)*exp(exp(3)-3)^2*exp(1/3*(-2*x^4+18*x^2)*exp(exp(3)-3)^2)+1,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [C]  time = 0.85, size = 258, normalized size = 8.90 \begin {gather*} \frac {3}{2} \, \sqrt {3} \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{3} \, \sqrt {3} \sqrt {2} x^{2} e^{\left (e^{3} - 3\right )} - \frac {3}{2} \, \sqrt {3} \sqrt {2} e^{\left (e^{3} - 3\right )}\right ) e^{\left (e^{3} + \frac {27}{2} \, e^{\left (2 \, e^{3} - 6\right )} - 3\right )} - \frac {3 \, \sqrt {\frac {2}{3}} {\left (\frac {3 \, \sqrt {\pi } {\left (2 \, x^{2} e^{\left (2 \, e^{3} - 6\right )} - 9 \, e^{\left (2 \, e^{3} - 6\right )}\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{6}} \sqrt {{\left (2 \, x^{2} e^{\left (2 \, e^{3} - 6\right )} - 9 \, e^{\left (2 \, e^{3} - 6\right )}\right )}^{2}} e^{\left (-e^{3} + 3\right )}\right ) - 1\right )} e^{\left (3 \, e^{3} - 9\right )}}{\sqrt {{\left (2 \, x^{2} e^{\left (2 \, e^{3} - 6\right )} - 9 \, e^{\left (2 \, e^{3} - 6\right )}\right )}^{2}} \left (-e^{\left (2 \, e^{3} - 6\right )}\right )^{\frac {3}{2}}} - \frac {\sqrt {\frac {2}{3}} e^{\left (-\frac {1}{6} \, {\left (2 \, x^{2} e^{\left (2 \, e^{3} - 6\right )} - 9 \, e^{\left (2 \, e^{3} - 6\right )}\right )}^{2} e^{\left (-2 \, e^{3} + 6\right )} + 2 \, e^{3} - 6\right )}}{\left (-e^{\left (2 \, e^{3} - 6\right )}\right )^{\frac {3}{2}}}\right )} e^{\left (2 \, e^{3} + \frac {27}{2} \, e^{\left (2 \, e^{3} - 6\right )} - 6\right )}}{2 \, \sqrt {-e^{\left (2 \, e^{3} - 6\right )}}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-8*x^3+36*x)*exp(exp(3)-3)^2*exp(1/3*(-2*x^4+18*x^2)*exp(exp(3)-3)^2)+1,x, algorithm="maxima")

[Out]

3/2*sqrt(3)*sqrt(2)*sqrt(pi)*erf(1/3*sqrt(3)*sqrt(2)*x^2*e^(e^3 - 3) - 3/2*sqrt(3)*sqrt(2)*e^(e^3 - 3))*e^(e^3
+ 27/2*e^(2*e^3 - 6) - 3) - 3/2*sqrt(2/3)*(3*sqrt(pi)*(2*x^2*e^(2*e^3 - 6) - 9*e^(2*e^3 - 6))*(erf(sqrt(1/6)*
sqrt((2*x^2*e^(2*e^3 - 6) - 9*e^(2*e^3 - 6))^2)*e^(-e^3 + 3)) - 1)*e^(3*e^3 - 9)/(sqrt((2*x^2*e^(2*e^3 - 6) -
9*e^(2*e^3 - 6))^2)*(-e^(2*e^3 - 6))^(3/2)) - sqrt(2/3)*e^(-1/6*(2*x^2*e^(2*e^3 - 6) - 9*e^(2*e^3 - 6))^2*e^(-
2*e^3 + 6) + 2*e^3 - 6)/(-e^(2*e^3 - 6))^(3/2))*e^(2*e^3 + 27/2*e^(2*e^3 - 6) - 6)/sqrt(-e^(2*e^3 - 6)) + x

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mupad [B]  time = 0.23, size = 28, normalized size = 0.97 \begin {gather*} x+{\mathrm {e}}^{6\,x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^3}\,{\mathrm {e}}^{-6}-\frac {2\,x^4\,{\mathrm {e}}^{2\,{\mathrm {e}}^3}\,{\mathrm {e}}^{-6}}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((exp(2*exp(3) - 6)*(18*x^2 - 2*x^4))/3)*exp(2*exp(3) - 6)*(36*x - 8*x^3))/3 + 1,x)

[Out]

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

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sympy [A]  time = 0.16, size = 22, normalized size = 0.76 \begin {gather*} x + e^{\left (- \frac {2 x^{4}}{3} + 6 x^{2}\right ) e^{-6 + 2 e^{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-8*x**3+36*x)*exp(exp(3)-3)**2*exp(1/3*(-2*x**4+18*x**2)*exp(exp(3)-3)**2)+1,x)

[Out]

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

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