3.37.21 \(\int \frac {1}{3} e^{\frac {1}{3} (2 e^{e^{20}} x-9 x^2)} (-3-2 e^{e^{20}} x+18 x^2) \, dx\)

Optimal. Leaf size=28 \[ \frac {1}{\log \left (e^{-\frac {e^{-\frac {2}{3} e^{e^{20}} x+3 x^2}}{x}}\right )} \]

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Rubi [A]  time = 0.06, antiderivative size = 46, normalized size of antiderivative = 1.64, number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {12, 2288} \begin {gather*} -\frac {e^{\frac {1}{3} \left (2 e^{e^{20}} x-9 x^2\right )} \left (e^{e^{20}} x-9 x^2\right )}{e^{e^{20}}-9 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((2*E^E^20*x - 9*x^2)/3)*(-3 - 2*E^E^20*x + 18*x^2))/3,x]

[Out]

-((E^((2*E^E^20*x - 9*x^2)/3)*(E^E^20*x - 9*x^2))/(E^E^20 - 9*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 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int e^{\frac {1}{3} \left (2 e^{e^{20}} x-9 x^2\right )} \left (-3-2 e^{e^{20}} x+18 x^2\right ) \, dx\\ &=-\frac {e^{\frac {1}{3} \left (2 e^{e^{20}} x-9 x^2\right )} \left (e^{e^{20}} x-9 x^2\right )}{e^{e^{20}}-9 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 21, normalized size = 0.75 \begin {gather*} -e^{\frac {2 e^{e^{20}} x}{3}-3 x^2} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*E^E^20*x - 9*x^2)/3)*(-3 - 2*E^E^20*x + 18*x^2))/3,x]

[Out]

-(E^((2*E^E^20*x)/3 - 3*x^2)*x)

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fricas [A]  time = 0.73, size = 16, normalized size = 0.57 \begin {gather*} -x e^{\left (-3 \, x^{2} + \frac {2}{3} \, x e^{\left (e^{20}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-2*x*exp(exp(20))+18*x^2-3)/exp(-2/3*x*exp(exp(20))+3*x^2),x, algorithm="fricas")

[Out]

-x*e^(-3*x^2 + 2/3*x*e^(e^20))

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giac [A]  time = 0.21, size = 40, normalized size = 1.43 \begin {gather*} -\frac {1}{9} \, {\left (9 \, x + e^{\left (e^{20}\right )}\right )} e^{\left (-3 \, x^{2} + \frac {2}{3} \, x e^{\left (e^{20}\right )}\right )} + \frac {1}{9} \, e^{\left (-3 \, x^{2} + \frac {2}{3} \, x e^{\left (e^{20}\right )} + e^{20}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-2*x*exp(exp(20))+18*x^2-3)/exp(-2/3*x*exp(exp(20))+3*x^2),x, algorithm="giac")

[Out]

-1/9*(9*x + e^(e^20))*e^(-3*x^2 + 2/3*x*e^(e^20)) + 1/9*e^(-3*x^2 + 2/3*x*e^(e^20) + e^20)

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maple [A]  time = 0.06, size = 17, normalized size = 0.61

method	result	size
risch	\(-x \,{\mathrm e}^{-\frac {x \left (-2 \,{\mathrm e}^{{\mathrm e}^{20}}+9 x \right )}{3}}\)	\(17\)
gosper	\(-x \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{{\mathrm e}^{20}}}{3}-3 x^{2}}\)	\(19\)
norman	\(-x \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{{\mathrm e}^{20}}}{3}-3 x^{2}}\)	\(19\)
default	\(-x \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{{\mathrm e}^{20}}}{3}-3 x^{2}}+\frac {2 \,{\mathrm e}^{{\mathrm e}^{20}} \left (-\frac {{\mathrm e}^{\frac {2 x \,{\mathrm e}^{{\mathrm e}^{20}}}{3}-3 x^{2}}}{6}+\frac {{\mathrm e}^{{\mathrm e}^{20}} \sqrt {\pi }\, {\mathrm e}^{\frac {{\mathrm e}^{2 \,{\mathrm e}^{20}}}{27}} \sqrt {3}\, \erf \left (\sqrt {3}\, x -\frac {{\mathrm e}^{{\mathrm e}^{20}} \sqrt {3}}{9}\right )}{54}\right )}{3}+\frac {{\mathrm e}^{\frac {2 x \,{\mathrm e}^{{\mathrm e}^{20}}}{3}-3 x^{2}+{\mathrm e}^{20}}}{9}-\frac {{\mathrm e}^{{\mathrm e}^{20}} \sqrt {\pi }\, {\mathrm e}^{{\mathrm e}^{20}+\frac {{\mathrm e}^{2 \,{\mathrm e}^{20}}}{27}} \sqrt {3}\, \erf \left (\sqrt {3}\, x -\frac {{\mathrm e}^{{\mathrm e}^{20}} \sqrt {3}}{9}\right )}{81}\)	\(127\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(-2*x*exp(exp(20))+18*x^2-3)/exp(-2/3*x*exp(exp(20))+3*x^2),x,method=_RETURNVERBOSE)

[Out]

-x*exp(-1/3*x*(-2*exp(exp(20))+9*x))

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maxima [C]  time = 0.70, size = 267, normalized size = 9.54 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} x - \frac {1}{9} \, \sqrt {3} e^{\left (e^{20}\right )}\right ) e^{\left (\frac {1}{27} \, e^{\left (2 \, e^{20}\right )}\right )} - \frac {1}{81} i \, \sqrt {3} {\left (\frac {i \, \sqrt {3} \sqrt {\frac {1}{3}} \sqrt {\pi } {\left (9 \, x - e^{\left (e^{20}\right )}\right )} {\left (\operatorname {erf}\left (\frac {1}{3} \, \sqrt {\frac {1}{3}} \sqrt {{\left (9 \, x - e^{\left (e^{20}\right )}\right )}^{2}}\right ) - 1\right )} e^{\left (2 \, e^{20}\right )}}{\sqrt {{\left (9 \, x - e^{\left (e^{20}\right )}\right )}^{2}}} - \frac {27 i \, \sqrt {3} \sqrt {\frac {1}{3}} {\left (9 \, x - e^{\left (e^{20}\right )}\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {1}{27} \, {\left (9 \, x - e^{\left (e^{20}\right )}\right )}^{2}\right )}{{\left ({\left (9 \, x - e^{\left (e^{20}\right )}\right )}^{2}\right )}^{\frac {3}{2}}} - 6 i \, \sqrt {3} e^{\left (-\frac {1}{27} \, {\left (9 \, x - e^{\left (e^{20}\right )}\right )}^{2} + e^{20}\right )}\right )} e^{\left (\frac {1}{27} \, e^{\left (2 \, e^{20}\right )}\right )} + \frac {1}{81} i \, \sqrt {3} {\left (\frac {i \, \sqrt {3} \sqrt {\frac {1}{3}} \sqrt {\pi } {\left (9 \, x - e^{\left (e^{20}\right )}\right )} {\left (\operatorname {erf}\left (\frac {1}{3} \, \sqrt {\frac {1}{3}} \sqrt {{\left (9 \, x - e^{\left (e^{20}\right )}\right )}^{2}}\right ) - 1\right )} e^{\left (e^{20}\right )}}{\sqrt {{\left (9 \, x - e^{\left (e^{20}\right )}\right )}^{2}}} - 3 i \, \sqrt {3} e^{\left (-\frac {1}{27} \, {\left (9 \, x - e^{\left (e^{20}\right )}\right )}^{2}\right )}\right )} e^{\left (e^{20} + \frac {1}{27} \, e^{\left (2 \, e^{20}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-2*x*exp(exp(20))+18*x^2-3)/exp(-2/3*x*exp(exp(20))+3*x^2),x, algorithm="maxima")

[Out]

-1/6*sqrt(3)*sqrt(pi)*erf(sqrt(3)*x - 1/9*sqrt(3)*e^(e^20))*e^(1/27*e^(2*e^20)) - 1/81*I*sqrt(3)*(I*sqrt(3)*sq
rt(1/3)*sqrt(pi)*(9*x - e^(e^20))*(erf(1/3*sqrt(1/3)*sqrt((9*x - e^(e^20))^2)) - 1)*e^(2*e^20)/sqrt((9*x - e^(
e^20))^2) - 27*I*sqrt(3)*sqrt(1/3)*(9*x - e^(e^20))^3*gamma(3/2, 1/27*(9*x - e^(e^20))^2)/((9*x - e^(e^20))^2)
^(3/2) - 6*I*sqrt(3)*e^(-1/27*(9*x - e^(e^20))^2 + e^20))*e^(1/27*e^(2*e^20)) + 1/81*I*sqrt(3)*(I*sqrt(3)*sqrt
(1/3)*sqrt(pi)*(9*x - e^(e^20))*(erf(1/3*sqrt(1/3)*sqrt((9*x - e^(e^20))^2)) - 1)*e^(e^20)/sqrt((9*x - e^(e^20
))^2) - 3*I*sqrt(3)*e^(-1/27*(9*x - e^(e^20))^2))*e^(e^20 + 1/27*e^(2*e^20))

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mupad [B]  time = 0.13, size = 16, normalized size = 0.57 \begin {gather*} -x\,{\mathrm {e}}^{\frac {2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{20}}}{3}}\,{\mathrm {e}}^{-3\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp((2*x*exp(exp(20)))/3 - 3*x^2)*((2*x*exp(exp(20)))/3 - 6*x^2 + 1),x)

[Out]

-x*exp((2*x*exp(exp(20)))/3)*exp(-3*x^2)

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sympy [A]  time = 0.13, size = 19, normalized size = 0.68 \begin {gather*} - x e^{- 3 x^{2} + \frac {2 x e^{e^{20}}}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-2*x*exp(exp(20))+18*x**2-3)/exp(-2/3*x*exp(exp(20))+3*x**2),x)

[Out]

-x*exp(-3*x**2 + 2*x*exp(exp(20))/3)

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