3.61.89 \(\int \frac {1}{9} e^{3^{e^{-x} x}} (e^x (2 x+4 x^2+x^3)+3^{e^{-x} x} (x^2-x^4) \log (3)) \, dx\)

Optimal. Leaf size=23 \[ \frac {1}{9} e^{3^{e^{-x} x}+x} x^2 (1+x) \]

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Rubi [A]  time = 0.07, antiderivative size = 40, normalized size of antiderivative = 1.74, number of steps used = 2, number of rules used = 2, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {12, 2288} \begin {gather*} \frac {e^{3^{e^{-x} x}} \left (x^2-x^4\right )}{9 \left (e^{-x}-e^{-x} x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^3^(x/E^x)*(E^x*(2*x + 4*x^2 + x^3) + 3^(x/E^x)*(x^2 - x^4)*Log[3]))/9,x]

[Out]

(E^3^(x/E^x)*(x^2 - x^4))/(9*(E^(-x) - x/E^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}{9} \int e^{3^{e^{-x} x}} \left (e^x \left (2 x+4 x^2+x^3\right )+3^{e^{-x} x} \left (x^2-x^4\right ) \log (3)\right ) \, dx\\ &=\frac {e^{3^{e^{-x} x}} \left (x^2-x^4\right )}{9 \left (e^{-x}-e^{-x} x\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 23, normalized size = 1.00 \begin {gather*} \frac {1}{9} e^{3^{e^{-x} x}+x} x^2 (1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^3^(x/E^x)*(E^x*(2*x + 4*x^2 + x^3) + 3^(x/E^x)*(x^2 - x^4)*Log[3]))/9,x]

[Out]

(E^(3^(x/E^x) + x)*x^2*(1 + x))/9

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fricas [A]  time = 0.50, size = 20, normalized size = 0.87 \begin {gather*} \frac {1}{9} \, {\left (x^{3} + x^{2}\right )} e^{\left (3^{x e^{\left (-x\right )}} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((-x^4+x^2)*log(3)*exp(x*log(3)/exp(x))+(x^3+4*x^2+2*x)*exp(x))*exp(exp(x*log(3)/exp(x))),x, alg
orithm="fricas")

[Out]

1/9*(x^3 + x^2)*e^(3^(x*e^(-x)) + x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {1}{9} \, {\left ({\left (x^{4} - x^{2}\right )} 3^{x e^{\left (-x\right )}} \log \relax (3) - {\left (x^{3} + 4 \, x^{2} + 2 \, x\right )} e^{x}\right )} e^{\left (3^{x e^{\left (-x\right )}}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((-x^4+x^2)*log(3)*exp(x*log(3)/exp(x))+(x^3+4*x^2+2*x)*exp(x))*exp(exp(x*log(3)/exp(x))),x, alg
orithm="giac")

[Out]

integrate(-1/9*((x^4 - x^2)*3^(x*e^(-x))*log(3) - (x^3 + 4*x^2 + 2*x)*e^x)*e^(3^(x*e^(-x))), x)

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maple [A]  time = 0.06, size = 20, normalized size = 0.87

method	result	size
risch	\(\frac {\left (x +1\right ) x^{2} {\mathrm e}^{x +3^{x \,{\mathrm e}^{-x}}}}{9}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*((-x^4+x^2)*ln(3)*exp(x*ln(3)/exp(x))+(x^3+4*x^2+2*x)*exp(x))*exp(exp(x*ln(3)/exp(x))),x,method=_RETUR
NVERBOSE)

[Out]

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

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maxima [A]  time = 0.49, size = 20, normalized size = 0.87 \begin {gather*} \frac {1}{9} \, {\left (x^{3} + x^{2}\right )} e^{\left (3^{x e^{\left (-x\right )}} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((-x^4+x^2)*log(3)*exp(x*log(3)/exp(x))+(x^3+4*x^2+2*x)*exp(x))*exp(exp(x*log(3)/exp(x))),x, alg
orithm="maxima")

[Out]

1/9*(x^3 + x^2)*e^(3^(x*e^(-x)) + x)

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mupad [B]  time = 4.59, size = 19, normalized size = 0.83 \begin {gather*} \frac {x^2\,{\mathrm {e}}^{3^{x\,{\mathrm {e}}^{-x}}}\,{\mathrm {e}}^x\,\left (x+1\right )}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(x*exp(-x)*log(3)))*(exp(x)*(2*x + 4*x^2 + x^3) + exp(x*exp(-x)*log(3))*log(3)*(x^2 - x^4)))/9,x)

[Out]

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

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sympy [A]  time = 0.30, size = 22, normalized size = 0.96 \begin {gather*} \frac {\left (x^{3} + x^{2}\right ) e^{x} e^{e^{x e^{- x} \log {\relax (3 )}}}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((-x**4+x**2)*ln(3)*exp(x*ln(3)/exp(x))+(x**3+4*x**2+2*x)*exp(x))*exp(exp(x*ln(3)/exp(x))),x)

[Out]

(x**3 + x**2)*exp(x)*exp(exp(x*exp(-x)*log(3)))/9

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