3.83.27 \(\int \frac {1}{81} e^{2 e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} x-2 x^2} (81-324 x^2+e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} (162 x+e^{\frac {x^4}{81}} (972 x^2+48 x^6)+e^{\frac {2 x^4}{81}} (324 x^3+16 x^7))) \, dx\)

Optimal. Leaf size=30 \[ 5+e^{-2 x \left (-e^{\left (3+e^{\frac {x^4}{81}} x\right )^2}+x\right )} x \]

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Rubi [B]  time = 1.58, antiderivative size = 248, normalized size of antiderivative = 8.27, number of steps used = 2, number of rules used = 2, integrand size = 127, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {12, 2288} \begin {gather*} -\frac {\left (162 x^2-e^{6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2+9} \left (2 e^{\frac {2 x^4}{81}} \left (4 x^7+81 x^3\right )+6 e^{\frac {x^4}{81}} \left (4 x^6+81 x^2\right )+81 x\right )\right ) \exp \left (2 e^{6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2+9} x-2 x^2\right )}{81 e^{6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2+9}+2 e^{6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2+9} \left (12 e^{\frac {x^4}{81}} x^4+81 e^{\frac {2 x^4}{81}} x+243 e^{\frac {x^4}{81}}+4 e^{\frac {2 x^4}{81}} x^5\right ) x-162 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2*E^(9 + 6*E^(x^4/81)*x + E^((2*x^4)/81)*x^2)*x - 2*x^2)*(81 - 324*x^2 + E^(9 + 6*E^(x^4/81)*x + E^((2
*x^4)/81)*x^2)*(162*x + E^(x^4/81)*(972*x^2 + 48*x^6) + E^((2*x^4)/81)*(324*x^3 + 16*x^7))))/81,x]

[Out]

-((E^(2*E^(9 + 6*E^(x^4/81)*x + E^((2*x^4)/81)*x^2)*x - 2*x^2)*(162*x^2 - E^(9 + 6*E^(x^4/81)*x + E^((2*x^4)/8
1)*x^2)*(81*x + 6*E^(x^4/81)*(81*x^2 + 4*x^6) + 2*E^((2*x^4)/81)*(81*x^3 + 4*x^7))))/(81*E^(9 + 6*E^(x^4/81)*x
+ E^((2*x^4)/81)*x^2) - 162*x + 2*E^(9 + 6*E^(x^4/81)*x + E^((2*x^4)/81)*x^2)*x*(243*E^(x^4/81) + 81*E^((2*x^
4)/81)*x + 12*E^(x^4/81)*x^4 + 4*E^((2*x^4)/81)*x^5)))

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}{81} \int \exp \left (2 e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} x-2 x^2\right ) \left (81-324 x^2+e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} \left (162 x+e^{\frac {x^4}{81}} \left (972 x^2+48 x^6\right )+e^{\frac {2 x^4}{81}} \left (324 x^3+16 x^7\right )\right )\right ) \, dx\\ &=-\frac {\exp \left (2 e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} x-2 x^2\right ) \left (162 x^2-e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} \left (81 x+6 e^{\frac {x^4}{81}} \left (81 x^2+4 x^6\right )+2 e^{\frac {2 x^4}{81}} \left (81 x^3+4 x^7\right )\right )\right )}{81 e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2}-162 x+2 e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} x \left (243 e^{\frac {x^4}{81}}+81 e^{\frac {2 x^4}{81}} x+12 e^{\frac {x^4}{81}} x^4+4 e^{\frac {2 x^4}{81}} x^5\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 28, normalized size = 0.93 \begin {gather*} e^{2 \left (e^{\left (3+e^{\frac {x^4}{81}} x\right )^2}-x\right ) x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*E^(9 + 6*E^(x^4/81)*x + E^((2*x^4)/81)*x^2)*x - 2*x^2)*(81 - 324*x^2 + E^(9 + 6*E^(x^4/81)*x +
E^((2*x^4)/81)*x^2)*(162*x + E^(x^4/81)*(972*x^2 + 48*x^6) + E^((2*x^4)/81)*(324*x^3 + 16*x^7))))/81,x]

[Out]

E^(2*(E^(3 + E^(x^4/81)*x)^2 - x)*x)*x

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fricas [A]  time = 0.65, size = 34, normalized size = 1.13 \begin {gather*} x e^{\left (-2 \, x^{2} + 2 \, x e^{\left (x^{2} e^{\left (\frac {2}{81} \, x^{4}\right )} + 6 \, x e^{\left (\frac {1}{81} \, x^{4}\right )} + 9\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*(((16*x^7+324*x^3)*exp(1/81*x^4)^2+(48*x^6+972*x^2)*exp(1/81*x^4)+162*x)*exp(x^2*exp(1/81*x^4)^
2+6*x*exp(1/81*x^4)+9)-324*x^2+81)/exp(-2*x*exp(x^2*exp(1/81*x^4)^2+6*x*exp(1/81*x^4)+9)+2*x^2),x, algorithm="
fricas")

[Out]

x*e^(-2*x^2 + 2*x*e^(x^2*e^(2/81*x^4) + 6*x*e^(1/81*x^4) + 9))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {1}{81} \, {\left (324 \, x^{2} - 2 \, {\left (2 \, {\left (4 \, x^{7} + 81 \, x^{3}\right )} e^{\left (\frac {2}{81} \, x^{4}\right )} + 6 \, {\left (4 \, x^{6} + 81 \, x^{2}\right )} e^{\left (\frac {1}{81} \, x^{4}\right )} + 81 \, x\right )} e^{\left (x^{2} e^{\left (\frac {2}{81} \, x^{4}\right )} + 6 \, x e^{\left (\frac {1}{81} \, x^{4}\right )} + 9\right )} - 81\right )} e^{\left (-2 \, x^{2} + 2 \, x e^{\left (x^{2} e^{\left (\frac {2}{81} \, x^{4}\right )} + 6 \, x e^{\left (\frac {1}{81} \, x^{4}\right )} + 9\right )}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*(((16*x^7+324*x^3)*exp(1/81*x^4)^2+(48*x^6+972*x^2)*exp(1/81*x^4)+162*x)*exp(x^2*exp(1/81*x^4)^
2+6*x*exp(1/81*x^4)+9)-324*x^2+81)/exp(-2*x*exp(x^2*exp(1/81*x^4)^2+6*x*exp(1/81*x^4)+9)+2*x^2),x, algorithm="
giac")

[Out]

integrate(-1/81*(324*x^2 - 2*(2*(4*x^7 + 81*x^3)*e^(2/81*x^4) + 6*(4*x^6 + 81*x^2)*e^(1/81*x^4) + 81*x)*e^(x^2
*e^(2/81*x^4) + 6*x*e^(1/81*x^4) + 9) - 81)*e^(-2*x^2 + 2*x*e^(x^2*e^(2/81*x^4) + 6*x*e^(1/81*x^4) + 9)), x)

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maple [A]  time = 0.11, size = 33, normalized size = 1.10

method	result	size
risch	\(x \,{\mathrm e}^{-2 x \left (-{\mathrm e}^{x^{2} {\mathrm e}^{\frac {2 x^{4}}{81}}+6 x \,{\mathrm e}^{\frac {x^{4}}{81}}+9}+x \right )}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/81*(((16*x^7+324*x^3)*exp(1/81*x^4)^2+(48*x^6+972*x^2)*exp(1/81*x^4)+162*x)*exp(x^2*exp(1/81*x^4)^2+6*x*
exp(1/81*x^4)+9)-324*x^2+81)/exp(-2*x*exp(x^2*exp(1/81*x^4)^2+6*x*exp(1/81*x^4)+9)+2*x^2),x,method=_RETURNVERB
OSE)

[Out]

x*exp(-2*x*(-exp(x^2*exp(2/81*x^4)+6*x*exp(1/81*x^4)+9)+x))

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maxima [A]  time = 0.46, size = 34, normalized size = 1.13 \begin {gather*} x e^{\left (-2 \, x^{2} + 2 \, x e^{\left (x^{2} e^{\left (\frac {2}{81} \, x^{4}\right )} + 6 \, x e^{\left (\frac {1}{81} \, x^{4}\right )} + 9\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*(((16*x^7+324*x^3)*exp(1/81*x^4)^2+(48*x^6+972*x^2)*exp(1/81*x^4)+162*x)*exp(x^2*exp(1/81*x^4)^
2+6*x*exp(1/81*x^4)+9)-324*x^2+81)/exp(-2*x*exp(x^2*exp(1/81*x^4)^2+6*x*exp(1/81*x^4)+9)+2*x^2),x, algorithm="
maxima")

[Out]

x*e^(-2*x^2 + 2*x*e^(x^2*e^(2/81*x^4) + 6*x*e^(1/81*x^4) + 9))

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mupad [B]  time = 0.26, size = 35, normalized size = 1.17 \begin {gather*} x\,{\mathrm {e}}^{-2\,x^2}\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{\frac {2\,x^4}{81}}}\,{\mathrm {e}}^9\,{\mathrm {e}}^{6\,x\,{\mathrm {e}}^{\frac {x^4}{81}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x*exp(6*x*exp(x^4/81) + x^2*exp((2*x^4)/81) + 9) - 2*x^2)*((exp(6*x*exp(x^4/81) + x^2*exp((2*x^4)/81
) + 9)*(162*x + exp((2*x^4)/81)*(324*x^3 + 16*x^7) + exp(x^4/81)*(972*x^2 + 48*x^6)))/81 - 4*x^2 + 1),x)

[Out]

x*exp(-2*x^2)*exp(2*x*exp(x^2*exp((2*x^4)/81))*exp(9)*exp(6*x*exp(x^4/81)))

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sympy [A]  time = 7.11, size = 36, normalized size = 1.20 \begin {gather*} x e^{- 2 x^{2} + 2 x e^{x^{2} e^{\frac {2 x^{4}}{81}} + 6 x e^{\frac {x^{4}}{81}} + 9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*(((16*x**7+324*x**3)*exp(1/81*x**4)**2+(48*x**6+972*x**2)*exp(1/81*x**4)+162*x)*exp(x**2*exp(1/
81*x**4)**2+6*x*exp(1/81*x**4)+9)-324*x**2+81)/exp(-2*x*exp(x**2*exp(1/81*x**4)**2+6*x*exp(1/81*x**4)+9)+2*x**
2),x)

[Out]

x*exp(-2*x**2 + 2*x*exp(x**2*exp(2*x**4/81) + 6*x*exp(x**4/81) + 9))

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