3.66.67 \(\int \frac {1}{3} e^{\frac {1}{9} (e^2+e^{6 x}+9 e^{2 e^{5 e^x} x}+2 e^{1+3 x}+e^{e^{5 e^x} x} (6 e+6 e^{3 x}))} (2 e^{6 x}+2 e^{1+3 x}+e^{5 e^x+2 e^{5 e^x} x} (6+30 e^x x)+e^{e^{5 e^x} x} (6 e^{3 x}+e^{5 e^x} (2 e+10 e^{1+x} x+e^{3 x} (2+10 e^x x)))) \, dx\)

Optimal. Leaf size=27 \[ e^{\left (e^{e^{5 e^x} x}+\frac {1}{3} \left (e+e^{3 x}\right )\right )^2} \]

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Rubi [A]  time = 3.83, antiderivative size = 28, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, integrand size = 163, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {12, 6688, 6706} \begin {gather*} e^{\frac {1}{9} \left (e^{3 x}+3 e^{e^{5 e^x} x}+e\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((E^2 + E^(6*x) + 9*E^(2*E^(5*E^x)*x) + 2*E^(1 + 3*x) + E^(E^(5*E^x)*x)*(6*E + 6*E^(3*x)))/9)*(2*E^(6*x
) + 2*E^(1 + 3*x) + E^(5*E^x + 2*E^(5*E^x)*x)*(6 + 30*E^x*x) + E^(E^(5*E^x)*x)*(6*E^(3*x) + E^(5*E^x)*(2*E + 1
0*E^(1 + x)*x + E^(3*x)*(2 + 10*E^x*x)))))/3,x]

[Out]

E^((E + E^(3*x) + 3*E^(E^(5*E^x)*x))^2/9)

Rule 12

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

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

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 \exp \left (\frac {1}{9} \left (e^2+e^{6 x}+9 e^{2 e^{5 e^x} x}+2 e^{1+3 x}+e^{e^{5 e^x} x} \left (6 e+6 e^{3 x}\right )\right )\right ) \left (2 e^{6 x}+2 e^{1+3 x}+e^{5 e^x+2 e^{5 e^x} x} \left (6+30 e^x x\right )+e^{e^{5 e^x} x} \left (6 e^{3 x}+e^{5 e^x} \left (2 e+10 e^{1+x} x+e^{3 x} \left (2+10 e^x x\right )\right )\right )\right ) \, dx\\ &=\frac {1}{3} \int 2 e^{\frac {1}{9} \left (e+e^{3 x}+3 e^{e^{5 e^x} x}\right )^2} \left (e+e^{3 x}+3 e^{e^{5 e^x} x}\right ) \left (e^{3 x}+e^{5 e^x+e^{5 e^x} x}+5 e^{5 e^x+x+e^{5 e^x} x} x\right ) \, dx\\ &=\frac {2}{3} \int e^{\frac {1}{9} \left (e+e^{3 x}+3 e^{e^{5 e^x} x}\right )^2} \left (e+e^{3 x}+3 e^{e^{5 e^x} x}\right ) \left (e^{3 x}+e^{5 e^x+e^{5 e^x} x}+5 e^{5 e^x+x+e^{5 e^x} x} x\right ) \, dx\\ &=e^{\frac {1}{9} \left (e+e^{3 x}+3 e^{e^{5 e^x} x}\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 28, normalized size = 1.04 \begin {gather*} e^{\frac {1}{9} \left (e+e^{3 x}+3 e^{e^{5 e^x} x}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((E^2 + E^(6*x) + 9*E^(2*E^(5*E^x)*x) + 2*E^(1 + 3*x) + E^(E^(5*E^x)*x)*(6*E + 6*E^(3*x)))/9)*(2*
E^(6*x) + 2*E^(1 + 3*x) + E^(5*E^x + 2*E^(5*E^x)*x)*(6 + 30*E^x*x) + E^(E^(5*E^x)*x)*(6*E^(3*x) + E^(5*E^x)*(2
*E + 10*E^(1 + x)*x + E^(3*x)*(2 + 10*E^x*x)))))/3,x]

[Out]

E^((E + E^(3*x) + 3*E^(E^(5*E^x)*x))^2/9)

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fricas [B]  time = 0.76, size = 54, normalized size = 2.00 \begin {gather*} e^{\left (\frac {1}{9} \, {\left (6 \, {\left (e^{7} + e^{\left (3 \, x + 6\right )}\right )} e^{\left (x e^{\left (5 \, e^{x}\right )}\right )} + e^{8} + 9 \, e^{\left (2 \, x e^{\left (5 \, e^{x}\right )} + 6\right )} + e^{\left (6 \, x + 6\right )} + 2 \, e^{\left (3 \, x + 7\right )}\right )} e^{\left (-6\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((30*exp(x)*x+6)*exp(5*exp(x))*exp(x*exp(5*exp(x)))^2+(((10*exp(x)*x+2)*exp(3*x)+10*x*exp(1)*exp
(x)+2*exp(1))*exp(5*exp(x))+6*exp(3*x))*exp(x*exp(5*exp(x)))+2*exp(3*x)^2+2*exp(1)*exp(3*x))*exp(exp(x*exp(5*e
xp(x)))^2+1/9*(6*exp(3*x)+6*exp(1))*exp(x*exp(5*exp(x)))+1/9*exp(3*x)^2+2/9*exp(1)*exp(3*x)+1/9*exp(1)^2),x, a
lgorithm="fricas")

[Out]

e^(1/9*(6*(e^7 + e^(3*x + 6))*e^(x*e^(5*e^x)) + e^8 + 9*e^(2*x*e^(5*e^x) + 6) + e^(6*x + 6) + 2*e^(3*x + 7))*e
^(-6))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2}{3} \, {\left (3 \, {\left (5 \, x e^{x} + 1\right )} e^{\left (2 \, x e^{\left (5 \, e^{x}\right )} + 5 \, e^{x}\right )} + {\left ({\left ({\left (5 \, x e^{x} + 1\right )} e^{\left (3 \, x\right )} + 5 \, x e^{\left (x + 1\right )} + e\right )} e^{\left (5 \, e^{x}\right )} + 3 \, e^{\left (3 \, x\right )}\right )} e^{\left (x e^{\left (5 \, e^{x}\right )}\right )} + e^{\left (6 \, x\right )} + e^{\left (3 \, x + 1\right )}\right )} e^{\left (\frac {2}{3} \, {\left (e + e^{\left (3 \, x\right )}\right )} e^{\left (x e^{\left (5 \, e^{x}\right )}\right )} + \frac {1}{9} \, e^{2} + e^{\left (2 \, x e^{\left (5 \, e^{x}\right )}\right )} + \frac {1}{9} \, e^{\left (6 \, x\right )} + \frac {2}{9} \, e^{\left (3 \, x + 1\right )}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((30*exp(x)*x+6)*exp(5*exp(x))*exp(x*exp(5*exp(x)))^2+(((10*exp(x)*x+2)*exp(3*x)+10*x*exp(1)*exp
(x)+2*exp(1))*exp(5*exp(x))+6*exp(3*x))*exp(x*exp(5*exp(x)))+2*exp(3*x)^2+2*exp(1)*exp(3*x))*exp(exp(x*exp(5*e
xp(x)))^2+1/9*(6*exp(3*x)+6*exp(1))*exp(x*exp(5*exp(x)))+1/9*exp(3*x)^2+2/9*exp(1)*exp(3*x)+1/9*exp(1)^2),x, a
lgorithm="giac")

[Out]

integrate(2/3*(3*(5*x*e^x + 1)*e^(2*x*e^(5*e^x) + 5*e^x) + (((5*x*e^x + 1)*e^(3*x) + 5*x*e^(x + 1) + e)*e^(5*e
^x) + 3*e^(3*x))*e^(x*e^(5*e^x)) + e^(6*x) + e^(3*x + 1))*e^(2/3*(e + e^(3*x))*e^(x*e^(5*e^x)) + 1/9*e^2 + e^(
2*x*e^(5*e^x)) + 1/9*e^(6*x) + 2/9*e^(3*x + 1)), x)

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maple [B]  time = 0.19, size = 54, normalized size = 2.00

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{2 x \,{\mathrm e}^{5 \,{\mathrm e}^{x}}}+\frac {2 \,{\mathrm e}^{x \,{\mathrm e}^{5 \,{\mathrm e}^{x}}+1}}{3}+\frac {2 \,{\mathrm e}^{x \left ({\mathrm e}^{5 \,{\mathrm e}^{x}}+3\right )}}{3}+\frac {{\mathrm e}^{6 x}}{9}+\frac {2 \,{\mathrm e}^{3 x +1}}{9}+\frac {{\mathrm e}^{2}}{9}}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((30*exp(x)*x+6)*exp(5*exp(x))*exp(x*exp(5*exp(x)))^2+(((10*exp(x)*x+2)*exp(3*x)+10*x*exp(1)*exp(x)+2*
exp(1))*exp(5*exp(x))+6*exp(3*x))*exp(x*exp(5*exp(x)))+2*exp(3*x)^2+2*exp(1)*exp(3*x))*exp(exp(x*exp(5*exp(x))
)^2+1/9*(6*exp(3*x)+6*exp(1))*exp(x*exp(5*exp(x)))+1/9*exp(3*x)^2+2/9*exp(1)*exp(3*x)+1/9*exp(1)^2),x,method=_
RETURNVERBOSE)

[Out]

exp(exp(2*x*exp(5*exp(x)))+2/3*exp(x*exp(5*exp(x))+1)+2/3*exp(x*(exp(5*exp(x))+3))+1/9*exp(6*x)+2/9*exp(3*x+1)
+1/9*exp(2))

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maxima [B]  time = 0.75, size = 55, normalized size = 2.04 \begin {gather*} e^{\left (\frac {1}{9} \, e^{2} + e^{\left (2 \, x e^{\left (5 \, e^{x}\right )}\right )} + \frac {2}{3} \, e^{\left (x e^{\left (5 \, e^{x}\right )} + 3 \, x\right )} + \frac {2}{3} \, e^{\left (x e^{\left (5 \, e^{x}\right )} + 1\right )} + \frac {1}{9} \, e^{\left (6 \, x\right )} + \frac {2}{9} \, e^{\left (3 \, x + 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((30*exp(x)*x+6)*exp(5*exp(x))*exp(x*exp(5*exp(x)))^2+(((10*exp(x)*x+2)*exp(3*x)+10*x*exp(1)*exp
(x)+2*exp(1))*exp(5*exp(x))+6*exp(3*x))*exp(x*exp(5*exp(x)))+2*exp(3*x)^2+2*exp(1)*exp(3*x))*exp(exp(x*exp(5*e
xp(x)))^2+1/9*(6*exp(3*x)+6*exp(1))*exp(x*exp(5*exp(x)))+1/9*exp(3*x)^2+2/9*exp(1)*exp(3*x)+1/9*exp(1)^2),x, a
lgorithm="maxima")

[Out]

e^(1/9*e^2 + e^(2*x*e^(5*e^x)) + 2/3*e^(x*e^(5*e^x) + 3*x) + 2/3*e^(x*e^(5*e^x) + 1) + 1/9*e^(6*x) + 2/9*e^(3*
x + 1))

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mupad [B]  time = 4.51, size = 60, normalized size = 2.22 \begin {gather*} {\mathrm {e}}^{\frac {{\mathrm {e}}^{6\,x}}{9}}\,{\mathrm {e}}^{\frac {2\,\mathrm {e}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{5\,{\mathrm {e}}^x}}}{3}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^2}{9}}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{5\,{\mathrm {e}}^x}}}\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{3\,x}\,\mathrm {e}}{9}}\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{5\,{\mathrm {e}}^x}}}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(6*x)/9 + exp(2)/9 + exp(2*x*exp(5*exp(x))) + (2*exp(3*x)*exp(1))/9 + (exp(x*exp(5*exp(x)))*(6*exp
(3*x) + 6*exp(1)))/9)*(2*exp(6*x) + 2*exp(3*x)*exp(1) + exp(x*exp(5*exp(x)))*(6*exp(3*x) + exp(5*exp(x))*(2*ex
p(1) + exp(3*x)*(10*x*exp(x) + 2) + 10*x*exp(1)*exp(x))) + exp(5*exp(x))*exp(2*x*exp(5*exp(x)))*(30*x*exp(x) +
6)))/3,x)

[Out]

exp(exp(6*x)/9)*exp((2*exp(1)*exp(x*exp(5*exp(x))))/3)*exp(exp(2)/9)*exp(exp(2*x*exp(5*exp(x))))*exp((2*exp(3*
x)*exp(1))/9)*exp((2*exp(3*x)*exp(x*exp(5*exp(x))))/3)

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sympy [B]  time = 11.20, size = 61, normalized size = 2.26 \begin {gather*} e^{\left (\frac {2 e^{3 x}}{3} + \frac {2 e}{3}\right ) e^{x e^{5 e^{x}}} + \frac {e^{6 x}}{9} + \frac {2 e e^{3 x}}{9} + e^{2 x e^{5 e^{x}}} + \frac {e^{2}}{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((30*exp(x)*x+6)*exp(5*exp(x))*exp(x*exp(5*exp(x)))**2+(((10*exp(x)*x+2)*exp(3*x)+10*x*exp(1)*ex
p(x)+2*exp(1))*exp(5*exp(x))+6*exp(3*x))*exp(x*exp(5*exp(x)))+2*exp(3*x)**2+2*exp(1)*exp(3*x))*exp(exp(x*exp(5
*exp(x)))**2+1/9*(6*exp(3*x)+6*exp(1))*exp(x*exp(5*exp(x)))+1/9*exp(3*x)**2+2/9*exp(1)*exp(3*x)+1/9*exp(1)**2)
,x)

[Out]

exp((2*exp(3*x)/3 + 2*E/3)*exp(x*exp(5*exp(x))) + exp(6*x)/9 + 2*E*exp(3*x)/9 + exp(2*x*exp(5*exp(x))) + exp(2
)/9)

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