3.60.63 \(\int e^{81+e^{2 x} (4+8 e^3+4 e^6)+18 x+x^2+4 e^6 x^2+e^3 (36 x+4 x^2)+e^x (36+4 x+8 e^6 x+e^3 (36+12 x))} (18+e^{2 x} (8+16 e^3+8 e^6)+2 x+8 e^6 x+e^3 (36+8 x)+e^x (40+4 x+e^6 (8+8 x)+e^3 (48+12 x))) \, dx\)

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

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Rubi [A]  time = 2.13, antiderivative size = 25, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, integrand size = 139, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6, 6688, 12, 6706} \begin {gather*} e^{\left (\left (1+2 e^3\right ) x+2 \left (1+e^3\right ) e^x+9\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(81 + E^(2*x)*(4 + 8*E^3 + 4*E^6) + 18*x + x^2 + 4*E^6*x^2 + E^3*(36*x + 4*x^2) + E^x*(36 + 4*x + 8*E^6*
x + E^3*(36 + 12*x)))*(18 + E^(2*x)*(8 + 16*E^3 + 8*E^6) + 2*x + 8*E^6*x + E^3*(36 + 8*x) + E^x*(40 + 4*x + E^
6*(8 + 8*x) + E^3*(48 + 12*x))),x]

[Out]

E^(9 + 2*E^x*(1 + E^3) + (1 + 2*E^3)*x)^2

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 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} &=\int \exp \left (81+e^{2 x} \left (4+8 e^3+4 e^6\right )+18 x+x^2+4 e^6 x^2+e^3 \left (36 x+4 x^2\right )+e^x \left (36+4 x+8 e^6 x+e^3 (36+12 x)\right )\right ) \left (18+e^{2 x} \left (8+16 e^3+8 e^6\right )+\left (2+8 e^6\right ) x+e^3 (36+8 x)+e^x \left (40+4 x+e^6 (8+8 x)+e^3 (48+12 x)\right )\right ) \, dx\\ &=\int 2 e^{\left (9+2 e^x \left (1+e^3\right )+\left (1+2 e^3\right ) x\right )^2} \left (1+2 e^3+2 e^x \left (1+e^3\right )\right ) \left (9+2 e^x \left (1+e^3\right )+\left (1+2 e^3\right ) x\right ) \, dx\\ &=2 \int e^{\left (9+2 e^x \left (1+e^3\right )+\left (1+2 e^3\right ) x\right )^2} \left (1+2 e^3+2 e^x \left (1+e^3\right )\right ) \left (9+2 e^x \left (1+e^3\right )+\left (1+2 e^3\right ) x\right ) \, dx\\ &=e^{\left (9+2 e^x \left (1+e^3\right )+\left (1+2 e^3\right ) x\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.30, size = 25, normalized size = 1.09 \begin {gather*} e^{\left (9+2 e^x+2 e^{3+x}+x+2 e^3 x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(81 + E^(2*x)*(4 + 8*E^3 + 4*E^6) + 18*x + x^2 + 4*E^6*x^2 + E^3*(36*x + 4*x^2) + E^x*(36 + 4*x +
8*E^6*x + E^3*(36 + 12*x)))*(18 + E^(2*x)*(8 + 16*E^3 + 8*E^6) + 2*x + 8*E^6*x + E^3*(36 + 8*x) + E^x*(40 + 4*
x + E^6*(8 + 8*x) + E^3*(48 + 12*x))),x]

[Out]

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

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fricas [B]  time = 1.00, size = 60, normalized size = 2.61 \begin {gather*} e^{\left (4 \, x^{2} e^{6} + x^{2} + 4 \, {\left (x^{2} + 9 \, x\right )} e^{3} + 4 \, {\left (e^{6} + 2 \, e^{3} + 1\right )} e^{\left (2 \, x\right )} + 4 \, {\left (2 \, x e^{6} + 3 \, {\left (x + 3\right )} e^{3} + x + 9\right )} e^{x} + 18 \, x + 81\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(3)^2+16*exp(3)+8)*exp(x)^2+((8*x+8)*exp(3)^2+(12*x+48)*exp(3)+4*x+40)*exp(x)+8*x*exp(3)^2+(8
*x+36)*exp(3)+2*x+18)*exp((4*exp(3)^2+8*exp(3)+4)*exp(x)^2+(8*x*exp(3)^2+(12*x+36)*exp(3)+4*x+36)*exp(x)+4*x^2
*exp(3)^2+(4*x^2+36*x)*exp(3)+x^2+18*x+81),x, algorithm="fricas")

[Out]

e^(4*x^2*e^6 + x^2 + 4*(x^2 + 9*x)*e^3 + 4*(e^6 + 2*e^3 + 1)*e^(2*x) + 4*(2*x*e^6 + 3*(x + 3)*e^3 + x + 9)*e^x
+ 18*x + 81)

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giac [B]  time = 0.51, size = 79, normalized size = 3.43 \begin {gather*} e^{\left (4 \, x^{2} e^{6} + 4 \, x^{2} e^{3} + x^{2} + 36 \, x e^{3} + 8 \, x e^{\left (x + 6\right )} + 12 \, x e^{\left (x + 3\right )} + 4 \, x e^{x} + 18 \, x + 4 \, e^{\left (2 \, x\right )} + 4 \, e^{\left (2 \, x + 6\right )} + 8 \, e^{\left (2 \, x + 3\right )} + 36 \, e^{\left (x + 3\right )} + 36 \, e^{x} + 81\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(3)^2+16*exp(3)+8)*exp(x)^2+((8*x+8)*exp(3)^2+(12*x+48)*exp(3)+4*x+40)*exp(x)+8*x*exp(3)^2+(8
*x+36)*exp(3)+2*x+18)*exp((4*exp(3)^2+8*exp(3)+4)*exp(x)^2+(8*x*exp(3)^2+(12*x+36)*exp(3)+4*x+36)*exp(x)+4*x^2
*exp(3)^2+(4*x^2+36*x)*exp(3)+x^2+18*x+81),x, algorithm="giac")

[Out]

e^(4*x^2*e^6 + 4*x^2*e^3 + x^2 + 36*x*e^3 + 8*x*e^(x + 6) + 12*x*e^(x + 3) + 4*x*e^x + 18*x + 4*e^(2*x) + 4*e^
(2*x + 6) + 8*e^(2*x + 3) + 36*e^(x + 3) + 36*e^x + 81)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \left (\left (8 \,{\mathrm e}^{6}+16 \,{\mathrm e}^{3}+8\right ) {\mathrm e}^{2 x}+\left (\left (8 x +8\right ) {\mathrm e}^{6}+\left (12 x +48\right ) {\mathrm e}^{3}+4 x +40\right ) {\mathrm e}^{x}+8 x \,{\mathrm e}^{6}+\left (8 x +36\right ) {\mathrm e}^{3}+2 x +18\right ) {\mathrm e}^{\left (8 \,{\mathrm e}^{3}+4 \,{\mathrm e}^{6}+4\right ) {\mathrm e}^{2 x}+\left (8 x \,{\mathrm e}^{6}+\left (12 x +36\right ) {\mathrm e}^{3}+4 x +36\right ) {\mathrm e}^{x}+4 x^{2} {\mathrm e}^{6}+\left (4 x^{2}+36 x \right ) {\mathrm e}^{3}+x^{2}+18 x +81}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*exp(3)^2+16*exp(3)+8)*exp(x)^2+((8*x+8)*exp(3)^2+(12*x+48)*exp(3)+4*x+40)*exp(x)+8*x*exp(3)^2+(8*x+36)
*exp(3)+2*x+18)*exp((4*exp(3)^2+8*exp(3)+4)*exp(x)^2+(8*x*exp(3)^2+(12*x+36)*exp(3)+4*x+36)*exp(x)+4*x^2*exp(3
)^2+(4*x^2+36*x)*exp(3)+x^2+18*x+81),x)

[Out]

int(((8*exp(3)^2+16*exp(3)+8)*exp(x)^2+((8*x+8)*exp(3)^2+(12*x+48)*exp(3)+4*x+40)*exp(x)+8*x*exp(3)^2+(8*x+36)
*exp(3)+2*x+18)*exp((4*exp(3)^2+8*exp(3)+4)*exp(x)^2+(8*x*exp(3)^2+(12*x+36)*exp(3)+4*x+36)*exp(x)+4*x^2*exp(3
)^2+(4*x^2+36*x)*exp(3)+x^2+18*x+81),x)

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maxima [B]  time = 0.93, size = 79, normalized size = 3.43 \begin {gather*} e^{\left (4 \, x^{2} e^{6} + 4 \, x^{2} e^{3} + x^{2} + 36 \, x e^{3} + 8 \, x e^{\left (x + 6\right )} + 12 \, x e^{\left (x + 3\right )} + 4 \, x e^{x} + 18 \, x + 4 \, e^{\left (2 \, x\right )} + 4 \, e^{\left (2 \, x + 6\right )} + 8 \, e^{\left (2 \, x + 3\right )} + 36 \, e^{\left (x + 3\right )} + 36 \, e^{x} + 81\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(3)^2+16*exp(3)+8)*exp(x)^2+((8*x+8)*exp(3)^2+(12*x+48)*exp(3)+4*x+40)*exp(x)+8*x*exp(3)^2+(8
*x+36)*exp(3)+2*x+18)*exp((4*exp(3)^2+8*exp(3)+4)*exp(x)^2+(8*x*exp(3)^2+(12*x+36)*exp(3)+4*x+36)*exp(x)+4*x^2
*exp(3)^2+(4*x^2+36*x)*exp(3)+x^2+18*x+81),x, algorithm="maxima")

[Out]

e^(4*x^2*e^6 + 4*x^2*e^3 + x^2 + 36*x*e^3 + 8*x*e^(x + 6) + 12*x*e^(x + 3) + 4*x*e^x + 18*x + 4*e^(2*x) + 4*e^
(2*x + 6) + 8*e^(2*x + 3) + 36*e^(x + 3) + 36*e^x + 81)

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mupad [B]  time = 4.49, size = 92, normalized size = 4.00 \begin {gather*} {\mathrm {e}}^{4\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^3}\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^6}\,{\mathrm {e}}^{36\,{\mathrm {e}}^3\,{\mathrm {e}}^x}\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{18\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{81}\,{\mathrm {e}}^{4\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^6}\,{\mathrm {e}}^{8\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^3}\,{\mathrm {e}}^{8\,x\,{\mathrm {e}}^6\,{\mathrm {e}}^x}\,{\mathrm {e}}^{12\,x\,{\mathrm {e}}^3\,{\mathrm {e}}^x}\,{\mathrm {e}}^{36\,{\mathrm {e}}^x}\,{\mathrm {e}}^{36\,x\,{\mathrm {e}}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(18*x + exp(3)*(36*x + 4*x^2) + exp(2*x)*(8*exp(3) + 4*exp(6) + 4) + 4*x^2*exp(6) + exp(x)*(4*x + 8*x*e
xp(6) + exp(3)*(12*x + 36) + 36) + x^2 + 81)*(2*x + 8*x*exp(6) + exp(x)*(4*x + exp(6)*(8*x + 8) + exp(3)*(12*x
+ 48) + 40) + exp(2*x)*(16*exp(3) + 8*exp(6) + 8) + exp(3)*(8*x + 36) + 18),x)

[Out]

exp(4*exp(2*x))*exp(4*x^2*exp(3))*exp(4*x^2*exp(6))*exp(36*exp(3)*exp(x))*exp(4*x*exp(x))*exp(18*x)*exp(x^2)*e
xp(81)*exp(4*exp(2*x)*exp(6))*exp(8*exp(2*x)*exp(3))*exp(8*x*exp(6)*exp(x))*exp(12*x*exp(3)*exp(x))*exp(36*exp
(x))*exp(36*x*exp(3))

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sympy [B]  time = 0.43, size = 70, normalized size = 3.04 \begin {gather*} e^{x^{2} + 4 x^{2} e^{6} + 18 x + \left (4 x^{2} + 36 x\right ) e^{3} + \left (4 x + 8 x e^{6} + \left (12 x + 36\right ) e^{3} + 36\right ) e^{x} + \left (4 + 8 e^{3} + 4 e^{6}\right ) e^{2 x} + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(3)**2+16*exp(3)+8)*exp(x)**2+((8*x+8)*exp(3)**2+(12*x+48)*exp(3)+4*x+40)*exp(x)+8*x*exp(3)**
2+(8*x+36)*exp(3)+2*x+18)*exp((4*exp(3)**2+8*exp(3)+4)*exp(x)**2+(8*x*exp(3)**2+(12*x+36)*exp(3)+4*x+36)*exp(x
)+4*x**2*exp(3)**2+(4*x**2+36*x)*exp(3)+x**2+18*x+81),x)

[Out]

exp(x**2 + 4*x**2*exp(6) + 18*x + (4*x**2 + 36*x)*exp(3) + (4*x + 8*x*exp(6) + (12*x + 36)*exp(3) + 36)*exp(x)
+ (4 + 8*exp(3) + 4*exp(6))*exp(2*x) + 81)

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