3.42.78 \(\int (-3+2 e^{3+2 e^{e^{3+x}}+e^{3+x}+x}+2 x+e^{e^{e^{3+x}}} (2+e^{3+e^{3+x}+x} (-4+2 x))) \, dx\)

Optimal. Leaf size=19 \[ 2+x+\left (-2+e^{e^{e^{3+x}}}+x\right )^2+\log (5) \]

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Rubi [A]  time = 0.07, antiderivative size = 34, normalized size of antiderivative = 1.79, number of steps used = 5, number of rules used = 3, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2282, 2194, 2288} \begin {gather*} x^2-3 x+e^{2 e^{e^{x+3}}}-2 e^{e^{e^{x+3}}} (2-x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[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} &=-3 x+x^2+2 \int e^{3+2 e^{e^{3+x}}+e^{3+x}+x} \, dx+\int e^{e^{e^{3+x}}} \left (2+e^{3+e^{3+x}+x} (-4+2 x)\right ) \, dx\\ &=-2 e^{e^{e^{3+x}}} (2-x)-3 x+x^2+2 \operatorname {Subst}\left (\int e^{3+2 e^{e^3 x}+e^3 x} \, dx,x,e^x\right )\\ &=-2 e^{e^{e^{3+x}}} (2-x)-3 x+x^2+\frac {2 \operatorname {Subst}\left (\int e^{3+2 x} \, dx,x,e^{e^{3+x}}\right )}{e^3}\\ &=e^{2 e^{e^{3+x}}}-2 e^{e^{e^{3+x}}} (2-x)-3 x+x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 32, normalized size = 1.68 \begin {gather*} e^{2 e^{e^{3+x}}}+2 e^{e^{e^{3+x}}} (-2+x)-3 x+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.92, size = 26, normalized size = 1.37 \begin {gather*} x^{2} + 2 \, {\left (x - 2\right )} e^{\left (e^{\left (e^{\left (x + 3\right )}\right )}\right )} - 3 \, x + e^{\left (2 \, e^{\left (e^{\left (x + 3\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(3+x)*exp(exp(3+x))*exp(exp(exp(3+x)))^2+((2*x-4)*exp(3+x)*exp(exp(3+x))+2)*exp(exp(exp(3+x)))+
2*x-3,x, algorithm="fricas")

[Out]

x^2 + 2*(x - 2)*e^(e^(e^(x + 3))) - 3*x + e^(2*e^(e^(x + 3)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(3+x)*exp(exp(3+x))*exp(exp(exp(3+x)))^2+((2*x-4)*exp(3+x)*exp(exp(3+x))+2)*exp(exp(exp(3+x)))+
2*x-3,x, algorithm="giac")

[Out]

integrate(2*((x - 2)*e^(x + e^(x + 3) + 3) + 1)*e^(e^(e^(x + 3))) + 2*x + 2*e^(x + e^(x + 3) + 2*e^(e^(x + 3))
+ 3) - 3, x)

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maple [A]  time = 0.08, size = 28, normalized size = 1.47

method	result	size
risch	\({\mathrm e}^{2 \,{\mathrm e}^{{\mathrm e}^{3+x}}}+\left (2 x -4\right ) {\mathrm e}^{{\mathrm e}^{{\mathrm e}^{3+x}}}+x^{2}-3 x\)	\(28\)
default	\(-3 x +2 x \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{3+x}}}-4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{3+x}}}+x^{2}+{\mathrm e}^{2 \,{\mathrm e}^{{\mathrm e}^{3+x}}}\)	\(33\)
norman	\(-3 x +2 x \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{3+x}}}-4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{3+x}}}+x^{2}+{\mathrm e}^{2 \,{\mathrm e}^{{\mathrm e}^{3+x}}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*exp(3+x)*exp(exp(3+x))*exp(exp(exp(3+x)))^2+((2*x-4)*exp(3+x)*exp(exp(3+x))+2)*exp(exp(exp(3+x)))+2*x-3,
x,method=_RETURNVERBOSE)

[Out]

exp(2*exp(exp(3+x)))+(2*x-4)*exp(exp(exp(3+x)))+x^2-3*x

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maxima [A]  time = 0.36, size = 26, normalized size = 1.37 \begin {gather*} x^{2} + 2 \, {\left (x - 2\right )} e^{\left (e^{\left (e^{\left (x + 3\right )}\right )}\right )} - 3 \, x + e^{\left (2 \, e^{\left (e^{\left (x + 3\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(3+x)*exp(exp(3+x))*exp(exp(exp(3+x)))^2+((2*x-4)*exp(3+x)*exp(exp(3+x))+2)*exp(exp(exp(3+x)))+
2*x-3,x, algorithm="maxima")

[Out]

x^2 + 2*(x - 2)*e^(e^(e^(x + 3))) - 3*x + e^(2*e^(e^(x + 3)))

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mupad [B]  time = 0.10, size = 35, normalized size = 1.84 \begin {gather*} {\mathrm {e}}^{2\,{\mathrm {e}}^{{\mathrm {e}}^3\,{\mathrm {e}}^x}}-4\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^3\,{\mathrm {e}}^x}}-3\,x+2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^3\,{\mathrm {e}}^x}}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x + exp(exp(exp(x + 3)))*(exp(x + 3)*exp(exp(x + 3))*(2*x - 4) + 2) + 2*exp(x + 3)*exp(2*exp(exp(x + 3))
)*exp(exp(x + 3)) - 3,x)

[Out]

exp(2*exp(exp(3)*exp(x))) - 4*exp(exp(exp(3)*exp(x))) - 3*x + 2*x*exp(exp(exp(3)*exp(x))) + x^2

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sympy [A]  time = 3.67, size = 29, normalized size = 1.53 \begin {gather*} x^{2} - 3 x + \left (2 x - 4\right ) e^{e^{e^{x + 3}}} + e^{2 e^{e^{x + 3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(3+x)*exp(exp(3+x))*exp(exp(exp(3+x)))**2+((2*x-4)*exp(3+x)*exp(exp(3+x))+2)*exp(exp(exp(3+x)))
+2*x-3,x)

[Out]

x**2 - 3*x + (2*x - 4)*exp(exp(exp(x + 3))) + exp(2*exp(exp(x + 3)))

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