3.55.43 \(\int (e^{e^x+x}+2 x) \, dx\)

Optimal. Leaf size=10 \[ -3+e^{e^x}+x^2 \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 9, normalized size of antiderivative = 0.90, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2282, 2194} \begin {gather*} x^2+e^{e^x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(E^x + x) + 2*x,x]

[Out]

E^E^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]]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x^2+\int e^{e^x+x} \, dx\\ &=x^2+\operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )\\ &=e^{e^x}+x^2\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 9, normalized size = 0.90 \begin {gather*} e^{e^x}+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(E^x + x) + 2*x,x]

[Out]

E^E^x + x^2

________________________________________________________________________________________

fricas [B]  time = 0.70, size = 17, normalized size = 1.70 \begin {gather*} {\left (x^{2} e^{x} + e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*exp(exp(x))+2*x,x, algorithm="fricas")

[Out]

(x^2*e^x + e^(x + e^x))*e^(-x)

________________________________________________________________________________________

giac [A]  time = 0.12, size = 7, normalized size = 0.70 \begin {gather*} x^{2} + e^{\left (e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*exp(exp(x))+2*x,x, algorithm="giac")

[Out]

x^2 + e^(e^x)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 8, normalized size = 0.80

method	result	size
default	\(x^{2}+{\mathrm e}^{{\mathrm e}^{x}}\)	\(8\)
norman	\(x^{2}+{\mathrm e}^{{\mathrm e}^{x}}\)	\(8\)
risch	\(x^{2}+{\mathrm e}^{{\mathrm e}^{x}}\)	\(8\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*exp(exp(x))+2*x,x,method=_RETURNVERBOSE)

[Out]

x^2+exp(exp(x))

________________________________________________________________________________________

maxima [A]  time = 0.37, size = 7, normalized size = 0.70 \begin {gather*} x^{2} + e^{\left (e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*exp(exp(x))+2*x,x, algorithm="maxima")

[Out]

x^2 + e^(e^x)

________________________________________________________________________________________

mupad [B]  time = 3.36, size = 7, normalized size = 0.70 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^x}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x + exp(exp(x))*exp(x),x)

[Out]

exp(exp(x)) + x^2

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 7, normalized size = 0.70 \begin {gather*} x^{2} + e^{e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*exp(exp(x))+2*x,x)

[Out]

x**2 + exp(exp(x))

________________________________________________________________________________________