3.90.43 \(\int (2+10 e^{2-10 e^{2+x}+x}) \, dx\)

Optimal. Leaf size=16 \[ 3-e^{-10 e^{2+x}}+2 x \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 0.94, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2282, 2194} \begin {gather*} 2 x-e^{-10 e^{x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2 + 10*E^(2 - 10*E^(2 + x) + x),x]

[Out]

-E^(-10*E^(2 + x)) + 2*x

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} &=2 x+10 \int e^{2-10 e^{2+x}+x} \, dx\\ &=2 x+10 \operatorname {Subst}\left (\int e^{2-10 e^2 x} \, dx,x,e^x\right )\\ &=-e^{-10 e^{2+x}}+2 x\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 15, normalized size = 0.94 \begin {gather*} -e^{-10 e^{2+x}}+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2 + 10*E^(2 - 10*E^(2 + x) + x),x]

[Out]

-E^(-10*E^(2 + x)) + 2*x

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 27, normalized size = 1.69 \begin {gather*} {\left (2 \, x e^{\left (x + 2\right )} - e^{\left (x - 10 \, e^{\left (x + 2\right )} + 2\right )}\right )} e^{\left (-x - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(10*exp(2+x)*exp(-10*exp(2+x))+2,x, algorithm="fricas")

[Out]

(2*x*e^(x + 2) - e^(x - 10*e^(x + 2) + 2))*e^(-x - 2)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 13, normalized size = 0.81 \begin {gather*} 2 \, x - e^{\left (-10 \, e^{\left (x + 2\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(10*exp(2+x)*exp(-10*exp(2+x))+2,x, algorithm="giac")

[Out]

2*x - e^(-10*e^(x + 2))

________________________________________________________________________________________

maple [A]  time = 0.06, size = 14, normalized size = 0.88

method	result	size
default	\(2 x -{\mathrm e}^{-10 \,{\mathrm e}^{2+x}}\)	\(14\)
norman	\(2 x -{\mathrm e}^{-10 \,{\mathrm e}^{2+x}}\)	\(14\)
risch	\(2 x -{\mathrm e}^{-10 \,{\mathrm e}^{2+x}}\)	\(14\)
derivativedivides	\(2 \ln \left (-10 \,{\mathrm e}^{2+x}\right )-{\mathrm e}^{-10 \,{\mathrm e}^{2+x}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(10*exp(2+x)*exp(-10*exp(2+x))+2,x,method=_RETURNVERBOSE)

[Out]

2*x-exp(-10*exp(2+x))

________________________________________________________________________________________

maxima [A]  time = 0.34, size = 13, normalized size = 0.81 \begin {gather*} 2 \, x - e^{\left (-10 \, e^{\left (x + 2\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(10*exp(2+x)*exp(-10*exp(2+x))+2,x, algorithm="maxima")

[Out]

2*x - e^(-10*e^(x + 2))

________________________________________________________________________________________

mupad [B]  time = 0.10, size = 13, normalized size = 0.81 \begin {gather*} 2\,x-{\mathrm {e}}^{-10\,{\mathrm {e}}^{x+2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(10*exp(-10*exp(x + 2))*exp(x + 2) + 2,x)

[Out]

2*x - exp(-10*exp(x + 2))

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 12, normalized size = 0.75 \begin {gather*} 2 x - e^{- 10 e^{x + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(10*exp(2+x)*exp(-10*exp(2+x))+2,x)

[Out]

2*x - exp(-10*exp(x + 2))

________________________________________________________________________________________