3.33.100 $\int e^{-2-e^5+e^x+x}(1+e^x)dx$

Optimal. Leaf size=13 $$e^{-2-e^5+e^x+x}$$

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Rubi [B]  time = 0.04, antiderivative size = 33, normalized size of antiderivative = 2.54, number of steps used = 3, number of rules used = 3, integrand size = 19, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.158, Rules used = {2282, 2176, 2194} $$\begin{array}{c}{e}^{e^x-2-e^5}(e^x+1)-e^{e^x-2-e^5}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[E^(-2 - E^5 + E^x + x)*(1 + E^x),x]

[Out]

-E^(-2 - E^5 + E^x) + E^(-2 - E^5 + E^x)*(1 + E^x)

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\mathrm{Subst}(\int e^{-2-e^5+x}(1+x)dx,x,e^x)\\ & =e^{-2-e^5+e^x}(1+e^x)-\mathrm{Subst}(\int e^{-2-e^5+x}dx,x,e^x)\\ & =-e^{-2-e^5+e^x}+e^{-2-e^5+e^x}(1+e^x)\end{array}\end{array}$$

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Mathematica [A]  time = 0.01, size = 13, normalized size = 1.00 $$\begin{array}{c}{e}^{-2-e^5+e^x+x}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[E^(-2 - E^5 + E^x + x)*(1 + E^x),x]

[Out]

E^(-2 - E^5 + E^x + x)

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fricas [A]  time = 0.85, size = 10, normalized size = 0.77 $$\begin{array}{c}{e}^{(x-e^5+e^x-2)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+1)*exp(exp(x)-exp(5)+x-2),x, algorithm="fricas")

[Out]

e^(x - e^5 + e^x - 2)

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giac [A]  time = 0.24, size = 10, normalized size = 0.77 $$\begin{array}{c}{e}^{(x-e^5+e^x-2)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+1)*exp(exp(x)-exp(5)+x-2),x, algorithm="giac")

[Out]

e^(x - e^5 + e^x - 2)

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maple [A]  time = 0.02, size = 11, normalized size = 0.85

method	result	size
derivativedivides	${\mathrm{e}}^{{\mathrm{e}}^x-{\mathrm{e}}^5+x-2}$	$11$
default	${\mathrm{e}}^{{\mathrm{e}}^x-{\mathrm{e}}^5+x-2}$	$11$
norman	${\mathrm{e}}^{{\mathrm{e}}^x-{\mathrm{e}}^5+x-2}$	$11$
risch	${\mathrm{e}}^{{\mathrm{e}}^x-{\mathrm{e}}^5+x-2}$	$11$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)+1)*exp(exp(x)-exp(5)+x-2),x,method=_RETURNVERBOSE)

[Out]

exp(exp(x)-exp(5)+x-2)

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maxima [A]  time = 0.74, size = 10, normalized size = 0.77 $$\begin{array}{c}{e}^{(x-e^5+e^x-2)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+1)*exp(exp(x)-exp(5)+x-2),x, algorithm="maxima")

[Out]

e^(x - e^5 + e^x - 2)

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mupad [B]  time = 1.99, size = 13, normalized size = 1.00 $$\begin{array}{c}{\mathrm{e}}^{-{\mathrm{e}}^5}{\mathrm{e}}^{{\mathrm{e}}^x}{\mathrm{e}}^{-2}{\mathrm{e}}^x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x - exp(5) + exp(x) - 2)*(exp(x) + 1),x)

[Out]

exp(-exp(5))*exp(exp(x))*exp(-2)*exp(x)

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sympy [A]  time = 0.14, size = 10, normalized size = 0.77 $$\begin{array}{c}{e}^{x+e^x-e^5-2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+1)*exp(exp(x)-exp(5)+x-2),x)

[Out]

exp(x + exp(x) - exp(5) - 2)

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