3.25.100 \(\int e^{-4+e^{e^e}-x} (1+e^{4-e^{e^e}+x}-x) \, dx\)

Optimal. Leaf size=17 \[ 5+x+e^{-4+e^{e^e}-x} x \]

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Rubi [A]  time = 0.12, antiderivative size = 33, normalized size of antiderivative = 1.94, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6688, 2176, 2194} \begin {gather*} -e^{-x+e^{e^e}-4} (1-x)+e^{-x+e^{e^e}-4}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-4 + E^E^E - x)*(1 + E^(4 - E^E^E + x) - x),x]

[Out]

E^(-4 + E^E^E - x) - E^(-4 + E^E^E - x)*(1 - x) + 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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-e^{-4+e^{e^e}-x} (-1+x)\right ) \, dx\\ &=x-\int e^{-4+e^{e^e}-x} (-1+x) \, dx\\ &=-e^{-4+e^{e^e}-x} (1-x)+x-\int e^{-4+e^{e^e}-x} \, dx\\ &=e^{-4+e^{e^e}-x}-e^{-4+e^{e^e}-x} (1-x)+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 16, normalized size = 0.94 \begin {gather*} x+e^{-4+e^{e^e}-x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-4 + E^E^E - x)*(1 + E^(4 - E^E^E + x) - x),x]

[Out]

x + E^(-4 + E^E^E - x)*x

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fricas [A]  time = 0.81, size = 14, normalized size = 0.82 \begin {gather*} x + e^{\left (-x + e^{\left (e^{e}\right )} + \log \relax (x) - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(-exp(exp(exp(1)))-log(x)+4+x)-x+1)/x/exp(-exp(exp(exp(1)))-log(x)+4+x),x, algorithm="fricas")

[Out]

x + e^(-x + e^(e^e) + log(x) - 4)

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giac [A]  time = 1.48, size = 14, normalized size = 0.82 \begin {gather*} x e^{\left (-x + e^{\left (e^{e}\right )} - 4\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(-exp(exp(exp(1)))-log(x)+4+x)-x+1)/x/exp(-exp(exp(exp(1)))-log(x)+4+x),x, algorithm="giac")

[Out]

x*e^(-x + e^(e^e) - 4) + x

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maple [A]  time = 0.03, size = 15, normalized size = 0.88

method	result	size
risch	\(x +x \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{{\mathrm e}}}-4-x}\)	\(15\)
norman	\(\left (1+x \,{\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{{\mathrm e}}}-\ln \relax (x )+4+x}\right ) x \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{{\mathrm e}}}-4-x}\)	\(36\)
default	\(x +{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{{\mathrm e}}}-4-x} {\mathrm e}^{{\mathrm e}^{{\mathrm e}}}-\left ({\mathrm e}^{{\mathrm e}^{{\mathrm e}}}-4-x \right ) {\mathrm e}^{{\mathrm e}^{{\mathrm e}^{{\mathrm e}}}-4-x}-4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{{\mathrm e}}}-4-x}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*exp(-exp(exp(exp(1)))-ln(x)+4+x)-x+1)/x/exp(-exp(exp(exp(1)))-ln(x)+4+x),x,method=_RETURNVERBOSE)

[Out]

x+x*exp(exp(exp(exp(1)))-4-x)

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maxima [B]  time = 0.39, size = 34, normalized size = 2.00 \begin {gather*} {\left (x e^{\left (e^{\left (e^{e}\right )}\right )} + e^{\left (e^{\left (e^{e}\right )}\right )}\right )} e^{\left (-x - 4\right )} + x - e^{\left (-x + e^{\left (e^{e}\right )} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(-exp(exp(exp(1)))-log(x)+4+x)-x+1)/x/exp(-exp(exp(exp(1)))-log(x)+4+x),x, algorithm="maxima")

[Out]

(x*e^(e^(e^e)) + e^(e^(e^e)))*e^(-x - 4) + x - e^(-x + e^(e^e) - 4)

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mupad [B]  time = 1.38, size = 16, normalized size = 0.94 \begin {gather*} x\,\left ({\mathrm {e}}^{-x}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{\mathrm {e}}}}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(exp(exp(1))) - x + log(x) - 4)*(x*exp(x - exp(exp(exp(1))) - log(x) + 4) - x + 1))/x,x)

[Out]

x*(exp(-x)*exp(-4)*exp(exp(exp(exp(1)))) + 1)

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sympy [A]  time = 0.10, size = 14, normalized size = 0.82 \begin {gather*} x e^{- x - 4 + e^{e^{e}}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(-exp(exp(exp(1)))-ln(x)+4+x)-x+1)/x/exp(-exp(exp(exp(1)))-ln(x)+4+x),x)

[Out]

x*exp(-x - 4 + exp(exp(E))) + x

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