3.44.23 \(\int e^{-7+e^{e+2 x}-x} (e (1-x)+2 e^{1+e+2 x} x) \, dx\)

Optimal. Leaf size=16 \[ e^{-6+e^{e+2 x}-x} x \]

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Rubi [B]  time = 0.18, antiderivative size = 40, normalized size of antiderivative = 2.50, number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {6741, 2288} \begin {gather*} \frac {e^{-x+e^{2 x+e}-6} \left (x-2 e^{2 x+e} x\right )}{1-2 e^{2 x+e}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-7 + E^(E + 2*x) - x)*(E*(1 - x) + 2*E^(1 + E + 2*x)*x),x]

[Out]

(E^(-6 + E^(E + 2*x) - x)*(x - 2*E^(E + 2*x)*x))/(1 - 2*E^(E + 2*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]

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{-6+e^{e+2 x}-x} \left (1-x+2 e^{e+2 x} x\right ) \, dx\\ &=\frac {e^{-6+e^{e+2 x}-x} \left (x-2 e^{e+2 x} x\right )}{1-2 e^{e+2 x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 16, normalized size = 1.00 \begin {gather*} e^{-6+e^{e+2 x}-x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-7 + E^(E + 2*x) - x)*(E*(1 - x) + 2*E^(1 + E + 2*x)*x),x]

[Out]

E^(-6 + E^(E + 2*x) - x)*x

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fricas [A]  time = 0.64, size = 26, normalized size = 1.62 \begin {gather*} x e^{\left (-{\left ({\left (x + 7\right )} e - e^{\left (2 \, x + e + 1\right )}\right )} e^{\left (-1\right )} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(1)*exp(exp(1)+2*x)+(-x+1)*exp(1))/exp(-exp(exp(1)+2*x)+x+7),x, algorithm="fricas")

[Out]

x*e^(-((x + 7)*e - e^(2*x + e + 1))*e^(-1) + 1)

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giac [A]  time = 0.22, size = 15, normalized size = 0.94 \begin {gather*} x e^{\left (-x + e^{\left (2 \, x + e\right )} - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(1)*exp(exp(1)+2*x)+(-x+1)*exp(1))/exp(-exp(exp(1)+2*x)+x+7),x, algorithm="giac")

[Out]

x*e^(-x + e^(2*x + e) - 6)

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maple [A]  time = 0.06, size = 16, normalized size = 1.00

method	result	size
risch	\(x \,{\mathrm e}^{-6+{\mathrm e}^{{\mathrm e}+2 x}-x}\)	\(16\)
norman	\({\mathrm e} \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}+2 x}-x -7} x\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*exp(1)*exp(exp(1)+2*x)+(1-x)*exp(1))/exp(-exp(exp(1)+2*x)+x+7),x,method=_RETURNVERBOSE)

[Out]

x*exp(-6+exp(exp(1)+2*x)-x)

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maxima [A]  time = 0.43, size = 15, normalized size = 0.94 \begin {gather*} x e^{\left (-x + e^{\left (2 \, x + e\right )} - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(1)*exp(exp(1)+2*x)+(-x+1)*exp(1))/exp(-exp(exp(1)+2*x)+x+7),x, algorithm="maxima")

[Out]

x*e^(-x + e^(2*x + e) - 6)

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mupad [B]  time = 3.10, size = 17, normalized size = 1.06 \begin {gather*} x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-6}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{\mathrm {e}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(exp(2*x + exp(1)) - x - 7)*(exp(1)*(x - 1) - 2*x*exp(2*x + exp(1))*exp(1)),x)

[Out]

x*exp(-x)*exp(-6)*exp(exp(2*x)*exp(exp(1)))

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sympy [A]  time = 0.17, size = 17, normalized size = 1.06 \begin {gather*} e x e^{- x + e^{2 x + e} - 7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(1)*exp(exp(1)+2*x)+(-x+1)*exp(1))/exp(-exp(exp(1)+2*x)+x+7),x)

[Out]

E*x*exp(-x + exp(2*x + E) - 7)

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