3.29.13 \(\int (2-e^{-3+x}+e^{e^4} (2-e^{-3+x})-2 x) \, dx\)

Optimal. Leaf size=26 \[ -x^2+\left (1+e^{e^4}\right ) \left (2-e^{-3+x}+2 x\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.31, number of steps used = 4, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2194} \begin {gather*} -x^2+2 e^{e^4} x+2 x-e^{x-3}-e^{x+e^4-3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2 - E^(-3 + x) + E^E^4*(2 - E^(-3 + x)) - 2*x,x]

[Out]

-E^(-3 + x) - E^(-3 + E^4 + x) + 2*x + 2*E^E^4*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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=2 x-x^2+e^{e^4} \int \left (2-e^{-3+x}\right ) \, dx-\int e^{-3+x} \, dx\\ &=-e^{-3+x}+2 x+2 e^{e^4} x-x^2-e^{e^4} \int e^{-3+x} \, dx\\ &=-e^{-3+x}-e^{-3+e^4+x}+2 x+2 e^{e^4} x-x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 34, normalized size = 1.31 \begin {gather*} -e^{-3+x}-e^{-3+e^4+x}+2 x+2 e^{e^4} x-x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2 - E^(-3 + x) + E^E^4*(2 - E^(-3 + x)) - 2*x,x]

[Out]

-E^(-3 + x) - E^(-3 + E^4 + x) + 2*x + 2*E^E^4*x - x^2

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fricas [A]  time = 0.44, size = 29, normalized size = 1.12 \begin {gather*} -x^{2} + {\left (2 \, x - e^{\left (x - 3\right )}\right )} e^{\left (e^{4}\right )} + 2 \, x - e^{\left (x - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x-3)+2)*exp(exp(4))-exp(x-3)-2*x+2,x, algorithm="fricas")

[Out]

-x^2 + (2*x - e^(x - 3))*e^(e^4) + 2*x - e^(x - 3)

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giac [A]  time = 0.33, size = 29, normalized size = 1.12 \begin {gather*} -x^{2} + {\left (2 \, x - e^{\left (x - 3\right )}\right )} e^{\left (e^{4}\right )} + 2 \, x - e^{\left (x - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x-3)+2)*exp(exp(4))-exp(x-3)-2*x+2,x, algorithm="giac")

[Out]

-x^2 + (2*x - e^(x - 3))*e^(e^4) + 2*x - e^(x - 3)

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maple [A]  time = 0.04, size = 28, normalized size = 1.08

method	result	size
norman	\(\left (-{\mathrm e}^{{\mathrm e}^{4}}-1\right ) {\mathrm e}^{x -3}+\left (2 \,{\mathrm e}^{{\mathrm e}^{4}}+2\right ) x -x^{2}\)	\(28\)
default	\(2 x +{\mathrm e}^{{\mathrm e}^{4}} \left (2 x -{\mathrm e}^{x -3}\right )-x^{2}-{\mathrm e}^{x -3}\)	\(30\)
risch	\(2 x \,{\mathrm e}^{{\mathrm e}^{4}}-{\mathrm e}^{x -3+{\mathrm e}^{4}}-{\mathrm e}^{x -3}-x^{2}+2 x\)	\(30\)
derivativedivides	\(-4 x +12+{\mathrm e}^{{\mathrm e}^{4}} \left (2 x -6-{\mathrm e}^{x -3}\right )-\left (x -3\right )^{2}-{\mathrm e}^{x -3}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(x-3)+2)*exp(exp(4))-exp(x-3)-2*x+2,x,method=_RETURNVERBOSE)

[Out]

(-exp(exp(4))-1)*exp(x-3)+(2*exp(exp(4))+2)*x-x^2

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maxima [A]  time = 0.40, size = 29, normalized size = 1.12 \begin {gather*} -x^{2} + {\left (2 \, x - e^{\left (x - 3\right )}\right )} e^{\left (e^{4}\right )} + 2 \, x - e^{\left (x - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x-3)+2)*exp(exp(4))-exp(x-3)-2*x+2,x, algorithm="maxima")

[Out]

-x^2 + (2*x - e^(x - 3))*e^(e^4) + 2*x - e^(x - 3)

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mupad [B]  time = 1.72, size = 26, normalized size = 1.00 \begin {gather*} x\,\left (2\,{\mathrm {e}}^{{\mathrm {e}}^4}+2\right )-{\mathrm {e}}^{x-3}\,\left ({\mathrm {e}}^{{\mathrm {e}}^4}+1\right )-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2 - exp(x - 3) - exp(exp(4))*(exp(x - 3) - 2) - 2*x,x)

[Out]

x*(2*exp(exp(4)) + 2) - exp(x - 3)*(exp(exp(4)) + 1) - x^2

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sympy [A]  time = 0.10, size = 26, normalized size = 1.00 \begin {gather*} - x^{2} + x \left (2 + 2 e^{e^{4}}\right ) + \left (- e^{e^{4}} - 1\right ) e^{x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x-3)+2)*exp(exp(4))-exp(x-3)-2*x+2,x)

[Out]

-x**2 + x*(2 + 2*exp(exp(4))) + (-exp(exp(4)) - 1)*exp(x - 3)

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