3.99.66 \(\int (3+e^{2+e^3}+e^x (-1-x)-2 x) \, dx\)

Optimal. Leaf size=21 \[ -4+\left (3+e^{2+e^3}-e^x-x\right ) x \]

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Rubi [A]  time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.33, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2176, 2194} \begin {gather*} -x^2+\left (3+e^{2+e^3}\right ) x+e^x-e^x (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[3 + E^(2 + E^3) + E^x*(-1 - x) - 2*x,x]

[Out]

E^x + (3 + E^(2 + E^3))*x - x^2 - E^x*(1 + 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\left (3+e^{2+e^3}\right ) x-x^2+\int e^x (-1-x) \, dx\\ &=\left (3+e^{2+e^3}\right ) x-x^2-e^x (1+x)+\int e^x \, dx\\ &=e^x+\left (3+e^{2+e^3}\right ) x-x^2-e^x (1+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 1.14 \begin {gather*} 3 x+e^{2+e^3} x-e^x x-x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[3 + E^(2 + E^3) + E^x*(-1 - x) - 2*x,x]

[Out]

3*x + E^(2 + E^3)*x - E^x*x - x^2

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fricas [A]  time = 0.98, size = 21, normalized size = 1.00 \begin {gather*} -x^{2} - x e^{x} + x e^{\left (e^{3} + 2\right )} + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2+exp(3))+(-x-1)*exp(x)+3-2*x,x, algorithm="fricas")

[Out]

-x^2 - x*e^x + x*e^(e^3 + 2) + 3*x

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giac [A]  time = 0.15, size = 21, normalized size = 1.00 \begin {gather*} -x^{2} - x e^{x} + x e^{\left (e^{3} + 2\right )} + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2+exp(3))+(-x-1)*exp(x)+3-2*x,x, algorithm="giac")

[Out]

-x^2 - x*e^x + x*e^(e^3 + 2) + 3*x

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maple [A]  time = 0.03, size = 22, normalized size = 1.05

method	result	size
default	\(-{\mathrm e}^{x} x +3 x -x^{2}+x \,{\mathrm e}^{2+{\mathrm e}^{3}}\)	\(22\)
norman	\(\left ({\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{3}}+3\right ) x -x^{2}-{\mathrm e}^{x} x\)	\(22\)
risch	\(-{\mathrm e}^{x} x +3 x -x^{2}+x \,{\mathrm e}^{2+{\mathrm e}^{3}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2+exp(3))+(-x-1)*exp(x)+3-2*x,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.36, size = 21, normalized size = 1.00 \begin {gather*} -x^{2} - x e^{x} + x e^{\left (e^{3} + 2\right )} + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2+exp(3))+(-x-1)*exp(x)+3-2*x,x, algorithm="maxima")

[Out]

-x^2 - x*e^x + x*e^(e^3 + 2) + 3*x

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mupad [B]  time = 0.05, size = 15, normalized size = 0.71 \begin {gather*} -x\,\left (x-{\mathrm {e}}^{{\mathrm {e}}^3+2}+{\mathrm {e}}^x-3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(exp(3) + 2) - 2*x - exp(x)*(x + 1) + 3,x)

[Out]

-x*(x - exp(exp(3) + 2) + exp(x) - 3)

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sympy [A]  time = 0.10, size = 19, normalized size = 0.90 \begin {gather*} - x^{2} - x e^{x} + x \left (3 + e^{2} e^{e^{3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2+exp(3))+(-x-1)*exp(x)+3-2*x,x)

[Out]

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

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