3.73.17 \(\int (1-2 x+e^{4+e^2+e^x+2 x-x^2} (-1-2 x-e^x x+2 x^2)) \, dx\)

Optimal. Leaf size=25 \[ x-x \left (e^{4+e^2+e^x+2 x-x^2}+x\right ) \]

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Rubi [B]  time = 0.12, antiderivative size = 51, normalized size of antiderivative = 2.04, number of steps used = 2, number of rules used = 1, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {2288} \begin {gather*} -x^2-\frac {e^{-x^2+2 x+e^x+e^2+4} \left (-2 x^2+e^x x+2 x\right )}{-2 x+e^x+2}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 - 2*x + E^(4 + E^2 + E^x + 2*x - x^2)*(-1 - 2*x - E^x*x + 2*x^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x-x^2+\int e^{4+e^2+e^x+2 x-x^2} \left (-1-2 x-e^x x+2 x^2\right ) \, dx\\ &=x-x^2-\frac {e^{4+e^2+e^x+2 x-x^2} \left (2 x+e^x x-2 x^2\right )}{2+e^x-2 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 27, normalized size = 1.08 \begin {gather*} \left (1-e^{4+e^2+e^x+2 x-x^2}-x\right ) x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 - 2*x + E^(4 + E^2 + E^x + 2*x - x^2)*(-1 - 2*x - E^x*x + 2*x^2),x]

[Out]

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

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fricas [A]  time = 0.77, size = 25, normalized size = 1.00 \begin {gather*} -x^{2} - x e^{\left (-x^{2} + 2 \, x + e^{2} + e^{x} + 4\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)*x+2*x^2-2*x-1)*exp(exp(x)+exp(2)-x^2+2*x+4)+1-2*x,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.22, size = 25, normalized size = 1.00 \begin {gather*} -x^{2} - x e^{\left (-x^{2} + 2 \, x + e^{2} + e^{x} + 4\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)*x+2*x^2-2*x-1)*exp(exp(x)+exp(2)-x^2+2*x+4)+1-2*x,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 26, normalized size = 1.04

method	result	size
default	\(x -{\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{2}-x^{2}+2 x +4} x -x^{2}\)	\(26\)
norman	\(x -{\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{2}-x^{2}+2 x +4} x -x^{2}\)	\(26\)
risch	\(x -{\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{2}-x^{2}+2 x +4} x -x^{2}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(x)*x+2*x^2-2*x-1)*exp(exp(x)+exp(2)-x^2+2*x+4)+1-2*x,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.36, size = 25, normalized size = 1.00 \begin {gather*} -x^{2} - x e^{\left (-x^{2} + 2 \, x + e^{2} + e^{x} + 4\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)*x+2*x^2-2*x-1)*exp(exp(x)+exp(2)-x^2+2*x+4)+1-2*x,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.11, size = 28, normalized size = 1.12 \begin {gather*} x-x^2-x\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^4\,{\mathrm {e}}^{-x^2}\,{\mathrm {e}}^{{\mathrm {e}}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1 - exp(2*x + exp(2) + exp(x) - x^2 + 4)*(2*x + x*exp(x) - 2*x^2 + 1) - 2*x,x)

[Out]

x - x^2 - x*exp(2*x)*exp(exp(x))*exp(4)*exp(-x^2)*exp(exp(2))

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sympy [A]  time = 0.18, size = 22, normalized size = 0.88 \begin {gather*} - x^{2} - x e^{- x^{2} + 2 x + e^{x} + 4 + e^{2}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)*x+2*x**2-2*x-1)*exp(exp(x)+exp(2)-x**2+2*x+4)+1-2*x,x)

[Out]

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

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