3.40.13 \(\int e^{-x} (e^x (-71+8 x)+e^{e^{-x} (2 x+x^2+x^3)} (-2-2 x^2+x^3)) \, dx\) [3913]

Optimal. Leaf size=26 \[ -e^{e^{-x} x \left (2+x+x^2\right )}+4 (-9+x)^2+x \]

[Out]

x+4*(x-9)^2-exp((x^2+x+2)/exp(x)*x)

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Rubi [F]
time = 0.58, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int e^{-x} \left (e^x (-71+8 x)+e^{e^{-x} \left (2 x+x^2+x^3\right )} \left (-2-2 x^2+x^3\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(-71 + 8*x) + E^((2*x + x^2 + x^3)/E^x)*(-2 - 2*x^2 + x^3))/E^x,x]

[Out]

-71*x + 4*x^2 - 2*Defer[Int][E^(-x + (x*(2 + x + x^2))/E^x), x] - 2*Defer[Int][E^(-x + (x*(2 + x + x^2))/E^x)*
x^2, x] + Defer[Int][E^(-x + (x*(2 + x + x^2))/E^x)*x^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-71+8 x+e^{-x+e^{-x} x \left (2+x+x^2\right )} \left (-2-2 x^2+x^3\right )\right ) \, dx\\ &=-71 x+4 x^2+\int e^{-x+e^{-x} x \left (2+x+x^2\right )} \left (-2-2 x^2+x^3\right ) \, dx\\ &=-71 x+4 x^2+\int \left (-2 e^{-x+e^{-x} x \left (2+x+x^2\right )}-2 e^{-x+e^{-x} x \left (2+x+x^2\right )} x^2+e^{-x+e^{-x} x \left (2+x+x^2\right )} x^3\right ) \, dx\\ &=-71 x+4 x^2-2 \int e^{-x+e^{-x} x \left (2+x+x^2\right )} \, dx-2 \int e^{-x+e^{-x} x \left (2+x+x^2\right )} x^2 \, dx+\int e^{-x+e^{-x} x \left (2+x+x^2\right )} x^3 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.07, size = 26, normalized size = 1.00 \begin {gather*} -e^{e^{-x} x \left (2+x+x^2\right )}-71 x+4 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-71 + 8*x) + E^((2*x + x^2 + x^3)/E^x)*(-2 - 2*x^2 + x^3))/E^x,x]

[Out]

-E^((x*(2 + x + x^2))/E^x) - 71*x + 4*x^2

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Maple [A]
time = 0.29, size = 25, normalized size = 0.96

method	result	size
risch	\(4 x^{2}-71 x -{\mathrm e}^{x \left (x^{2}+x +2\right ) {\mathrm e}^{-x}}\)	\(25\)
norman	\(\left (-71 \,{\mathrm e}^{x} x +4 \,{\mathrm e}^{x} x^{2}-{\mathrm e}^{x} {\mathrm e}^{\left (x^{3}+x^{2}+2 x \right ) {\mathrm e}^{-x}}\right ) {\mathrm e}^{-x}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3-2*x^2-2)*exp((x^3+x^2+2*x)/exp(x))+(8*x-71)*exp(x))/exp(x),x,method=_RETURNVERBOSE)

[Out]

4*x^2-71*x-exp(x*(x^2+x+2)*exp(-x))

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Maxima [A]
time = 0.41, size = 36, normalized size = 1.38 \begin {gather*} 4 \, x^{2} - 71 \, x - e^{\left (x^{3} e^{\left (-x\right )} + x^{2} e^{\left (-x\right )} + 2 \, x e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-2*x^2-2)*exp((x^3+x^2+2*x)/exp(x))+(8*x-71)*exp(x))/exp(x),x, algorithm="maxima")

[Out]

4*x^2 - 71*x - e^(x^3*e^(-x) + x^2*e^(-x) + 2*x*e^(-x))

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Fricas [A]
time = 0.33, size = 27, normalized size = 1.04 \begin {gather*} 4 \, x^{2} - 71 \, x - e^{\left ({\left (x^{3} + x^{2} + 2 \, x\right )} e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-2*x^2-2)*exp((x^3+x^2+2*x)/exp(x))+(8*x-71)*exp(x))/exp(x),x, algorithm="fricas")

[Out]

4*x^2 - 71*x - e^((x^3 + x^2 + 2*x)*e^(-x))

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Sympy [A]
time = 0.09, size = 22, normalized size = 0.85 \begin {gather*} 4 x^{2} - 71 x - e^{\left (x^{3} + x^{2} + 2 x\right ) e^{- x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**3-2*x**2-2)*exp((x**3+x**2+2*x)/exp(x))+(8*x-71)*exp(x))/exp(x),x)

[Out]

4*x**2 - 71*x - exp((x**3 + x**2 + 2*x)*exp(-x))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-2*x^2-2)*exp((x^3+x^2+2*x)/exp(x))+(8*x-71)*exp(x))/exp(x),x, algorithm="giac")

[Out]

integrate(((x^3 - 2*x^2 - 2)*e^((x^3 + x^2 + 2*x)*e^(-x)) + (8*x - 71)*e^x)*e^(-x), x)

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Mupad [B]
time = 2.31, size = 37, normalized size = 1.42 \begin {gather*} 4\,x^2-71\,x-{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{x^3\,{\mathrm {e}}^{-x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-x)*(exp(exp(-x)*(2*x + x^2 + x^3))*(2*x^2 - x^3 + 2) - exp(x)*(8*x - 71)),x)

[Out]

4*x^2 - 71*x - exp(2*x*exp(-x))*exp(x^2*exp(-x))*exp(x^3*exp(-x))

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