3.23.84 \(\int e^{-1+\frac {-4 e x+e^2 x^2+e^{2-x} (4 x^2+x^3)}{e}} (-4 e+2 e^2 x+e^{2-x} (8 x-x^2-x^3)) \, dx\)

Optimal. Leaf size=21 \[ e^{x \left (-4+e^{1-x} x \left (4+e^x+x\right )\right )} \]

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Rubi [F]  time = 1.66, 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 \exp \left (-1+\frac {-4 e x+e^2 x^2+e^{2-x} \left (4 x^2+x^3\right )}{e}\right ) \left (-4 e+2 e^2 x+e^{2-x} \left (8 x-x^2-x^3\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(-1 + (-4*E*x + E^2*x^2 + E^(2 - x)*(4*x^2 + x^3))/E)*(-4*E + 2*E^2*x + E^(2 - x)*(8*x - x^2 - x^3)),x]

[Out]

-4*Defer[Int][E^(-4*x + E*x^2 + E^(1 - x)*x^2*(4 + x)), x] + 8*Defer[Int][E^(1 - 5*x + E*x^2 + E^(1 - x)*x^2*(
4 + x))*x, x] + 2*Defer[Int][E^(1 - 4*x + E*x^2 + E^(1 - x)*x^2*(4 + x))*x, x] - Defer[Int][E^(1 - 5*x + E*x^2
+ E^(1 - x)*x^2*(4 + x))*x^2, x] - Defer[Int][E^(1 - 5*x + E*x^2 + E^(1 - x)*x^2*(4 + x))*x^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{-5 x+e x^2+e^{1-x} x^2 (4+x)} \left (-4 e^x+2 e^{1+x} x-e x \left (-8+x+x^2\right )\right ) \, dx\\ &=\int \left (-4 e^{-4 x+e x^2+e^{1-x} x^2 (4+x)}+2 e^{1-4 x+e x^2+e^{1-x} x^2 (4+x)} x-e^{1-5 x+e x^2+e^{1-x} x^2 (4+x)} x \left (-8+x+x^2\right )\right ) \, dx\\ &=2 \int e^{1-4 x+e x^2+e^{1-x} x^2 (4+x)} x \, dx-4 \int e^{-4 x+e x^2+e^{1-x} x^2 (4+x)} \, dx-\int e^{1-5 x+e x^2+e^{1-x} x^2 (4+x)} x \left (-8+x+x^2\right ) \, dx\\ &=2 \int e^{1-4 x+e x^2+e^{1-x} x^2 (4+x)} x \, dx-4 \int e^{-4 x+e x^2+e^{1-x} x^2 (4+x)} \, dx-\int \left (-8 e^{1-5 x+e x^2+e^{1-x} x^2 (4+x)} x+e^{1-5 x+e x^2+e^{1-x} x^2 (4+x)} x^2+e^{1-5 x+e x^2+e^{1-x} x^2 (4+x)} x^3\right ) \, dx\\ &=2 \int e^{1-4 x+e x^2+e^{1-x} x^2 (4+x)} x \, dx-4 \int e^{-4 x+e x^2+e^{1-x} x^2 (4+x)} \, dx+8 \int e^{1-5 x+e x^2+e^{1-x} x^2 (4+x)} x \, dx-\int e^{1-5 x+e x^2+e^{1-x} x^2 (4+x)} x^2 \, dx-\int e^{1-5 x+e x^2+e^{1-x} x^2 (4+x)} x^3 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.70, size = 21, normalized size = 1.00 \begin {gather*} e^{x \left (-4+e x+e^{1-x} x (4+x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-1 + (-4*E*x + E^2*x^2 + E^(2 - x)*(4*x^2 + x^3))/E)*(-4*E + 2*E^2*x + E^(2 - x)*(8*x - x^2 - x^3
)),x]

[Out]

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

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fricas [B]  time = 0.98, size = 38, normalized size = 1.81 \begin {gather*} e^{\left ({\left (x^{2} e^{2} - {\left (4 \, x + 1\right )} e + {\left (x^{3} + 4 \, x^{2}\right )} e^{\left (-x + 2\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(2-x)*exp(x)+(-x^3-x^2+8*x)*exp(2-x)-4*exp(1))*exp((x^2*exp(2-x)*exp(x)+(x^3+4*x^2)*exp(2-x)
-4*x*exp(1))/exp(1))/exp(1),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.34, size = 32, normalized size = 1.52 \begin {gather*} e^{\left (x^{3} e^{\left (-x + 1\right )} + x^{2} e + 4 \, x^{2} e^{\left (-x + 1\right )} - 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(2-x)*exp(x)+(-x^3-x^2+8*x)*exp(2-x)-4*exp(1))*exp((x^2*exp(2-x)*exp(x)+(x^3+4*x^2)*exp(2-x)
-4*x*exp(1))/exp(1))/exp(1),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.16, size = 35, normalized size = 1.67

method	result	size
norman	\({\mathrm e}^{\left (x^{2} {\mathrm e}^{2}+\left (x^{3}+4 x^{2}\right ) {\mathrm e}^{2} {\mathrm e}^{-x}-4 x \,{\mathrm e}\right ) {\mathrm e}^{-1}}\)	\(35\)
risch	\({\mathrm e}^{-x \left (-x^{2} {\mathrm e}^{2-x}-{\mathrm e}^{2} x -4 x \,{\mathrm e}^{2-x}+4 \,{\mathrm e}\right ) {\mathrm e}^{-1}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*exp(2-x)*exp(x)+(-x^3-x^2+8*x)*exp(2-x)-4*exp(1))*exp((x^2*exp(2-x)*exp(x)+(x^3+4*x^2)*exp(2-x)-4*x*e
xp(1))/exp(1))/exp(1),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.63, size = 32, normalized size = 1.52 \begin {gather*} e^{\left (x^{3} e^{\left (-x + 1\right )} + x^{2} e + 4 \, x^{2} e^{\left (-x + 1\right )} - 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(2-x)*exp(x)+(-x^3-x^2+8*x)*exp(2-x)-4*exp(1))*exp((x^2*exp(2-x)*exp(x)+(x^3+4*x^2)*exp(2-x)
-4*x*exp(1))/exp(1))/exp(1),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.44, size = 35, normalized size = 1.67 \begin {gather*} {\mathrm {e}}^{x^2\,\mathrm {e}}\,{\mathrm {e}}^{-4\,x}\,{\mathrm {e}}^{x^3\,{\mathrm {e}}^{-x}\,\mathrm {e}}\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^{-x}\,\mathrm {e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-1)*exp(exp(-1)*(exp(2 - x)*(4*x^2 + x^3) - 4*x*exp(1) + x^2*exp(2 - x)*exp(x)))*(4*exp(1) + exp(2 -
x)*(x^2 - 8*x + x^3) - 2*x*exp(2 - x)*exp(x)),x)

[Out]

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

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sympy [A]  time = 0.29, size = 32, normalized size = 1.52 \begin {gather*} e^{\frac {x^{2} e^{2} - 4 e x + \left (x^{3} + 4 x^{2}\right ) e^{2} e^{- x}}{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(2-x)*exp(x)+(-x**3-x**2+8*x)*exp(2-x)-4*exp(1))*exp((x**2*exp(2-x)*exp(x)+(x**3+4*x**2)*exp
(2-x)-4*x*exp(1))/exp(1))/exp(1),x)

[Out]

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

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