3.48.28 \(\int \frac {2+e^{e^{x^2}} (-2+e^{x^2} (-16 x-4 x^2))}{e} \, dx\)

Optimal. Leaf size=23 \[ -\frac {2 \left (e^3-x+e^{e^{x^2}} (4+x)\right )}{e} \]

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Rubi [A]  time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.22, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {12, 2288} \begin {gather*} \frac {2 x}{e}-\frac {2 e^{e^{x^2}-1} \left (x^2+4 x\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + E^E^x^2*(-2 + E^x^2*(-16*x - 4*x^2)))/E,x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=\frac {\int \left (2+e^{e^{x^2}} \left (-2+e^{x^2} \left (-16 x-4 x^2\right )\right )\right ) \, dx}{e}\\ &=\frac {2 x}{e}+\frac {\int e^{e^{x^2}} \left (-2+e^{x^2} \left (-16 x-4 x^2\right )\right ) \, dx}{e}\\ &=\frac {2 x}{e}-\frac {2 e^{-1+e^{x^2}} \left (4 x+x^2\right )}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 20, normalized size = 0.87 \begin {gather*} -\frac {2 \left (-x+e^{e^{x^2}} (4+x)\right )}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + E^E^x^2*(-2 + E^x^2*(-16*x - 4*x^2)))/E,x]

[Out]

(-2*(-x + E^E^x^2*(4 + x)))/E

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fricas [A]  time = 0.68, size = 17, normalized size = 0.74 \begin {gather*} -2 \, {\left ({\left (x + 4\right )} e^{\left (e^{\left (x^{2}\right )}\right )} - x\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^2-16*x)*exp(x^2)-2)*exp(exp(x^2))+2)/exp(1),x, algorithm="fricas")

[Out]

-2*((x + 4)*e^(e^(x^2)) - x)*e^(-1)

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giac [A]  time = 0.19, size = 38, normalized size = 1.65 \begin {gather*} -2 \, {\left ({\left (x e^{\left (x^{2} + e^{\left (x^{2}\right )}\right )} + 4 \, e^{\left (x^{2} + e^{\left (x^{2}\right )}\right )}\right )} e^{\left (-x^{2}\right )} - x\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^2-16*x)*exp(x^2)-2)*exp(exp(x^2))+2)/exp(1),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 20, normalized size = 0.87

method	result	size
risch	\(2 \,{\mathrm e}^{-1} x +\left (-2 x -8\right ) {\mathrm e}^{{\mathrm e}^{x^{2}}-1}\)	\(20\)
default	\({\mathrm e}^{-1} \left (2 x -2 \,{\mathrm e}^{{\mathrm e}^{x^{2}}} x -8 \,{\mathrm e}^{{\mathrm e}^{x^{2}}}\right )\)	\(25\)
norman	\(2 \,{\mathrm e}^{-1} x -8 \,{\mathrm e}^{-1} {\mathrm e}^{{\mathrm e}^{x^{2}}}-2 x \,{\mathrm e}^{-1} {\mathrm e}^{{\mathrm e}^{x^{2}}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-4*x^2-16*x)*exp(x^2)-2)*exp(exp(x^2))+2)/exp(1),x,method=_RETURNVERBOSE)

[Out]

2*exp(-1)*x+(-2*x-8)*exp(exp(x^2)-1)

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maxima [A]  time = 0.41, size = 17, normalized size = 0.74 \begin {gather*} -2 \, {\left ({\left (x + 4\right )} e^{\left (e^{\left (x^{2}\right )}\right )} - x\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^2-16*x)*exp(x^2)-2)*exp(exp(x^2))+2)/exp(1),x, algorithm="maxima")

[Out]

-2*((x + 4)*e^(e^(x^2)) - x)*e^(-1)

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mupad [B]  time = 3.24, size = 22, normalized size = 0.96 \begin {gather*} -2\,{\mathrm {e}}^{-1}\,\left (4\,{\mathrm {e}}^{{\mathrm {e}}^{x^2}}-x+x\,{\mathrm {e}}^{{\mathrm {e}}^{x^2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-1)*(exp(exp(x^2))*(exp(x^2)*(16*x + 4*x^2) + 2) - 2),x)

[Out]

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

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sympy [A]  time = 1.15, size = 22, normalized size = 0.96 \begin {gather*} \frac {2 x}{e} + \frac {\left (- 2 x - 8\right ) e^{e^{x^{2}}}}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x**2-16*x)*exp(x**2)-2)*exp(exp(x**2))+2)/exp(1),x)

[Out]

2*x*exp(-1) + (-2*x - 8)*exp(-1)*exp(exp(x**2))

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