3.81.93 \(\int (8-e^x+e^{2 e^x} (-16-32 e^x x)) \, dx\)

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

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Rubi [A]  time = 0.02, antiderivative size = 19, normalized size of antiderivative = 0.83, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2194, 2288} \begin {gather*} -16 e^{2 e^x} x+8 x-e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[8 - E^x + E^(2*E^x)*(-16 - 32*E^x*x),x]

[Out]

-E^x + 8*x - 16*E^(2*E^x)*x

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]

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} &=8 x-\int e^x \, dx+\int e^{2 e^x} \left (-16-32 e^x x\right ) \, dx\\ &=-e^x+8 x-16 e^{2 e^x} x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 0.83 \begin {gather*} -e^x+8 x-16 e^{2 e^x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[8 - E^x + E^(2*E^x)*(-16 - 32*E^x*x),x]

[Out]

-E^x + 8*x - 16*E^(2*E^x)*x

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fricas [A]  time = 0.74, size = 16, normalized size = 0.70 \begin {gather*} -16 \, x e^{\left (2 \, e^{x}\right )} + 8 \, x - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*exp(x)*x-16)*exp(2*exp(x))-exp(x)+8,x, algorithm="fricas")

[Out]

-16*x*e^(2*e^x) + 8*x - e^x

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giac [A]  time = 0.14, size = 16, normalized size = 0.70 \begin {gather*} -16 \, x e^{\left (2 \, e^{x}\right )} + 8 \, x - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*exp(x)*x-16)*exp(2*exp(x))-exp(x)+8,x, algorithm="giac")

[Out]

-16*x*e^(2*e^x) + 8*x - e^x

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maple [A]  time = 0.04, size = 17, normalized size = 0.74

method	result	size
default	\(8 x -16 \,{\mathrm e}^{2 \,{\mathrm e}^{x}} x -{\mathrm e}^{x}\)	\(17\)
norman	\(8 x -16 \,{\mathrm e}^{2 \,{\mathrm e}^{x}} x -{\mathrm e}^{x}\)	\(17\)
risch	\(8 x -16 \,{\mathrm e}^{2 \,{\mathrm e}^{x}} x -{\mathrm e}^{x}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-32*exp(x)*x-16)*exp(2*exp(x))-exp(x)+8,x,method=_RETURNVERBOSE)

[Out]

8*x-16*exp(2*exp(x))*x-exp(x)

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maxima [A]  time = 0.35, size = 16, normalized size = 0.70 \begin {gather*} -16 \, x e^{\left (2 \, e^{x}\right )} + 8 \, x - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*exp(x)*x-16)*exp(2*exp(x))-exp(x)+8,x, algorithm="maxima")

[Out]

-16*x*e^(2*e^x) + 8*x - e^x

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mupad [B]  time = 0.06, size = 16, normalized size = 0.70 \begin {gather*} 8\,x-{\mathrm {e}}^x-16\,x\,{\mathrm {e}}^{2\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(8 - exp(2*exp(x))*(32*x*exp(x) + 16) - exp(x),x)

[Out]

8*x - exp(x) - 16*x*exp(2*exp(x))

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sympy [A]  time = 0.15, size = 15, normalized size = 0.65 \begin {gather*} - 16 x e^{2 e^{x}} + 8 x - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*exp(x)*x-16)*exp(2*exp(x))-exp(x)+8,x)

[Out]

-16*x*exp(2*exp(x)) + 8*x - exp(x)

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