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

Optimal. Leaf size=23 \[ \frac {x+e^{e^4+x} (1-x) (-1+x) x}{x} \]

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Rubi [A]  time = 0.06, antiderivative size = 32, normalized size of antiderivative = 1.39, number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2196, 2194, 2176} \begin {gather*} -e^{x+e^4} x^2+2 e^{x+e^4} x-e^{x+e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(E^4 + x)*(1 - x^2),x]

[Out]

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

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 14, normalized size = 0.61 \begin {gather*} -e^{e^4+x} (-1+x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(E^4 + x)*(1 - x^2),x]

[Out]

-(E^(E^4 + x)*(-1 + x)^2)

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fricas [A]  time = 0.71, size = 15, normalized size = 0.65 \begin {gather*} -{\left (x^{2} - 2 \, x + 1\right )} e^{\left (x + e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^2 - 2*x + 1)*e^(x + e^4)

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giac [A]  time = 0.53, size = 15, normalized size = 0.65 \begin {gather*} -{\left (x^{2} - 2 \, x + 1\right )} e^{\left (x + e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^2 - 2*x + 1)*e^(x + e^4)

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maple [A]  time = 0.04, size = 13, normalized size = 0.57

method	result	size
gosper	\(-\left (x -1\right )^{2} {\mathrm e}^{x +{\mathrm e}^{4}}\)	\(13\)
risch	\(\left (-x^{2}+2 x -1\right ) {\mathrm e}^{x +{\mathrm e}^{4}}\)	\(17\)
norman	\(2 x \,{\mathrm e}^{x +{\mathrm e}^{4}}-x^{2} {\mathrm e}^{x +{\mathrm e}^{4}}-{\mathrm e}^{x +{\mathrm e}^{4}}\)	\(27\)
meijerg	\({\mathrm e}^{{\mathrm e}^{4}} \left (2-\frac {\left (3 x^{2}-6 x +6\right ) {\mathrm e}^{x}}{3}\right )-{\mathrm e}^{{\mathrm e}^{4}} \left (1-{\mathrm e}^{x}\right )\)	\(33\)
derivativedivides	\(-{\mathrm e}^{x +{\mathrm e}^{4}} \left (x +{\mathrm e}^{4}\right )^{2}+2 \left (x +{\mathrm e}^{4}\right ) {\mathrm e}^{x +{\mathrm e}^{4}}-{\mathrm e}^{x +{\mathrm e}^{4}}-{\mathrm e}^{x +{\mathrm e}^{4}} {\mathrm e}^{8}+2 \,{\mathrm e}^{4} \left (\left (x +{\mathrm e}^{4}\right ) {\mathrm e}^{x +{\mathrm e}^{4}}-{\mathrm e}^{x +{\mathrm e}^{4}}\right )\)	\(66\)
default	\(-{\mathrm e}^{x +{\mathrm e}^{4}} \left (x +{\mathrm e}^{4}\right )^{2}+2 \left (x +{\mathrm e}^{4}\right ) {\mathrm e}^{x +{\mathrm e}^{4}}-{\mathrm e}^{x +{\mathrm e}^{4}}-{\mathrm e}^{x +{\mathrm e}^{4}} {\mathrm e}^{8}+2 \,{\mathrm e}^{4} \left (\left (x +{\mathrm e}^{4}\right ) {\mathrm e}^{x +{\mathrm e}^{4}}-{\mathrm e}^{x +{\mathrm e}^{4}}\right )\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.34, size = 29, normalized size = 1.26 \begin {gather*} -{\left (x^{2} e^{\left (e^{4}\right )} - 2 \, x e^{\left (e^{4}\right )} + 2 \, e^{\left (e^{4}\right )}\right )} e^{x} + e^{\left (x + e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.06, size = 12, normalized size = 0.52 \begin {gather*} -{\mathrm {e}}^{x+{\mathrm {e}}^4}\,{\left (x-1\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(x + exp(4))*(x^2 - 1),x)

[Out]

-exp(x + exp(4))*(x - 1)^2

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sympy [A]  time = 0.08, size = 14, normalized size = 0.61 \begin {gather*} \left (- x^{2} + 2 x - 1\right ) e^{x + e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-x**2 + 2*x - 1)*exp(x + exp(4))

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