3.2.31 \(\int e^{-e^4 x-e^4 \log (2)} (4+e^{e^4 x+e^4 \log (2)} (-3+e^{-2+x} (-1-x))-4 e^4 x) \, dx\)

Optimal. Leaf size=29 \[ \left (-3-e^{-2+x}+4 e^{-e^4 (x+\log (2))}+\frac {5}{x}\right ) x \]

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Rubi [A]  time = 0.31, antiderivative size = 31, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 5, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2274, 12, 6742, 2194, 2176} \begin {gather*} -e^{x-2} x+2^{2-e^4} e^{-e^4 x} x-3 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-3*x - E^(-2 + x)*x + (2^(2 - E^4)*x)/E^(E^4*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 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 2274

Int[(u_.)*(F_)^((a_.)*(Log[z_]*(b_.) + (v_.))), x_Symbol] :> Int[u*F^(a*v)*z^(a*b*Log[F]), x] /; FreeQ[{F, a,
b}, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [A]  time = 0.18, size = 41, normalized size = 1.41 \begin {gather*} 2^{-e^4} \left (-3 2^{e^4} x-2^{e^4} e^{-2+x} x+4 e^{-e^4 x} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*2^E^4*x - 2^E^4*E^(-2 + x)*x + (4*x)/E^(E^4*x))/2^E^4

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fricas [A]  time = 0.89, size = 41, normalized size = 1.41 \begin {gather*} -{\left ({\left (x e^{\left (x - 2\right )} + 3 \, x\right )} e^{\left (x e^{4} + e^{4} \log \relax (2)\right )} - 4 \, x\right )} e^{\left (-x e^{4} - e^{4} \log \relax (2)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x-1)*exp(x-2)-3)*exp(exp(4)*log(2)+x*exp(4))-4*x*exp(4)+4)/exp(exp(4)*log(2)+x*exp(4)),x, algori
thm="fricas")

[Out]

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

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giac [B]  time = 0.49, size = 65, normalized size = 2.24 \begin {gather*} -{\left (3 \, x e^{10} - 4 \, x e^{\left (-x e^{4} - e^{4} \log \relax (2) + 10\right )} + x e^{\left (x + 8\right )} + \frac {4 \, e^{\left (-x e^{4} + 6\right )}}{2^{e^{4}}} - 4 \, e^{\left (-x e^{4} - e^{4} \log \relax (2) + 6\right )}\right )} e^{\left (-10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x-1)*exp(x-2)-3)*exp(exp(4)*log(2)+x*exp(4))-4*x*exp(4)+4)/exp(exp(4)*log(2)+x*exp(4)),x, algori
thm="giac")

[Out]

-(3*x*e^10 - 4*x*e^(-x*e^4 - e^4*log(2) + 10) + x*e^(x + 8) + 4*e^(-x*e^4 + 6)/2^e^4 - 4*e^(-x*e^4 - e^4*log(2
) + 6))*e^(-10)

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maple [A]  time = 0.06, size = 27, normalized size = 0.93

method	result	size
risch	\(-x \,{\mathrm e}^{x -2}-3 x +4 x 2^{-{\mathrm e}^{4}} {\mathrm e}^{-x \,{\mathrm e}^{4}}\)	\(27\)
norman	\(\left (4 x -3 x \,{\mathrm e}^{{\mathrm e}^{4} \ln \relax (2)+x \,{\mathrm e}^{4}}-{\mathrm e}^{x -2} {\mathrm e}^{{\mathrm e}^{4} \ln \relax (2)+x \,{\mathrm e}^{4}} x \right ) {\mathrm e}^{-{\mathrm e}^{4} \ln \relax (2)-x \,{\mathrm e}^{4}}\)	\(51\)
default	\(-3 x -4 \,2^{-{\mathrm e}^{4}} {\mathrm e}^{-4} {\mathrm e}^{-x \,{\mathrm e}^{4}}-{\mathrm e}^{x} {\mathrm e}^{-2}-{\mathrm e}^{-2} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )-4 \,2^{-{\mathrm e}^{4}} {\mathrm e}^{-4} \left (-x \,{\mathrm e}^{4} {\mathrm e}^{-x \,{\mathrm e}^{4}}-{\mathrm e}^{-x \,{\mathrm e}^{4}}\right )\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x-1)*exp(x-2)-3)*exp(exp(4)*ln(2)+x*exp(4))-4*x*exp(4)+4)/exp(exp(4)*ln(2)+x*exp(4)),x,method=_RETURNV
ERBOSE)

[Out]

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

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maxima [B]  time = 0.45, size = 58, normalized size = 2.00 \begin {gather*} {\left (x e^{4} + 1\right )} 2^{-e^{4} + 2} e^{\left (-x e^{4} - 4\right )} - {\left (x - 1\right )} e^{\left (x - 2\right )} - 3 \, x - 4 \, e^{\left (-x e^{4} - e^{4} \log \relax (2) - 4\right )} - e^{\left (x - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x-1)*exp(x-2)-3)*exp(exp(4)*log(2)+x*exp(4))-4*x*exp(4)+4)/exp(exp(4)*log(2)+x*exp(4)),x, algori
thm="maxima")

[Out]

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

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mupad [B]  time = 0.15, size = 23, normalized size = 0.79 \begin {gather*} -x\,\left ({\mathrm {e}}^{x-2}-\frac {4\,{\mathrm {e}}^{-x\,{\mathrm {e}}^4}}{2^{{\mathrm {e}}^4}}+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 7.50, size = 36, normalized size = 1.24 \begin {gather*} \frac {- \frac {2^{e^{4}} x e^{x}}{e^{2}} - 3 \cdot 2^{e^{4}} x + 4 x e^{- x e^{4}}}{2^{e^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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