3.34.13 \(\int e^{\frac {1}{2} (-12-8 e^2+2 x+x^2)} (5+e^{\frac {1}{2} (12+8 e^2-2 x-x^2)}+5 x) \, dx\)

Optimal. Leaf size=22 \[ -1+5 e^{-6-4 e^2+x+\frac {x^2}{2}}+x \]

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Rubi [A]  time = 0.19, antiderivative size = 24, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {6688, 2236} \begin {gather*} 5 e^{\frac {x^2}{2}+x-2 \left (3+2 e^2\right )}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^((-12 - 8*E^2 + 2*x + x^2)/2)*(5 + E^((12 + 8*E^2 - 2*x - x^2)/2) + 5*x),x]

[Out]

5*E^(-2*(3 + 2*E^2) + x + x^2/2) + x

Rule 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 6688

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

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 21, normalized size = 0.95 \begin {gather*} 5 e^{-6-4 e^2+x+\frac {x^2}{2}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^((-12 - 8*E^2 + 2*x + x^2)/2)*(5 + E^((12 + 8*E^2 - 2*x - x^2)/2) + 5*x),x]

[Out]

5*E^(-6 - 4*E^2 + x + x^2/2) + x

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fricas [A]  time = 0.60, size = 33, normalized size = 1.50 \begin {gather*} {\left (x e^{\left (-\frac {1}{2} \, x^{2} - x + 4 \, e^{2} + 6\right )} + 5\right )} e^{\left (\frac {1}{2} \, x^{2} + x - 4 \, e^{2} - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(4*exp(2)-1/2*x^2-x+6)+5*x+5)/exp(4*exp(2)-1/2*x^2-x+6),x, algorithm="fricas")

[Out]

(x*e^(-1/2*x^2 - x + 4*e^2 + 6) + 5)*e^(1/2*x^2 + x - 4*e^2 - 6)

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giac [A]  time = 0.21, size = 17, normalized size = 0.77 \begin {gather*} x + 5 \, e^{\left (\frac {1}{2} \, x^{2} + x - 4 \, e^{2} - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(4*exp(2)-1/2*x^2-x+6)+5*x+5)/exp(4*exp(2)-1/2*x^2-x+6),x, algorithm="giac")

[Out]

x + 5*e^(1/2*x^2 + x - 4*e^2 - 6)

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maple [A]  time = 0.03, size = 18, normalized size = 0.82

method	result	size
default	\(x +5 \,{\mathrm e}^{-4 \,{\mathrm e}^{2}+\frac {x^{2}}{2}+x -6}\)	\(18\)
risch	\(x +5 \,{\mathrm e}^{-4 \,{\mathrm e}^{2}+\frac {x^{2}}{2}+x -6}\)	\(18\)
norman	\(\left (5+x \,{\mathrm e}^{4 \,{\mathrm e}^{2}-\frac {x^{2}}{2}-x +6}\right ) {\mathrm e}^{-4 \,{\mathrm e}^{2}+\frac {x^{2}}{2}+x -6}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4*exp(2)-1/2*x^2-x+6)+5*x+5)/exp(4*exp(2)-1/2*x^2-x+6),x,method=_RETURNVERBOSE)

[Out]

x+5*exp(-4*exp(2)+1/2*x^2+x-6)

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maxima [C]  time = 0.84, size = 88, normalized size = 4.00 \begin {gather*} -\frac {5}{2} i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} i \, \sqrt {2} x + \frac {1}{2} i \, \sqrt {2}\right ) e^{\left (-4 \, e^{2} - \frac {13}{2}\right )} - \frac {5}{2} \, \sqrt {2} {\left (\frac {\sqrt {\pi } {\left (x + 1\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-{\left (x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (x + 1\right )}^{2}}} - \sqrt {2} e^{\left (\frac {1}{2} \, {\left (x + 1\right )}^{2}\right )}\right )} e^{\left (-4 \, e^{2} - \frac {13}{2}\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(4*exp(2)-1/2*x^2-x+6)+5*x+5)/exp(4*exp(2)-1/2*x^2-x+6),x, algorithm="maxima")

[Out]

-5/2*I*sqrt(2)*sqrt(pi)*erf(1/2*I*sqrt(2)*x + 1/2*I*sqrt(2))*e^(-4*e^2 - 13/2) - 5/2*sqrt(2)*(sqrt(pi)*(x + 1)
*(erf(sqrt(1/2)*sqrt(-(x + 1)^2)) - 1)/sqrt(-(x + 1)^2) - sqrt(2)*e^(1/2*(x + 1)^2))*e^(-4*e^2 - 13/2) + x

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mupad [B]  time = 0.12, size = 17, normalized size = 0.77 \begin {gather*} x+5\,{\mathrm {e}}^{\frac {x^2}{2}+x-4\,{\mathrm {e}}^2-6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x - 4*exp(2) + x^2/2 - 6)*(5*x + exp(4*exp(2) - x - x^2/2 + 6) + 5),x)

[Out]

x + 5*exp(x - 4*exp(2) + x^2/2 - 6)

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sympy [A]  time = 0.11, size = 17, normalized size = 0.77 \begin {gather*} x + 5 e^{\frac {x^{2}}{2} + x - 4 e^{2} - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(4*exp(2)-1/2*x**2-x+6)+5*x+5)/exp(4*exp(2)-1/2*x**2-x+6),x)

[Out]

x + 5*exp(x**2/2 + x - 4*exp(2) - 6)

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