3.2.64 \(\int e^{-3-5 x-2 x^2} \, dx\) [164]

Optimal. Leaf size=34 \[ \frac {\sqrt [8]{e} \sqrt {\pi } \text {erf}\left (\frac {5+4 x}{2 \sqrt {2}}\right )}{2 \sqrt {2}} \]

[Out]

1/4*exp(1/8)*Pi^(1/2)*erf(1/4*(5+4*x)*2^(1/2))*2^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2266, 2236} \begin {gather*} \frac {\sqrt [8]{e} \sqrt {\pi } \text {erf}\left (\frac {4 x+5}{2 \sqrt {2}}\right )}{2 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-3 - 5*x - 2*x^2),x]

[Out]

(E^(1/8)*Sqrt[Pi]*Erf[(5 + 4*x)/(2*Sqrt[2])])/(2*Sqrt[2])

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2266

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\sqrt [8]{e} \int e^{-\frac {1}{8} (-5-4 x)^2} \, dx\\ &=\frac {\sqrt [8]{e} \sqrt {\pi } \text {erf}\left (\frac {5+4 x}{2 \sqrt {2}}\right )}{2 \sqrt {2}}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 34, normalized size = 1.00 \begin {gather*} \frac {\sqrt [8]{e} \sqrt {\pi } \text {erf}\left (\frac {5+4 x}{2 \sqrt {2}}\right )}{2 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-3 - 5*x - 2*x^2),x]

[Out]

(E^(1/8)*Sqrt[Pi]*Erf[(5 + 4*x)/(2*Sqrt[2])])/(2*Sqrt[2])

________________________________________________________________________________________

Maple [A]
time = 0.01, size = 23, normalized size = 0.68

method	result	size
default	\(\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {1}{8}} \sqrt {2}\, \erf \left (\sqrt {2}\, x +\frac {5 \sqrt {2}}{4}\right )}{4}\)	\(23\)
risch	\(\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {1}{8}} \sqrt {2}\, \erf \left (\sqrt {2}\, x +\frac {5 \sqrt {2}}{4}\right )}{4}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-2*x^2-5*x-3),x,method=_RETURNVERBOSE)

[Out]

1/4*Pi^(1/2)*exp(1/8)*2^(1/2)*erf(2^(1/2)*x+5/4*2^(1/2))

________________________________________________________________________________________

Maxima [A]
time = 0.47, size = 22, normalized size = 0.65 \begin {gather*} \frac {1}{4} \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} x + \frac {5}{4} \, \sqrt {2}\right ) e^{\frac {1}{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-2*x^2-5*x-3),x, algorithm="maxima")

[Out]

1/4*sqrt(2)*sqrt(pi)*erf(sqrt(2)*x + 5/4*sqrt(2))*e^(1/8)

________________________________________________________________________________________

Fricas [A]
time = 0.58, size = 21, normalized size = 0.62 \begin {gather*} \frac {1}{4} \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{4} \, \sqrt {2} {\left (4 \, x + 5\right )}\right ) e^{\frac {1}{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-2*x^2-5*x-3),x, algorithm="fricas")

[Out]

1/4*sqrt(2)*sqrt(pi)*erf(1/4*sqrt(2)*(4*x + 5))*e^(1/8)

________________________________________________________________________________________

Sympy [A]
time = 0.34, size = 32, normalized size = 0.94 \begin {gather*} \frac {\sqrt {2} \sqrt {\pi } e^{\frac {1}{8}} \operatorname {erf}{\left (\sqrt {2} x + \frac {5 \sqrt {2}}{4} \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-2*x**2-5*x-3),x)

[Out]

sqrt(2)*sqrt(pi)*exp(1/8)*erf(sqrt(2)*x + 5*sqrt(2)/4)/4

________________________________________________________________________________________

Giac [A]
time = 0.48, size = 21, normalized size = 0.62 \begin {gather*} \frac {1}{4} \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{4} \, \sqrt {2} {\left (4 \, x + 5\right )}\right ) e^{\frac {1}{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-2*x^2-5*x-3),x, algorithm="giac")

[Out]

1/4*sqrt(2)*sqrt(pi)*erf(1/4*sqrt(2)*(4*x + 5))*e^(1/8)

________________________________________________________________________________________

Mupad [B]
time = 0.08, size = 22, normalized size = 0.65 \begin {gather*} \frac {\sqrt {2}\,\sqrt {\pi }\,{\mathrm {e}}^{1/8}\,\mathrm {erf}\left (\sqrt {2}\,x+\frac {5\,\sqrt {2}}{4}\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(- 5*x - 2*x^2 - 3),x)

[Out]

(2^(1/2)*pi^(1/2)*exp(1/8)*erf(2^(1/2)*x + (5*2^(1/2))/4))/4

________________________________________________________________________________________

Chatgpt [F] Failed to verify
time = 1.00, size = 10, normalized size = 0.29 \begin {gather*} -\frac {{\mathrm e}^{-2 \left (x +\frac {5}{4}\right )^{2}}}{4} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(exp(-2*x^2-5*x-3),x)

[Out]

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

________________________________________________________________________________________