3.153 \(\int \frac{e^{c-b^2 x^2}}{\text{Erfc}(b x)^2} \, dx\)

Optimal. Leaf size=21 \[ \frac{\sqrt{\pi } e^c}{2 b \text{Erfc}(b x)} \]

[Out]

(E^c*Sqrt[Pi])/(2*b*Erfc[b*x])

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Rubi [A]  time = 0.0277962, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {6374, 30} \[ \frac{\sqrt{\pi } e^c}{2 b \text{Erfc}(b x)} \]

Antiderivative was successfully verified.

[In]

Int[E^(c - b^2*x^2)/Erfc[b*x]^2,x]

[Out]

(E^c*Sqrt[Pi])/(2*b*Erfc[b*x])

Rule 6374

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(E^c*Sqrt[Pi])/(2*b), Subst[Int[x^n, x
], x, Erfc[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, -b^2]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{e^{c-b^2 x^2}}{\text{erfc}(b x)^2} \, dx &=-\frac{\left (e^c \sqrt{\pi }\right ) \operatorname{Subst}\left (\int \frac{1}{x^2} \, dx,x,\text{erfc}(b x)\right )}{2 b}\\ &=\frac{e^c \sqrt{\pi }}{2 b \text{erfc}(b x)}\\ \end{align*}

Mathematica [A]  time = 0.0060713, size = 21, normalized size = 1. \[ \frac{\sqrt{\pi } e^c}{2 b \text{Erfc}(b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(c - b^2*x^2)/Erfc[b*x]^2,x]

[Out]

(E^c*Sqrt[Pi])/(2*b*Erfc[b*x])

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Maple [F]  time = 0.148, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{-{b}^{2}{x}^{2}+c}}}{ \left ({\it erfc} \left ( bx \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-b^2*x^2+c)/erfc(b*x)^2,x)

[Out]

int(exp(-b^2*x^2+c)/erfc(b*x)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (-b^{2} x^{2} + c\right )}}{\operatorname{erfc}\left (b x\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b^2*x^2+c)/erfc(b*x)^2,x, algorithm="maxima")

[Out]

integrate(e^(-b^2*x^2 + c)/erfc(b*x)^2, x)

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Fricas [A]  time = 2.16569, size = 49, normalized size = 2.33 \begin{align*} -\frac{\sqrt{\pi } e^{c}}{2 \,{\left (b \operatorname{erf}\left (b x\right ) - b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b^2*x^2+c)/erfc(b*x)^2,x, algorithm="fricas")

[Out]

-1/2*sqrt(pi)*e^c/(b*erf(b*x) - b)

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Sympy [A]  time = 1.42661, size = 20, normalized size = 0.95 \begin{align*} \begin{cases} \frac{\sqrt{\pi } e^{c}}{2 b \operatorname{erfc}{\left (b x \right )}} & \text{for}\: b \neq 0 \\x e^{c} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b**2*x**2+c)/erfc(b*x)**2,x)

[Out]

Piecewise((sqrt(pi)*exp(c)/(2*b*erfc(b*x)), Ne(b, 0)), (x*exp(c), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (-b^{2} x^{2} + c\right )}}{\operatorname{erfc}\left (b x\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b^2*x^2+c)/erfc(b*x)^2,x, algorithm="giac")

[Out]

integrate(e^(-b^2*x^2 + c)/erfc(b*x)^2, x)