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

Optimal. Leaf size=20 \[ -\frac{\sqrt{\pi } e^c \log (\text{Erfc}(b x))}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.02752, antiderivative size = 20, 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, 29} \[ -\frac{\sqrt{\pi } e^c \log (\text{Erfc}(b x))}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[E^(c - b^2*x^2)/Erfc[b*x],x]

[Out]

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

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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.0107057, size = 20, normalized size = 1. \[ -\frac{\sqrt{\pi } e^c \log (\text{Erfc}(b x))}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{-{b}^{2}{x}^{2}+c}}}{{\it erfc} \left ( bx \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(exp(-b^2*x^2+c)/erfc(b*x),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 )}\,{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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.07644, size = 53, normalized size = 2.65 \begin{align*} -\frac{\sqrt{\pi } e^{c} \log \left (\operatorname{erf}\left (b x\right ) - 1\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(pi)*e^c*log(erf(b*x) - 1)/b

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Sympy [A]  time = 0.548293, size = 24, normalized size = 1.2 \begin{align*} \begin{cases} - \frac{\sqrt{\pi } e^{c} \log{\left (\operatorname{erfc}{\left (b x \right )} \right )}}{2 b} & \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),x)

[Out]

Piecewise((-sqrt(pi)*exp(c)*log(erfc(b*x))/(2*b), 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 )}\,{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),x, algorithm="giac")

[Out]

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