3.2.53 \(\int \frac {e^{c-b^2 x^2}}{\text {Erfc}(b x)^2} \, dx\) [153]

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

[Out]

1/2*exp(c)*Pi^(1/2)/b/erfc(b*x)

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Rubi [A]
time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, 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 = {6509, 30} \begin {gather*} \frac {\sqrt {\pi } e^c}{2 b \text {Erfc}(b x)} \end {gather*}

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 30

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

Rule 6509

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]

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 ) \text {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*}

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Mathematica [A]
time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} \frac {e^c \sqrt {\pi }}{2 b \text {Erfc}(b x)} \end {gather*}

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 = 0.08, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{-b^{2} x^{2}+c}}{\mathrm {erfc}\left (b x \right )^{2}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.39, size = 19, normalized size = 0.90 \begin {gather*} -\frac {\sqrt {\pi } e^{c}}{2 \, {\left (b \operatorname {erf}\left (b x\right ) - b\right )}} \end {gather*}

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 = 0.43, size = 20, normalized size = 0.95 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 0.16, size = 16, normalized size = 0.76 \begin {gather*} \frac {\sqrt {\pi }\,{\mathrm {e}}^c}{2\,b\,\mathrm {erfc}\left (b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(c - b^2*x^2)/erfc(b*x)^2,x)

[Out]

(pi^(1/2)*exp(c))/(2*b*erfc(b*x))

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