3.2.52 \(\int \frac {e^{c-b^2 x^2}}{\text {Erfc}(b x)} \, dx\) [152]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 20, 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, 29} \begin {gather*} -\frac {\sqrt {\pi } e^c \log (\text {Erfc}(b x))}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(c - b^2*x^2)/Erfc[b*x],x]

[Out]

-1/2*(E^c*Sqrt[Pi]*Log[Erfc[b*x]])/b

Rule 29

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

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)} \, dx &=-\frac {\left (e^c \sqrt {\pi }\right ) \text {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*}

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Mathematica [A]
time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} -\frac {e^c \sqrt {\pi } \log (\text {Erfc}(b x))}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(E^c*Sqrt[Pi]*Log[Erfc[b*x]])/b

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{-b^{2} x^{2}+c}}{\mathrm {erfc}\left (b x \right )}\, dx\]

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 = 0.00, 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),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 17, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {\pi } e^{c} \log \left (\operatorname {erf}\left (b x\right ) - 1\right )}{2 \, b} \end {gather*}

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 = 0.25, size = 24, normalized size = 1.20 \begin {gather*} \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 {gather*}

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 = 0.00, 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),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.18, size = 15, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {\pi }\,\ln \left (\mathrm {erfc}\left (b\,x\right )\right )\,{\mathrm {e}}^c}{2\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(c - b^2*x^2)/erfc(b*x),x)

[Out]

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

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