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

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

[Out]

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

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Rubi [A]  time = 0.0285661, 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 = {6373, 29} \[ \frac{\sqrt{\pi } e^c \log (\text{Erf}(b x))}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[E^(c - b^2*x^2)/Erf[b*x],x]

[Out]

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

Rule 6373

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[(E^c*Sqrt[Pi])/(2*b), Subst[Int[x^n, x],
x, Erf[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{erf}(b x)} \, dx &=\frac{\left (e^c \sqrt{\pi }\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\text{erf}(b x)\right )}{2 b}\\ &=\frac{e^c \sqrt{\pi } \log (\text{erf}(b x))}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.0105597, size = 20, normalized size = 1. \[ \frac{\sqrt{\pi } e^c \log (\text{Erf}(b x))}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(c - b^2*x^2)/Erf[b*x],x]

[Out]

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

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{-{b}^{2}{x}^{2}+c}}}{{\it Erf} \left ( bx \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0433, size = 20, normalized size = 1. \begin{align*} \frac{\sqrt{\pi } e^{c} \log \left (\operatorname{erf}\left (b x\right )\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(pi)*e^c*log(erf(b*x))/b

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Fricas [A]  time = 2.67479, size = 46, normalized size = 2.3 \begin{align*} \frac{\sqrt{\pi } e^{c} \log \left (\operatorname{erf}\left (b x\right )\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(pi)*e^c*log(erf(b*x))/b

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Sympy [A]  time = 0.915131, size = 17, normalized size = 0.85 \begin{align*} \frac{\sqrt{\pi } e^{c} \log{\left (\operatorname{erf}{\left (b x \right )} \right )}}{2 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(pi)*exp(c)*log(erf(b*x))/(2*b)

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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{erf}\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)/erf(b*x),x, algorithm="giac")

[Out]

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