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

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

[Out]

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

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Rubi [A]  time = 0.0276506, 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 = {6373, 30} \[ -\frac{\sqrt{\pi } e^c}{2 b \text{Erf}(b x)} \]

Antiderivative was successfully verified.

[In]

Int[E^(c - b^2*x^2)/Erf[b*x]^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0062499, size = 21, normalized size = 1. \[ -\frac{\sqrt{\pi } e^c}{2 b \text{Erf}(b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 17, normalized size = 0.8 \begin{align*} -{\frac{{{\rm e}^{c}}\sqrt{\pi }}{2\,b{\it Erf} \left ( bx \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01952, size = 22, normalized size = 1.05 \begin{align*} -\frac{\sqrt{\pi } e^{c}}{2 \, b \operatorname{erf}\left (b x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.65468, size = 43, normalized size = 2.05 \begin{align*} -\frac{\sqrt{\pi } e^{c}}{2 \, b \operatorname{erf}\left (b x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.99355, size = 17, normalized size = 0.81 \begin{align*} - \frac{\sqrt{\pi } e^{c}}{2 b \operatorname{erf}{\left (b x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )^{2}}\,{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)^2,x, algorithm="giac")

[Out]

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