3.51 \(\int \frac{e^{c-b^2 x^2}}{\text{Erf}(b x)^3} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\sqrt{\pi } e^c}{4 b \text{Erf}(b x)^2} \]

[Out]

-(E^c*Sqrt[Pi])/(4*b*Erf[b*x]^2)

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Rubi [A]  time = 0.0287007, 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} \[ -\frac{\sqrt{\pi } e^c}{4 b \text{Erf}(b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(c - b^2*x^2)/Erf[b*x]^3,x]

[Out]

-(E^c*Sqrt[Pi])/(4*b*Erf[b*x]^2)

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)^3} \, dx &=\frac{\left (e^c \sqrt{\pi }\right ) \operatorname{Subst}\left (\int \frac{1}{x^3} \, dx,x,\text{erf}(b x)\right )}{2 b}\\ &=-\frac{e^c \sqrt{\pi }}{4 b \text{erf}(b x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0057237, size = 21, normalized size = 1. \[ -\frac{\sqrt{\pi } e^c}{4 b \text{Erf}(b x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(c - b^2*x^2)/Erf[b*x]^3,x]

[Out]

-(E^c*Sqrt[Pi])/(4*b*Erf[b*x]^2)

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Maple [A]  time = 0.043, size = 17, normalized size = 0.8 \begin{align*} -{\frac{{{\rm e}^{c}}\sqrt{\pi }}{4\,b \left ({\it Erf} \left ( bx \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-b^2*x^2+c)/erf(b*x)^3,x)

[Out]

-1/4*exp(c)*Pi^(1/2)/b/erf(b*x)^2

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Maxima [A]  time = 1.0211, size = 22, normalized size = 1.05 \begin{align*} -\frac{\sqrt{\pi } e^{c}}{4 \, b \operatorname{erf}\left (b x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b^2*x^2+c)/erf(b*x)^3,x, algorithm="maxima")

[Out]

-1/4*sqrt(pi)*e^c/(b*erf(b*x)^2)

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Fricas [A]  time = 2.9458, size = 46, normalized size = 2.19 \begin{align*} -\frac{\sqrt{\pi } e^{c}}{4 \, b \operatorname{erf}\left (b x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b^2*x^2+c)/erf(b*x)^3,x, algorithm="fricas")

[Out]

-1/4*sqrt(pi)*e^c/(b*erf(b*x)^2)

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Sympy [A]  time = 2.74895, size = 19, normalized size = 0.9 \begin{align*} - \frac{\sqrt{\pi } e^{c}}{4 b \operatorname{erf}^{2}{\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)**3,x)

[Out]

-sqrt(pi)*exp(c)/(4*b*erf(b*x)**2)

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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 )^{3}}\,{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)^3,x, algorithm="giac")

[Out]

integrate(e^(-b^2*x^2 + c)/erf(b*x)^3, x)