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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, 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 = {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 30

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

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]

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 = 0.01, size = 21, normalized size = 1.00 \[ -\frac {\sqrt {\pi } e^c}{2 b \text {erf}(b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 16, normalized size = 0.76 \[ -\frac {\sqrt {\pi } e^{c}}{2 \, b \operatorname {erf}\left (b x\right )} \]

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))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (-b^{2} x^{2} + c\right )}}{\operatorname {erf}\left (b x\right )^{2}}\,{d x} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 17, normalized size = 0.81 \[ -\frac {{\mathrm e}^{c} \sqrt {\pi }}{2 b \erf \left (b x \right )} \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.53, size = 16, normalized size = 0.76 \[ -\frac {\sqrt {\pi } e^{c}}{2 \, b \operatorname {erf}\left (b x\right )} \]

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))

________________________________________________________________________________________

mupad [B]  time = 0.14, size = 16, normalized size = 0.76 \[ -\frac {\sqrt {\pi }\,{\mathrm {e}}^c}{2\,b\,\mathrm {erf}\left (b\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(c - b^2*x^2)/erf(b*x)^2,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.69, size = 17, normalized size = 0.81 \[ - \frac {\sqrt {\pi } e^{c}}{2 b \operatorname {erf}{\left (b x \right )}} \]

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))

________________________________________________________________________________________