3.256 \(\int \frac{e^{c+b^2 x^2}}{\text{Erfi}(b x)^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0261904, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6375, 30} \[ -\frac{\sqrt{\pi } e^c}{2 b \text{Erfi}(b x)} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + b^2*x^2)/Erfi[b*x]^2,x]

[Out]

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

Rule 6375

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

Mathematica [A]  time = 0.0062358, size = 21, normalized size = 1. \[ -\frac{\sqrt{\pi } e^c}{2 b \text{Erfi}(b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(c + b^2*x^2)/Erfi[b*x]^2,x]

[Out]

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

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{{b}^{2}{x}^{2}+c}}}{ \left ({\it erfi} \left ( bx \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (b^{2} x^{2} + c\right )}}{\operatorname{erfi}\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)/erfi(b*x)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.98027, size = 45, normalized size = 2.14 \begin{align*} -\frac{\sqrt{\pi } e^{c}}{2 \, b \operatorname{erfi}\left (b x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.39146, size = 24, normalized size = 1.14 \begin{align*} \begin{cases} - \frac{\sqrt{\pi } e^{c}}{2 b \operatorname{erfi}{\left (b x \right )}} & \text{for}\: b \neq 0 \\\tilde{\infty } x e^{c} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-sqrt(pi)*exp(c)/(2*b*erfi(b*x)), Ne(b, 0)), (zoo*x*exp(c), True))

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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{erfi}\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)/erfi(b*x)^2,x, algorithm="giac")

[Out]

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