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]

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

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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 = 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 30

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

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]

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*}

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

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.56, size = 16, normalized size = 0.76 \[ -\frac {\sqrt {\pi } e^{c}}{2 \, b \operatorname {erfi}\left (b x\right )} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{b^{2} x^{2}+c}}{\erfi \left (b x \right )^{2}}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.10, size = 16, normalized size = 0.76 \[ -\frac {\sqrt {\pi }\,{\mathrm {e}}^c}{2\,b\,\mathrm {erfi}\left (b\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(c + b^2*x^2)/erfi(b*x)^2,x)

[Out]

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

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sympy [A]  time = 0.97, size = 24, normalized size = 1.14 \[ \begin {cases} - \frac {\sqrt {\pi } e^{c}}{2 b \operatorname {erfi}{\left (b x \right )}} & \text {for}\: b \neq 0 \\\tilde {\infty } x e^{c} & \text {otherwise} \end {cases} \]

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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