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

Optimal. Leaf size=59 \[ -\frac{e^{b^2 x^2+c} \text{Erfi}(b x)}{x}+\frac{b e^c \text{ExpIntegralEi}\left (2 b^2 x^2\right )}{\sqrt{\pi }}+\frac{1}{2} \sqrt{\pi } b e^c \text{Erfi}(b x)^2 \]

[Out]

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

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Rubi [A]  time = 0.0816846, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {6393, 6375, 30, 2210} \[ -\frac{e^{b^2 x^2+c} \text{Erfi}(b x)}{x}+\frac{b e^c \text{Ei}\left (2 b^2 x^2\right )}{\sqrt{\pi }}+\frac{1}{2} \sqrt{\pi } b e^c \text{Erfi}(b x)^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6393

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m + 1)*E^(c + d*x^2)*Er
fi[a + b*x])/(m + 1), x] + (-Dist[(2*d)/(m + 1), Int[x^(m + 2)*E^(c + d*x^2)*Erfi[a + b*x], x], x] - Dist[(2*b
)/((m + 1)*Sqrt[Pi]), Int[x^(m + 1)*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x] &
& ILtQ[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]

Rule 30

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

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

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

Mathematica [A]  time = 0.0210227, size = 56, normalized size = 0.95 \[ \frac{1}{2} e^c \left (-\frac{2 e^{b^2 x^2} \text{Erfi}(b x)}{x}+\frac{2 b \text{ExpIntegralEi}\left (2 b^2 x^2\right )}{\sqrt{\pi }}+\sqrt{\pi } b \text{Erfi}(b x)^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.115, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{{b}^{2}{x}^{2}+c}}{\it erfi} \left ( bx \right ) }{{x}^{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)/x^2,x)

[Out]

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

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

[Out]

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

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Fricas [A]  time = 2.69375, size = 135, normalized size = 2.29 \begin{align*} -\frac{{\left (2 \, \pi \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - \sqrt{\pi }{\left (\pi b x \operatorname{erfi}\left (b x\right )^{2} + 2 \, b x{\rm Ei}\left (2 \, b^{2} x^{2}\right )\right )}\right )} e^{c}}{2 \, \pi x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{c} \int \frac{e^{b^{2} x^{2}} \operatorname{erfi}{\left (b x \right )}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(c)*Integral(exp(b**2*x**2)*erfi(b*x)/x**2, x)

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

[Out]

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