3.237 \(\int \text{Erfi}(b x)^2 \, dx\)

Optimal. Leaf size=54 \[ -\frac{2 e^{b^2 x^2} \text{Erfi}(b x)}{\sqrt{\pi } b}+x \text{Erfi}(b x)^2+\frac{\sqrt{\frac{2}{\pi }} \text{Erfi}\left (\sqrt{2} b x\right )}{b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0445048, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6354, 12, 6384, 2204} \[ -\frac{2 e^{b^2 x^2} \text{Erfi}(b x)}{\sqrt{\pi } b}+x \text{Erfi}(b x)^2+\frac{\sqrt{\frac{2}{\pi }} \text{Erfi}\left (\sqrt{2} b x\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[Erfi[b*x]^2,x]

[Out]

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

Rule 6354

Int[Erfi[(a_.) + (b_.)*(x_)]^2, x_Symbol] :> Simp[((a + b*x)*Erfi[a + b*x]^2)/b, x] - Dist[4/Sqrt[Pi], Int[(a
+ b*x)*E^(a + b*x)^2*Erfi[a + b*x], x], x] /; FreeQ[{a, b}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6384

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

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rubi steps

\begin{align*} \int \text{erfi}(b x)^2 \, dx &=x \text{erfi}(b x)^2-\frac{4 \int b e^{b^2 x^2} x \text{erfi}(b x) \, dx}{\sqrt{\pi }}\\ &=x \text{erfi}(b x)^2-\frac{(4 b) \int e^{b^2 x^2} x \text{erfi}(b x) \, dx}{\sqrt{\pi }}\\ &=-\frac{2 e^{b^2 x^2} \text{erfi}(b x)}{b \sqrt{\pi }}+x \text{erfi}(b x)^2+\frac{4 \int e^{2 b^2 x^2} \, dx}{\pi }\\ &=-\frac{2 e^{b^2 x^2} \text{erfi}(b x)}{b \sqrt{\pi }}+x \text{erfi}(b x)^2+\frac{\sqrt{\frac{2}{\pi }} \text{erfi}\left (\sqrt{2} b x\right )}{b}\\ \end{align*}

Mathematica [A]  time = 0.011072, size = 54, normalized size = 1. \[ -\frac{2 e^{b^2 x^2} \text{Erfi}(b x)}{\sqrt{\pi } b}+x \text{Erfi}(b x)^2+\frac{\sqrt{\frac{2}{\pi }} \text{Erfi}\left (\sqrt{2} b x\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Erfi[b*x]^2,x]

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(b*x)^2,x)

[Out]

int(erfi(b*x)^2,x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)^2,x, algorithm="maxima")

[Out]

integrate(erfi(b*x)^2, x)

________________________________________________________________________________________

Fricas [A]  time = 2.43849, size = 169, normalized size = 3.13 \begin{align*} \frac{\pi b^{2} x \operatorname{erfi}\left (b x\right )^{2} - 2 \, \sqrt{\pi } b \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} + \sqrt{2} \sqrt{\pi } \sqrt{b^{2}} \operatorname{erfi}\left (\sqrt{2} \sqrt{b^{2}} x\right )}{\pi b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)**2,x)

[Out]

Integral(erfi(b*x)**2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)^2,x, algorithm="giac")

[Out]

integrate(erfi(b*x)^2, x)