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]

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

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Rubi [A]  time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, 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 12

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

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 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]

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*}

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Mathematica [A]  time = 0.02, size = 54, normalized size = 1.00 \[ -\frac {2 e^{b^2 x^2} \text {erfi}(b x)}{\sqrt {\pi } b}+x \text {erfi}(b x)^2+\frac {\sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} b x\right )}{b} \]

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

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fricas [A]  time = 0.45, size = 61, normalized size = 1.13 \[ \frac {\pi b^{2} x \operatorname {erfi}\left (b x\right )^{2} - 2 \, \sqrt {\pi } b \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} + \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erfi}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{\pi b^{2}} \]

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)

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

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {erfi}\left (b\,x\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(erfi(b*x)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {erfi}^{2}{\left (b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(erfi(b*x)**2, x)

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