\(\int \text {erfi}(b x)^2 \, dx\) [237]

Optimal result

Integrand size = 6, antiderivative size = 54 \[ \int \text {erfi}(b x)^2 \, dx=-\frac {2 e^{b^2 x^2} \text {erfi}(b x)}{b \sqrt {\pi }}+x \text {erfi}(b x)^2+\frac {\sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} b x\right )}{b} \]

[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)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6489, 12, 6519, 2235} \[ \int \text {erfi}(b x)^2 \, dx=-\frac {2 e^{b^2 x^2} \text {erfi}(b x)}{\sqrt {\pi } b}+x \text {erfi}(b x)^2+\frac {\sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} b x\right )}{b} \]

[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 2235

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 6489

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 6519

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*} \text {integral}& = 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] (verified)

Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \text {erfi}(b x)^2 \, dx=-\frac {2 e^{b^2 x^2} \text {erfi}(b x)}{b \sqrt {\pi }}+x \text {erfi}(b x)^2+\frac {\sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} b x\right )}{b} \]

[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]

\[\int \operatorname {erfi}\left (b x \right )^{2}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.22 \[ \int \text {erfi}(b x)^2 \, dx=\frac {\pi b^{2} x \operatorname {erfi}\left (b x\right )^{2} - 2 \, \sqrt {\pi } b \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - \sqrt {2} \sqrt {\pi } \sqrt {-b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {-b^{2}} x\right )}{\pi b^{2}} \]

[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)*erf(sqrt(2)*sqrt(-b^2
)*x))/(pi*b^2)

Sympy [F]

\[ \int \text {erfi}(b x)^2 \, dx=\int \operatorname {erfi}^{2}{\left (b x \right )}\, dx \]

[In]

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

[Out]

Integral(erfi(b*x)**2, x)

Maxima [F]

\[ \int \text {erfi}(b x)^2 \, dx=\int { \operatorname {erfi}\left (b x\right )^{2} \,d x } \]

[In]

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

[Out]

integrate(erfi(b*x)^2, x)

Giac [F]

\[ \int \text {erfi}(b x)^2 \, dx=\int { \operatorname {erfi}\left (b x\right )^{2} \,d x } \]

[In]

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

[Out]

integrate(erfi(b*x)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \text {erfi}(b x)^2 \, dx=\int {\mathrm {erfi}\left (b\,x\right )}^2 \,d x \]

[In]

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

[Out]

int(erfi(b*x)^2, x)