\(\int \frac {\text {erfi}(b x)^2}{x^3} \, dx\) [232]

Optimal result

Integrand size = 10, antiderivative size = 65 \[ \int \frac {\text {erfi}(b x)^2}{x^3} \, dx=-\frac {2 b e^{b^2 x^2} \text {erfi}(b x)}{\sqrt {\pi } x}+b^2 \text {erfi}(b x)^2-\frac {\text {erfi}(b x)^2}{2 x^2}+\frac {2 b^2 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{\pi } \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6501, 6528, 6510, 30, 2241} \[ \int \frac {\text {erfi}(b x)^2}{x^3} \, dx=-\frac {2 b e^{b^2 x^2} \text {erfi}(b x)}{\sqrt {\pi } x}+b^2 \text {erfi}(b x)^2+\frac {2 b^2 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{\pi }-\frac {\text {erfi}(b x)^2}{2 x^2} \]

[In]

Int[Erfi[b*x]^2/x^3,x]

[Out]

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

Rule 30

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

Rule 2241

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]

Rule 6501

Int[Erfi[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erfi[b*x]^2/(m + 1)), x] - Dist[4*(b/(Sqrt[Pi]
*(m + 1))), Int[x^(m + 1)*E^(b^2*x^2)*Erfi[b*x], x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])

Rule 6510

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 6528

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {erfi}(b x)^2}{2 x^2}+\frac {(2 b) \int \frac {e^{b^2 x^2} \text {erfi}(b x)}{x^2} \, dx}{\sqrt {\pi }} \\ & = -\frac {2 b e^{b^2 x^2} \text {erfi}(b x)}{\sqrt {\pi } x}-\frac {\text {erfi}(b x)^2}{2 x^2}+\frac {\left (4 b^2\right ) \int \frac {e^{2 b^2 x^2}}{x} \, dx}{\pi }+\frac {\left (4 b^3\right ) \int e^{b^2 x^2} \text {erfi}(b x) \, dx}{\sqrt {\pi }} \\ & = -\frac {2 b e^{b^2 x^2} \text {erfi}(b x)}{\sqrt {\pi } x}-\frac {\text {erfi}(b x)^2}{2 x^2}+\frac {2 b^2 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{\pi }+\left (2 b^2\right ) \text {Subst}(\int x \, dx,x,\text {erfi}(b x)) \\ & = -\frac {2 b e^{b^2 x^2} \text {erfi}(b x)}{\sqrt {\pi } x}+b^2 \text {erfi}(b x)^2-\frac {\text {erfi}(b x)^2}{2 x^2}+\frac {2 b^2 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{\pi } \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92 \[ \int \frac {\text {erfi}(b x)^2}{x^3} \, dx=-\frac {2 b e^{b^2 x^2} \text {erfi}(b x)}{\sqrt {\pi } x}+\left (b^2-\frac {1}{2 x^2}\right ) \text {erfi}(b x)^2+\frac {2 b^2 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{\pi } \]

[In]

Integrate[Erfi[b*x]^2/x^3,x]

[Out]

(-2*b*E^(b^2*x^2)*Erfi[b*x])/(Sqrt[Pi]*x) + (b^2 - 1/(2*x^2))*Erfi[b*x]^2 + (2*b^2*ExpIntegralEi[2*b^2*x^2])/P
i

Maple [F]

\[\int \frac {\operatorname {erfi}\left (b x \right )^{2}}{x^{3}}d x\]

[In]

int(erfi(b*x)^2/x^3,x)

[Out]

int(erfi(b*x)^2/x^3,x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.98 \[ \int \frac {\text {erfi}(b x)^2}{x^3} \, dx=\frac {4 \, b^{2} x^{2} {\rm Ei}\left (2 \, b^{2} x^{2}\right ) - 4 \, \sqrt {\pi } b x \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - {\left (\pi - 2 \, \pi b^{2} x^{2}\right )} \operatorname {erfi}\left (b x\right )^{2}}{2 \, \pi x^{2}} \]

[In]

integrate(erfi(b*x)^2/x^3,x, algorithm="fricas")

[Out]

1/2*(4*b^2*x^2*Ei(2*b^2*x^2) - 4*sqrt(pi)*b*x*erfi(b*x)*e^(b^2*x^2) - (pi - 2*pi*b^2*x^2)*erfi(b*x)^2)/(pi*x^2
)

Sympy [F]

\[ \int \frac {\text {erfi}(b x)^2}{x^3} \, dx=\int \frac {\operatorname {erfi}^{2}{\left (b x \right )}}{x^{3}}\, dx \]

[In]

integrate(erfi(b*x)**2/x**3,x)

[Out]

Integral(erfi(b*x)**2/x**3, x)

Maxima [F]

\[ \int \frac {\text {erfi}(b x)^2}{x^3} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right )^{2}}{x^{3}} \,d x } \]

[In]

integrate(erfi(b*x)^2/x^3,x, algorithm="maxima")

[Out]

integrate(erfi(b*x)^2/x^3, x)

Giac [F]

\[ \int \frac {\text {erfi}(b x)^2}{x^3} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right )^{2}}{x^{3}} \,d x } \]

[In]

integrate(erfi(b*x)^2/x^3,x, algorithm="giac")

[Out]

integrate(erfi(b*x)^2/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {erfi}(b x)^2}{x^3} \, dx=\int \frac {{\mathrm {erfi}\left (b\,x\right )}^2}{x^3} \,d x \]

[In]

int(erfi(b*x)^2/x^3,x)

[Out]

int(erfi(b*x)^2/x^3, x)