3.218 \(\int \frac {\text {erfi}(b x)}{x^2} \, dx\)

Optimal. Leaf size=25 \[ \frac {b \text {Ei}\left (b^2 x^2\right )}{\sqrt {\pi }}-\frac {\text {erfi}(b x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6363, 2210} \[ \frac {b \text {Ei}\left (b^2 x^2\right )}{\sqrt {\pi }}-\frac {\text {Erfi}(b x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rule 6363

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 25, normalized size = 1.00 \[ \frac {b \text {Ei}\left (b^2 x^2\right )}{\sqrt {\pi }}-\frac {\text {erfi}(b x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, size = 29, normalized size = 1.16 \[ \frac {\sqrt {\pi } b x {\rm Ei}\left (b^{2} x^{2}\right ) - \pi \operatorname {erfi}\left (b x\right )}{\pi x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfi(b*x)/x^2, x)

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maple [A]  time = 0.00, size = 31, normalized size = 1.24 \[ b \left (-\frac {\erfi \left (b x \right )}{b x}-\frac {\Ei \left (1, -b^{2} x^{2}\right )}{\sqrt {\pi }}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.35, size = 23, normalized size = 0.92 \[ \frac {b {\rm Ei}\left (b^{2} x^{2}\right )}{\sqrt {\pi }} - \frac {\operatorname {erfi}\left (b x\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*Ei(b^2*x^2)/sqrt(pi) - erfi(b*x)/x

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mupad [B]  time = 0.18, size = 23, normalized size = 0.92 \[ \frac {b\,\mathrm {ei}\left (b^2\,x^2\right )}{\sqrt {\pi }}-\frac {\mathrm {erfi}\left (b\,x\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*ei(b^2*x^2))/pi^(1/2) - erfi(b*x)/x

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sympy [C]  time = 0.98, size = 32, normalized size = 1.28 \[ - \frac {b \operatorname {E}_{1}\left (b^{2} x^{2} e^{i \pi }\right )}{\sqrt {\pi }} - \frac {i \operatorname {erfc}{\left (i b x \right )}}{x} + \frac {i}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*expint(1, b**2*x**2*exp_polar(I*pi))/sqrt(pi) - I*erfc(I*b*x)/x + I/x

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