3.3.18 \(\int \frac {\text {Erfi}(b x)}{x^2} \, dx\) [218]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {6498, 2241} \begin {gather*} \frac {b \text {Ei}\left (b^2 x^2\right )}{\sqrt {\pi }}-\frac {\text {Erfi}(b x)}{x} \end {gather*}

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 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 6498

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 = 0.01, size = 25, normalized size = 1.00 \begin {gather*} -\frac {\text {Erfi}(b x)}{x}+\frac {b \text {Ei}\left (b^2 x^2\right )}{\sqrt {\pi }} \end {gather*}

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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Maple [A]
time = 0.62, size = 31, normalized size = 1.24

method	result	size
derivativedivides	\(b \left (-\frac {\erfi \left (b x \right )}{b x}-\frac {\expIntegral \left (1, -b^{2} x^{2}\right )}{\sqrt {\pi }}\right )\)	\(31\)
default	\(b \left (-\frac {\erfi \left (b x \right )}{b x}-\frac {\expIntegral \left (1, -b^{2} x^{2}\right )}{\sqrt {\pi }}\right )\)	\(31\)
meijerg	\(\frac {b \left (-\frac {2 \sqrt {\pi }\, \erfi \left (b x \right )}{b x}-2 \ln \left (-b^{2} x^{2}\right )-2 \expIntegral \left (1, -b^{2} x^{2}\right )+4 \ln \left (x \right )+4 \ln \left (i b \right )\right )}{2 \sqrt {\pi }}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(b*x)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 23, normalized size = 0.92 \begin {gather*} \frac {b {\rm Ei}\left (b^{2} x^{2}\right )}{\sqrt {\pi }} - \frac {\operatorname {erfi}\left (b x\right )}{x} \end {gather*}

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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Fricas [A]
time = 0.36, size = 29, normalized size = 1.16 \begin {gather*} \frac {\sqrt {\pi } b x {\rm Ei}\left (b^{2} x^{2}\right ) - \pi \operatorname {erfi}\left (b x\right )}{\pi x} \end {gather*}

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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Sympy [C] Result contains complex when optimal does not.
time = 0.56, size = 32, normalized size = 1.28 \begin {gather*} - \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} \end {gather*}

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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