\(\int \frac {\text {erf}(b x)}{x^2} \, dx\) [12]

Optimal result

Integrand size = 8, antiderivative size = 26 \[ \int \frac {\text {erf}(b x)}{x^2} \, dx=-\frac {\text {erf}(b x)}{x}+\frac {b \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{\sqrt {\pi }} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6496, 2241} \[ \int \frac {\text {erf}(b x)}{x^2} \, dx=\frac {b \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{\sqrt {\pi }}-\frac {\text {erf}(b x)}{x} \]

[In]

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

[Out]

-(Erf[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 6496

Int[Erf[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Erf[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*} \text {integral}& = -\frac {\text {erf}(b x)}{x}+\frac {(2 b) \int \frac {e^{-b^2 x^2}}{x} \, dx}{\sqrt {\pi }} \\ & = -\frac {\text {erf}(b x)}{x}+\frac {b \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{\sqrt {\pi }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\text {erf}(b x)}{x^2} \, dx=-\frac {\text {erf}(b x)}{x}+\frac {b \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{\sqrt {\pi }} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00

method	result	size
parts	\(-\frac {\operatorname {erf}\left (b x \right )}{x}-\frac {b \,\operatorname {Ei}_{1}\left (b^{2} x^{2}\right )}{\sqrt {\pi }}\)	\(26\)
derivativedivides	\(b \left (-\frac {\operatorname {erf}\left (b x \right )}{b x}-\frac {\operatorname {Ei}_{1}\left (b^{2} x^{2}\right )}{\sqrt {\pi }}\right )\)	\(30\)
default	\(b \left (-\frac {\operatorname {erf}\left (b x \right )}{b x}-\frac {\operatorname {Ei}_{1}\left (b^{2} x^{2}\right )}{\sqrt {\pi }}\right )\)	\(30\)
meijerg	\(\frac {b \left (-\frac {2 b^{2} x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {3}{2}\right ], \left [2, 2, \frac {5}{2}\right ], -b^{2} x^{2}\right )}{3}+2 \gamma -4+4 \ln \left (x \right )+4 \ln \left (b \right )\right )}{2 \sqrt {\pi }}\)	\(45\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {\text {erf}(b x)}{x^2} \, dx=\frac {\sqrt {\pi } b x {\rm Ei}\left (-b^{2} x^{2}\right ) - \pi \operatorname {erf}\left (b x\right )}{\pi x} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\text {erf}(b x)}{x^2} \, dx=- \frac {b \operatorname {E}_{1}\left (b^{2} x^{2}\right )}{\sqrt {\pi }} + \frac {\operatorname {erfc}{\left (b x \right )}}{x} - \frac {1}{x} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\text {erf}(b x)}{x^2} \, dx=\frac {b {\rm Ei}\left (-b^{2} x^{2}\right )}{\sqrt {\pi }} - \frac {\operatorname {erf}\left (b x\right )}{x} \]

[In]

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

[Out]

b*Ei(-b^2*x^2)/sqrt(pi) - erf(b*x)/x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\text {erf}(b x)}{x^2} \, dx=\frac {b {\rm Ei}\left (-b^{2} x^{2}\right )}{\sqrt {\pi }} - \frac {\operatorname {erf}\left (b x\right )}{x} \]

[In]

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

[Out]

b*Ei(-b^2*x^2)/sqrt(pi) - erf(b*x)/x

Mupad [B] (verification not implemented)

Time = 5.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\text {erf}(b x)}{x^2} \, dx=\frac {b\,\mathrm {ei}\left (-b^2\,x^2\right )}{\sqrt {\pi }}-\frac {\mathrm {erf}\left (b\,x\right )}{x} \]

[In]

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

[Out]

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