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

Optimal result

Integrand size = 8, antiderivative size = 25 \[ \int \frac {\text {erfi}(b x)}{x^2} \, dx=-\frac {\text {erfi}(b x)}{x}+\frac {b \operatorname {ExpIntegralEi}\left (b^2 x^2\right )}{\sqrt {\pi }} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 25, 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 = {6498, 2241} \[ \int \frac {\text {erfi}(b x)}{x^2} \, dx=\frac {b \operatorname {ExpIntegralEi}\left (b^2 x^2\right )}{\sqrt {\pi }}-\frac {\text {erfi}(b x)}{x} \]

[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*} \text {integral}& = -\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 \operatorname {ExpIntegralEi}\left (b^2 x^2\right )}{\sqrt {\pi }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\text {erfi}(b x)}{x^2} \, dx=-\frac {\text {erfi}(b x)}{x}+\frac {b \operatorname {ExpIntegralEi}\left (b^2 x^2\right )}{\sqrt {\pi }} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08

method	result	size
parts	\(-\frac {\operatorname {erfi}\left (b x \right )}{x}-\frac {b \,\operatorname {Ei}_{1}\left (-b^{2} x^{2}\right )}{\sqrt {\pi }}\)	\(27\)
derivativedivides	\(b \left (-\frac {\operatorname {erfi}\left (b x \right )}{b x}-\frac {\operatorname {Ei}_{1}\left (-b^{2} x^{2}\right )}{\sqrt {\pi }}\right )\)	\(31\)
default	\(b \left (-\frac {\operatorname {erfi}\left (b x \right )}{b x}-\frac {\operatorname {Ei}_{1}\left (-b^{2} x^{2}\right )}{\sqrt {\pi }}\right )\)	\(31\)
meijerg	\(\frac {b \left (-\frac {2 \sqrt {\pi }\, \operatorname {erfi}\left (b x \right )}{b x}-2 \ln \left (-b^{2} x^{2}\right )-2 \,\operatorname {Ei}_{1}\left (-b^{2} x^{2}\right )+4 \ln \left (x \right )+4 \ln \left (i b \right )\right )}{2 \sqrt {\pi }}\)	\(57\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {\text {erfi}(b x)}{x^2} \, dx=\frac {\sqrt {\pi } b x {\rm Ei}\left (b^{2} x^{2}\right ) - \pi \operatorname {erfi}\left (b x\right )}{\pi x} \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.56 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {\text {erfi}(b x)}{x^2} \, dx=- \frac {b \operatorname {E}_{1}\left (b^{2} x^{2} e^{i \pi }\right )}{\sqrt {\pi }} - \frac {i \operatorname {erfc}{\left (i b x \right )}}{x} + \frac {i}{x} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\text {erfi}(b x)}{x^2} \, dx=\frac {b {\rm Ei}\left (b^{2} x^{2}\right )}{\sqrt {\pi }} - \frac {\operatorname {erfi}\left (b x\right )}{x} \]

[In]

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

[Out]

b*Ei(b^2*x^2)/sqrt(pi) - erfi(b*x)/x

Giac [F]

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

[In]

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

[Out]

integrate(erfi(b*x)/x^2, x)

Mupad [B] (verification not implemented)

Time = 4.87 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\text {erfi}(b x)}{x^2} \, dx=\frac {b\,\mathrm {ei}\left (b^2\,x^2\right )}{\sqrt {\pi }}-\frac {\mathrm {erfi}\left (b\,x\right )}{x} \]

[In]

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

[Out]

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