\(\int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^2} \, dx\) [280]

Optimal result

Integrand size = 18, antiderivative size = 60 \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^2} \, dx=-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{x}-\frac {2 b^3 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{\sqrt {\pi }}+\frac {2 b \log (x)}{\sqrt {\pi }} \]

[Out]

-erfi(b*x)/exp(b^2*x^2)/x-2*b^3*x^2*hypergeom([1, 1],[3/2, 2],-b^2*x^2)/Pi^(1/2)+2*b*ln(x)/Pi^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6528, 6513, 29} \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^2} \, dx=-\frac {2 b^3 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{\sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{x}+\frac {2 b \log (x)}{\sqrt {\pi }} \]

[In]

Int[Erfi[b*x]/(E^(b^2*x^2)*x^2),x]

[Out]

-(Erfi[b*x]/(E^(b^2*x^2)*x)) - (2*b^3*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, -(b^2*x^2)])/Sqrt[Pi] + (2*b*Log
[x])/Sqrt[Pi]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 6513

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(b_.)*(x_)], x_Symbol] :> Simp[b*E^c*(x^2/Sqrt[Pi])*HypergeometricPFQ[{1, 1}
, {3/2, 2}, (-b^2)*x^2], x] /; FreeQ[{b, c, d}, 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 {e^{-b^2 x^2} \text {erfi}(b x)}{x}-\left (2 b^2\right ) \int e^{-b^2 x^2} \text {erfi}(b x) \, dx+\frac {(2 b) \int \frac {1}{x} \, dx}{\sqrt {\pi }} \\ & = -\frac {e^{-b^2 x^2} \text {erfi}(b x)}{x}-\frac {2 b^3 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{\sqrt {\pi }}+\frac {2 b \log (x)}{\sqrt {\pi }} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.43 \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^2} \, dx=-\frac {1}{2} b G_{2,3}^{2,1}\left (b^2 x^2|\begin {array}{c} 0,1 \\ 0,0,-\frac {1}{2} \\\end {array}\right ) \]

[In]

Integrate[Erfi[b*x]/(E^(b^2*x^2)*x^2),x]

[Out]

-1/2*(b*MeijerG[{{0}, {1}}, {{0, 0}, {-1/2}}, b^2*x^2])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(erfi(b*x)*e^(-b^2*x^2)/x^2, x)

Sympy [A] (verification not implemented)

Time = 3.70 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.68 \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^2} \, dx=- \frac {2 b^{3} x^{2} {{}_{2}F_{2}\left (\begin {matrix} 1, 1 \\ 2, \frac {5}{2} \end {matrix}\middle | {- b^{2} x^{2}} \right )}}{3 \sqrt {\pi }} + \frac {b \log {\left (b^{2} x^{2} \right )}}{\sqrt {\pi }} \]

[In]

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

[Out]

-2*b**3*x**2*hyper((1, 1), (2, 5/2), -b**2*x**2)/(3*sqrt(pi)) + b*log(b**2*x**2)/sqrt(pi)

Maxima [F]

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

[In]

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

[Out]

integrate(erfi(b*x)*e^(-b^2*x^2)/x^2, x)

Giac [F]

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

[In]

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

[Out]

integrate(erfi(b*x)*e^(-b^2*x^2)/x^2, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int((exp(-b^2*x^2)*erfi(b*x))/x^2, x)