\(\int e^{-b^2 x^2} \text {erfi}(b x) \, dx\) [279]

Optimal result

Integrand size = 15, antiderivative size = 27 \[ \int e^{-b^2 x^2} \text {erfi}(b x) \, dx=\frac {b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{\sqrt {\pi }} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {6513} \[ \int e^{-b^2 x^2} \text {erfi}(b x) \, dx=\frac {b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{\sqrt {\pi }} \]

[In]

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

[Out]

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

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]

Rubi steps \begin{align*} \text {integral}& = \frac {b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{\sqrt {\pi }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int e^{-b^2 x^2} \text {erfi}(b x) \, dx=\frac {b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{\sqrt {\pi }} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(erfi(b*x)*e^(-b^2*x^2), x)

Sympy [A] (verification not implemented)

Time = 4.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int e^{-b^2 x^2} \text {erfi}(b x) \, dx=\frac {b x^{2} {{}_{2}F_{2}\left (\begin {matrix} 1, 1 \\ \frac {3}{2}, 2 \end {matrix}\middle | {- b^{2} x^{2}} \right )}}{\sqrt {\pi }} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(erfi(b*x)*e^(-b^2*x^2), x)

Giac [F]

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

[In]

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

[Out]

integrate(erfi(b*x)*e^(-b^2*x^2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(exp(-b^2*x^2)*erfi(b*x), x)