\(\int e^{-b^2 x^2} x^2 \text {erfi}(b x) \, dx\) [278]

Optimal result

Integrand size = 18, antiderivative size = 70 \[ \int e^{-b^2 x^2} x^2 \text {erfi}(b x) \, dx=\frac {x^2}{2 b \sqrt {\pi }}-\frac {e^{-b^2 x^2} x \text {erfi}(b x)}{2 b^2}+\frac {x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{2 b \sqrt {\pi }} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 70, 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 = {6522, 6513, 30} \[ \int e^{-b^2 x^2} x^2 \text {erfi}(b x) \, dx=\frac {x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{2 \sqrt {\pi } b}-\frac {x e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {x^2}{2 \sqrt {\pi } b} \]

[In]

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

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 6522

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]/(2*d)), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erfi[a + b*x], x], x] - Dist[b/(d*S
qrt[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] && IGtQ[m, 1]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.51 \[ \int e^{-b^2 x^2} x^2 \text {erfi}(b x) \, dx=\frac {x^2 \left (1-\, _2F_2\left (1,1;\frac {1}{2},2;-b^2 x^2\right )\right )}{2 b \sqrt {\pi }} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 27.87 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.31 \[ \int e^{-b^2 x^2} x^2 \text {erfi}(b x) \, dx=\frac {b x^{4} {{}_{2}F_{2}\left (\begin {matrix} 1, 2 \\ \frac {3}{2}, 3 \end {matrix}\middle | {- b^{2} x^{2}} \right )}}{2 \sqrt {\pi }} \]

[In]

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

[Out]

b*x**4*hyper((1, 2), (3/2, 3), -b**2*x**2)/(2*sqrt(pi))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^2*exp(-b^2*x^2)*erfi(b*x), x)