\(\int e^{-b^2 x^2} x^3 \text {erfi}(b x) \, dx\) [271]

Optimal result

Integrand size = 18, antiderivative size = 71 \[ \int e^{-b^2 x^2} x^3 \text {erfi}(b x) \, dx=\frac {x}{b^3 \sqrt {\pi }}+\frac {x^3}{3 b \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erfi}(b x)}{2 b^2} \]

[Out]

-1/2*erfi(b*x)/b^4/exp(b^2*x^2)-1/2*x^2*erfi(b*x)/b^2/exp(b^2*x^2)+x/b^3/Pi^(1/2)+1/3*x^3/b/Pi^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6522, 6519, 8, 30} \[ \int e^{-b^2 x^2} x^3 \text {erfi}(b x) \, dx=\frac {x}{\sqrt {\pi } b^3}-\frac {x^2 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 b^4}+\frac {x^3}{3 \sqrt {\pi } b} \]

[In]

Int[(x^3*Erfi[b*x])/E^(b^2*x^2),x]

[Out]

x/(b^3*Sqrt[Pi]) + x^3/(3*b*Sqrt[Pi]) - Erfi[b*x]/(2*b^4*E^(b^2*x^2)) - (x^2*Erfi[b*x])/(2*b^2*E^(b^2*x^2))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

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

Rule 6519

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[E^(c + d*x^2)*(Erfi[a + b*x]/(2*
d)), x] - Dist[b/(d*Sqrt[Pi]), Int[E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2), x], x] /; FreeQ[{a, b, c, d}, x]

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^2 \text {erfi}(b x)}{2 b^2}+\frac {\int e^{-b^2 x^2} x \text {erfi}(b x) \, dx}{b^2}+\frac {\int x^2 \, dx}{b \sqrt {\pi }} \\ & = \frac {x^3}{3 b \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erfi}(b x)}{2 b^2}+\frac {\int 1 \, dx}{b^3 \sqrt {\pi }} \\ & = \frac {x}{b^3 \sqrt {\pi }}+\frac {x^3}{3 b \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erfi}(b x)}{2 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.72 \[ \int e^{-b^2 x^2} x^3 \text {erfi}(b x) \, dx=\frac {\frac {2 b x \left (3+b^2 x^2\right )}{\sqrt {\pi }}-3 e^{-b^2 x^2} \left (1+b^2 x^2\right ) \text {erfi}(b x)}{6 b^4} \]

[In]

Integrate[(x^3*Erfi[b*x])/E^(b^2*x^2),x]

[Out]

((2*b*x*(3 + b^2*x^2))/Sqrt[Pi] - (3*(1 + b^2*x^2)*Erfi[b*x])/E^(b^2*x^2))/(6*b^4)

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.01

method	result	size
default	\(\frac {\left (2 \,{\mathrm e}^{b^{2} x^{2}} b^{3} x^{3}-3 x^{2} \operatorname {erfi}\left (b x \right ) \sqrt {\pi }\, b^{2}+6 \,{\mathrm e}^{b^{2} x^{2}} b x -3 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }\right ) {\mathrm e}^{-b^{2} x^{2}}}{6 \sqrt {\pi }\, b^{4}}\)	\(72\)
parallelrisch	\(\frac {\left (2 \,{\mathrm e}^{b^{2} x^{2}} b^{3} x^{3}-3 x^{2} \operatorname {erfi}\left (b x \right ) \sqrt {\pi }\, b^{2}+6 \,{\mathrm e}^{b^{2} x^{2}} b x -3 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }\right ) {\mathrm e}^{-b^{2} x^{2}}}{6 \sqrt {\pi }\, b^{4}}\)	\(72\)

[In]

int(x^3*erfi(b*x)/exp(b^2*x^2),x,method=_RETURNVERBOSE)

[Out]

1/6*(2*exp(b^2*x^2)*b^3*x^3-3*x^2*erfi(b*x)*Pi^(1/2)*b^2+6*exp(b^2*x^2)*b*x-3*erfi(b*x)*Pi^(1/2))/Pi^(1/2)/b^4
/exp(b^2*x^2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.83 \[ \int e^{-b^2 x^2} x^3 \text {erfi}(b x) \, dx=\frac {{\left (2 \, \sqrt {\pi } {\left (b^{3} x^{3} + 3 \, b x\right )} e^{\left (b^{2} x^{2}\right )} - 3 \, {\left (\pi + \pi b^{2} x^{2}\right )} \operatorname {erfi}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{6 \, \pi b^{4}} \]

[In]

integrate(x^3*erfi(b*x)/exp(b^2*x^2),x, algorithm="fricas")

[Out]

1/6*(2*sqrt(pi)*(b^3*x^3 + 3*b*x)*e^(b^2*x^2) - 3*(pi + pi*b^2*x^2)*erfi(b*x))*e^(-b^2*x^2)/(pi*b^4)

Sympy [A] (verification not implemented)

Time = 17.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89 \[ \int e^{-b^2 x^2} x^3 \text {erfi}(b x) \, dx=\begin {cases} \frac {x^{3}}{3 \sqrt {\pi } b} - \frac {x^{2} e^{- b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{2 b^{2}} + \frac {x}{\sqrt {\pi } b^{3}} - \frac {e^{- b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{2 b^{4}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(x**3*erfi(b*x)/exp(b**2*x**2),x)

[Out]

Piecewise((x**3/(3*sqrt(pi)*b) - x**2*exp(-b**2*x**2)*erfi(b*x)/(2*b**2) + x/(sqrt(pi)*b**3) - exp(-b**2*x**2)
*erfi(b*x)/(2*b**4), Ne(b, 0)), (0, True))

Maxima [F]

\[ \int e^{-b^2 x^2} x^3 \text {erfi}(b x) \, dx=\int { x^{3} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]

[In]

integrate(x^3*erfi(b*x)/exp(b^2*x^2),x, algorithm="maxima")

[Out]

integrate(x^3*erfi(b*x)*e^(-b^2*x^2), x)

Giac [F]

\[ \int e^{-b^2 x^2} x^3 \text {erfi}(b x) \, dx=\int { x^{3} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]

[In]

integrate(x^3*erfi(b*x)/exp(b^2*x^2),x, algorithm="giac")

[Out]

integrate(x^3*erfi(b*x)*e^(-b^2*x^2), x)

Mupad [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.79 \[ \int e^{-b^2 x^2} x^3 \text {erfi}(b x) \, dx=\frac {\frac {b^2\,x^3}{3}+x}{b^3\,\sqrt {\pi }}-\mathrm {erfi}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{-b^2\,x^2}}{2\,b^4}+\frac {x^2\,{\mathrm {e}}^{-b^2\,x^2}}{2\,b^2}\right ) \]

[In]

int(x^3*exp(-b^2*x^2)*erfi(b*x),x)

[Out]

(x + (b^2*x^3)/3)/(b^3*pi^(1/2)) - erfi(b*x)*(exp(-b^2*x^2)/(2*b^4) + (x^2*exp(-b^2*x^2))/(2*b^2))