\(\int e^{-b^2 x^2} x \text {erfi}(b x) \, dx\) [272]

Optimal result

Integrand size = 16, antiderivative size = 32 \[ \int e^{-b^2 x^2} x \text {erfi}(b x) \, dx=\frac {x}{b \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6519, 8} \[ \int e^{-b^2 x^2} x \text {erfi}(b x) \, dx=\frac {x}{\sqrt {\pi } b}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2} \]

[In]

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

[Out]

x/(b*Sqrt[Pi]) - 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 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]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int e^{-b^2 x^2} x \text {erfi}(b x) \, dx=\frac {x}{b \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28

method	result	size
default	\(\frac {\left (2 \,{\mathrm e}^{b^{2} x^{2}} b x -\operatorname {erfi}\left (b x \right ) \sqrt {\pi }\right ) {\mathrm e}^{-b^{2} x^{2}}}{2 \sqrt {\pi }\, b^{2}}\)	\(41\)
parallelrisch	\(\frac {\left (2 \,{\mathrm e}^{b^{2} x^{2}} b x -\operatorname {erfi}\left (b x \right ) \sqrt {\pi }\right ) {\mathrm e}^{-b^{2} x^{2}}}{2 \sqrt {\pi }\, b^{2}}\)	\(41\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int e^{-b^2 x^2} x \text {erfi}(b x) \, dx=\frac {{\left (2 \, \sqrt {\pi } b x e^{\left (b^{2} x^{2}\right )} - \pi \operatorname {erfi}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{2 \, \pi b^{2}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 2.39 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int e^{-b^2 x^2} x \text {erfi}(b x) \, dx=\begin {cases} \frac {x}{\sqrt {\pi } b} - \frac {e^{- b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{2 b^{2}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.69 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int e^{-b^2 x^2} x \text {erfi}(b x) \, dx=\frac {x}{b\,\sqrt {\pi }}-\frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{2\,b^2} \]

[In]

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

[Out]

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