3.217 \(\int \text{Erfi}(b x) \, dx\)

Optimal. Leaf size=26 \[ x \text{Erfi}(b x)-\frac{e^{b^2 x^2}}{\sqrt{\pi } b} \]

[Out]

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

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Rubi [A]  time = 0.0047307, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6351} \[ x \text{Erfi}(b x)-\frac{e^{b^2 x^2}}{\sqrt{\pi } b} \]

Antiderivative was successfully verified.

[In]

Int[Erfi[b*x],x]

[Out]

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

Rule 6351

Int[Erfi[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((a + b*x)*Erfi[a + b*x])/b, x] - Simp[E^(a + b*x)^2/(b*Sqrt[P
i]), x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0072138, size = 26, normalized size = 1. \[ x \text{Erfi}(b x)-\frac{e^{b^2 x^2}}{\sqrt{\pi } b} \]

Antiderivative was successfully verified.

[In]

Integrate[Erfi[b*x],x]

[Out]

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

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Maple [A]  time = 0.04, size = 26, normalized size = 1. \begin{align*}{\frac{1}{b} \left ( bx{\it erfi} \left ( bx \right ) -{\frac{{{\rm e}^{{b}^{2}{x}^{2}}}}{\sqrt{\pi }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(b*x),x)

[Out]

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

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Maxima [A]  time = 1.01803, size = 34, normalized size = 1.31 \begin{align*} \frac{b x \operatorname{erfi}\left (b x\right ) - \frac{e^{\left (b^{2} x^{2}\right )}}{\sqrt{\pi }}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.26128, size = 68, normalized size = 2.62 \begin{align*} \frac{\pi b x \operatorname{erfi}\left (b x\right ) - \sqrt{\pi } e^{\left (b^{2} x^{2}\right )}}{\pi b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.148698, size = 22, normalized size = 0.85 \begin{align*} \begin{cases} x \operatorname{erfi}{\left (b x \right )} - \frac{e^{b^{2} x^{2}}}{\sqrt{\pi } b} & \text{for}\: b \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erfi}\left (b x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x),x, algorithm="giac")

[Out]

integrate(erfi(b*x), x)