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]

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

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Rubi [A]  time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*}

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Mathematica [A]  time = 0.01, size = 26, normalized size = 1.00 \[ x \text {erfi}(b x)-\frac {e^{b^2 x^2}}{\sqrt {\pi } b} \]

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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fricas [A]  time = 0.39, size = 29, normalized size = 1.12 \[ \frac {\pi b x \operatorname {erfi}\left (b x\right ) - \sqrt {\pi } e^{\left (b^{2} x^{2}\right )}}{\pi b} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {erfi}\left (b x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfi(b*x), x)

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maple [A]  time = 0.00, size = 26, normalized size = 1.00 \[ \frac {b x \erfi \left (b x \right )-\frac {{\mathrm e}^{b^{2} x^{2}}}{\sqrt {\pi }}}{b} \]

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 = 0.32, size = 25, normalized size = 0.96 \[ \frac {b x \operatorname {erfi}\left (b x\right ) - \frac {e^{\left (b^{2} x^{2}\right )}}{\sqrt {\pi }}}{b} \]

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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mupad [B]  time = 0.05, size = 23, normalized size = 0.88 \[ x\,\mathrm {erfi}\left (b\,x\right )-\frac {{\mathrm {e}}^{b^2\,x^2}}{b\,\sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(b*x),x)

[Out]

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

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sympy [A]  time = 0.13, size = 22, normalized size = 0.85 \[ \begin {cases} x \operatorname {erfi}{\left (b x \right )} - \frac {e^{b^{2} x^{2}}}{\sqrt {\pi } b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

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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