3.224 \(\int \text{Erfi}(a+b x) \, dx\)

Optimal. Leaf size=35 \[ \frac{(a+b x) \text{Erfi}(a+b x)}{b}-\frac{e^{(a+b x)^2}}{\sqrt{\pi } b} \]

[Out]

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

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Rubi [A]  time = 0.0057444, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6351} \[ \frac{(a+b x) \text{Erfi}(a+b x)}{b}-\frac{e^{(a+b x)^2}}{\sqrt{\pi } b} \]

Antiderivative was successfully verified.

[In]

Int[Erfi[a + b*x],x]

[Out]

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

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

Mathematica [A]  time = 0.0265767, size = 33, normalized size = 0.94 \[ \frac{(a+b x) \text{Erfi}(a+b x)-\frac{e^{(a+b x)^2}}{\sqrt{\pi }}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Erfi[a + b*x],x]

[Out]

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

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Maple [A]  time = 0.045, size = 31, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( \left ( bx+a \right ){\it erfi} \left ( bx+a \right ) -{\frac{{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}{\sqrt{\pi }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.0806, size = 41, normalized size = 1.17 \begin{align*} \frac{{\left (b x + a\right )} \operatorname{erfi}\left (b x + a\right ) - \frac{e^{\left ({\left (b x + a\right )}^{2}\right )}}{\sqrt{\pi }}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.43289, size = 107, normalized size = 3.06 \begin{align*} \frac{{\left (\pi b x + \pi a\right )} \operatorname{erfi}\left (b x + a\right ) - \sqrt{\pi } e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}}{\pi b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.409487, size = 51, normalized size = 1.46 \begin{align*} \begin{cases} \frac{a \operatorname{erfi}{\left (a + b x \right )}}{b} + x \operatorname{erfi}{\left (a + b x \right )} - \frac{e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{\sqrt{\pi } b} & \text{for}\: b \neq 0 \\x \operatorname{erfi}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfi(b*x + a), x)