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]

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

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.05, 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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fricas [A]  time = 0.41, size = 45, normalized size = 1.29 \[ \frac {{\left (\pi b x + \pi a\right )} \operatorname {erfi}\left (b x + a\right ) - \sqrt {\pi } e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}}{\pi b} \]

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

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)

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maple [A]  time = 0.00, size = 31, normalized size = 0.89 \[ \frac {\left (b x +a \right ) \erfi \left (b x +a \right )-\frac {{\mathrm e}^{\left (b x +a \right )^{2}}}{\sqrt {\pi }}}{b} \]

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 = 0.31, size = 30, normalized size = 0.86 \[ \frac {{\left (b x + a\right )} \operatorname {erfi}\left (b x + a\right ) - \frac {e^{\left ({\left (b x + a\right )}^{2}\right )}}{\sqrt {\pi }}}{b} \]

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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mupad [B]  time = 0.14, size = 46, normalized size = 1.31 \[ x\,\mathrm {erfi}\left (a+b\,x\right )+\frac {a\,\mathrm {erfi}\left (a+b\,x\right )}{b}-\frac {{\mathrm {e}}^{a^2}\,{\mathrm {e}}^{b^2\,x^2}\,{\mathrm {e}}^{2\,a\,b\,x}}{b\,\sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(a + b*x),x)

[Out]

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

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sympy [A]  time = 0.32, size = 51, normalized size = 1.46 \[ \begin {cases} \frac {a \operatorname {erfi}{\left (a + b x \right )}}{b} + x \operatorname {erfi}{\left (a + b x \right )} - \frac {e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{\sqrt {\pi } b} & \text {for}\: b \neq 0 \\x \operatorname {erfi}{\relax (a )} & \text {otherwise} \end {cases} \]

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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