3.243 \(\int \text{Erfi}(a+b x)^2 \, dx\)

Optimal. Leaf size=68 \[ \frac{(a+b x) \text{Erfi}(a+b x)^2}{b}-\frac{2 e^{(a+b x)^2} \text{Erfi}(a+b x)}{\sqrt{\pi } b}+\frac{\sqrt{\frac{2}{\pi }} \text{Erfi}\left (\sqrt{2} (a+b x)\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.134533, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6354, 6384, 2204} \[ \frac{(a+b x) \text{Erfi}(a+b x)^2}{b}-\frac{2 e^{(a+b x)^2} \text{Erfi}(a+b x)}{\sqrt{\pi } b}+\frac{\sqrt{\frac{2}{\pi }} \text{Erfi}\left (\sqrt{2} (a+b x)\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[Erfi[a + b*x]^2,x]

[Out]

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

Rule 6354

Int[Erfi[(a_.) + (b_.)*(x_)]^2, x_Symbol] :> Simp[((a + b*x)*Erfi[a + b*x]^2)/b, x] - Dist[4/Sqrt[Pi], Int[(a
+ b*x)*E^(a + b*x)^2*Erfi[a + b*x], x], x] /; FreeQ[{a, b}, x]

Rule 6384

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]

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rubi steps

\begin{align*} \int \text{erfi}(a+b x)^2 \, dx &=\frac{(a+b x) \text{erfi}(a+b x)^2}{b}-\frac{4 \int e^{(a+b x)^2} (a+b x) \text{erfi}(a+b x) \, dx}{\sqrt{\pi }}\\ &=\frac{(a+b x) \text{erfi}(a+b x)^2}{b}-\frac{4 \operatorname{Subst}\left (\int e^{x^2} x \text{erfi}(x) \, dx,x,a+b x\right )}{b \sqrt{\pi }}\\ &=-\frac{2 e^{(a+b x)^2} \text{erfi}(a+b x)}{b \sqrt{\pi }}+\frac{(a+b x) \text{erfi}(a+b x)^2}{b}+\frac{4 \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,a+b x\right )}{b \pi }\\ &=-\frac{2 e^{(a+b x)^2} \text{erfi}(a+b x)}{b \sqrt{\pi }}+\frac{(a+b x) \text{erfi}(a+b x)^2}{b}+\frac{\sqrt{\frac{2}{\pi }} \text{erfi}\left (\sqrt{2} (a+b x)\right )}{b}\\ \end{align*}

Mathematica [A]  time = 0.0410586, size = 64, normalized size = 0.94 \[ \frac{\sqrt{\pi } (a+b x) \text{Erfi}(a+b x)^2-2 e^{(a+b x)^2} \text{Erfi}(a+b x)+\sqrt{2} \text{Erfi}\left (\sqrt{2} (a+b x)\right )}{\sqrt{\pi } b} \]

Antiderivative was successfully verified.

[In]

Integrate[Erfi[a + b*x]^2,x]

[Out]

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

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int \left ({\it erfi} \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(b*x+a)^2,x)

[Out]

int(erfi(b*x+a)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erfi}\left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x+a)^2,x, algorithm="maxima")

[Out]

integrate(erfi(b*x + a)^2, x)

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Fricas [A]  time = 2.47335, size = 231, normalized size = 3.4 \begin{align*} -\frac{2 \, \sqrt{\pi } b \operatorname{erfi}\left (b x + a\right ) e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} -{\left (\pi b^{2} x + \pi a b\right )} \operatorname{erfi}\left (b x + a\right )^{2} - \sqrt{2} \sqrt{\pi } \sqrt{b^{2}} \operatorname{erfi}\left (\frac{\sqrt{2} \sqrt{b^{2}}{\left (b x + a\right )}}{b}\right )}{\pi b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erfi}^{2}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x+a)**2,x)

[Out]

Integral(erfi(a + b*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erfi}\left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x+a)^2,x, algorithm="giac")

[Out]

integrate(erfi(b*x + a)^2, x)