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]

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

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Rubi [A]  time = 0.13, antiderivative size = 68, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.06, 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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fricas [A]  time = 0.43, size = 90, normalized size = 1.32 \[ -\frac {2 \, \sqrt {\pi } b \operatorname {erfi}\left (b x + a\right ) e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} - {\left (\pi b^{2} x + \pi a b\right )} \operatorname {erfi}\left (b x + a\right )^{2} - \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erfi}\left (\frac {\sqrt {2} \sqrt {b^{2}} {\left (b x + a\right )}}{b}\right )}{\pi b^{2}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {erfi}\left (b x + a\right )^{2}\,{d x} \]

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)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[ \int \erfi \left (b x +a \right )^{2}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {erfi}\left (b x + a\right )^{2}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {erfi}\left (a+b\,x\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(a + b*x)^2,x)

[Out]

int(erfi(a + b*x)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {erfi}^{2}{\left (a + b x \right )}\, dx \]

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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