3.284 \(\int e^{c+b^2 x^2} x^3 \text{Erfi}(b x) \, dx\)

Optimal. Leaf size=97 \[ \frac{x^2 e^{b^2 x^2+c} \text{Erfi}(b x)}{2 b^2}-\frac{e^{b^2 x^2+c} \text{Erfi}(b x)}{2 b^4}+\frac{5 e^c \text{Erfi}\left (\sqrt{2} b x\right )}{8 \sqrt{2} b^4}-\frac{x e^{2 b^2 x^2+c}}{4 \sqrt{\pi } b^3} \]

[Out]

-(E^(c + 2*b^2*x^2)*x)/(4*b^3*Sqrt[Pi]) - (E^(c + b^2*x^2)*Erfi[b*x])/(2*b^4) + (E^(c + b^2*x^2)*x^2*Erfi[b*x]
)/(2*b^2) + (5*E^c*Erfi[Sqrt[2]*b*x])/(8*Sqrt[2]*b^4)

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Rubi [A]  time = 0.114576, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {6387, 6384, 2204, 2212} \[ \frac{x^2 e^{b^2 x^2+c} \text{Erfi}(b x)}{2 b^2}-\frac{e^{b^2 x^2+c} \text{Erfi}(b x)}{2 b^4}+\frac{5 e^c \text{Erfi}\left (\sqrt{2} b x\right )}{8 \sqrt{2} b^4}-\frac{x e^{2 b^2 x^2+c}}{4 \sqrt{\pi } b^3} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + b^2*x^2)*x^3*Erfi[b*x],x]

[Out]

-(E^(c + 2*b^2*x^2)*x)/(4*b^3*Sqrt[Pi]) - (E^(c + b^2*x^2)*Erfi[b*x])/(2*b^4) + (E^(c + b^2*x^2)*x^2*Erfi[b*x]
)/(2*b^2) + (5*E^c*Erfi[Sqrt[2]*b*x])/(8*Sqrt[2]*b^4)

Rule 6387

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m - 1)*E^(c + d*x^2)*Er
fi[a + b*x])/(2*d), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erfi[a + b*x], x], x] - Dist[b/(d*S
qrt[Pi]), Int[x^(m - 1)*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1]

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]

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rubi steps

\begin{align*} \int e^{c+b^2 x^2} x^3 \text{erfi}(b x) \, dx &=\frac{e^{c+b^2 x^2} x^2 \text{erfi}(b x)}{2 b^2}-\frac{\int e^{c+b^2 x^2} x \text{erfi}(b x) \, dx}{b^2}-\frac{\int e^{c+2 b^2 x^2} x^2 \, dx}{b \sqrt{\pi }}\\ &=-\frac{e^{c+2 b^2 x^2} x}{4 b^3 \sqrt{\pi }}-\frac{e^{c+b^2 x^2} \text{erfi}(b x)}{2 b^4}+\frac{e^{c+b^2 x^2} x^2 \text{erfi}(b x)}{2 b^2}+\frac{\int e^{c+2 b^2 x^2} \, dx}{4 b^3 \sqrt{\pi }}+\frac{\int e^{c+2 b^2 x^2} \, dx}{b^3 \sqrt{\pi }}\\ &=-\frac{e^{c+2 b^2 x^2} x}{4 b^3 \sqrt{\pi }}-\frac{e^{c+b^2 x^2} \text{erfi}(b x)}{2 b^4}+\frac{e^{c+b^2 x^2} x^2 \text{erfi}(b x)}{2 b^2}+\frac{5 e^c \text{erfi}\left (\sqrt{2} b x\right )}{8 \sqrt{2} b^4}\\ \end{align*}

Mathematica [A]  time = 0.0379454, size = 77, normalized size = 0.79 \[ \frac{e^c \left (8 \sqrt{\pi } e^{b^2 x^2} \left (b^2 x^2-1\right ) \text{Erfi}(b x)-4 b x e^{2 b^2 x^2}+5 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} b x\right )\right )}{16 \sqrt{\pi } b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(c + b^2*x^2)*x^3*Erfi[b*x],x]

[Out]

(E^c*(-4*b*E^(2*b^2*x^2)*x + 8*E^(b^2*x^2)*Sqrt[Pi]*(-1 + b^2*x^2)*Erfi[b*x] + 5*Sqrt[2*Pi]*Erfi[Sqrt[2]*b*x])
)/(16*b^4*Sqrt[Pi])

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Maple [F]  time = 0.227, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{b}^{2}{x}^{2}+c}}{x}^{3}{\it erfi} \left ( bx \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(b^2*x^2+c)*x^3*erfi(b*x),x)

[Out]

int(exp(b^2*x^2+c)*x^3*erfi(b*x),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b^2*x^2+c)*x^3*erfi(b*x),x, algorithm="maxima")

[Out]

integrate(x^3*erfi(b*x)*e^(b^2*x^2 + c), x)

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Fricas [A]  time = 2.66692, size = 213, normalized size = 2.2 \begin{align*} -\frac{4 \, \sqrt{\pi } b^{2} x e^{\left (2 \, b^{2} x^{2} + c\right )} - 5 \, \sqrt{2} \pi \sqrt{b^{2}} \operatorname{erfi}\left (\sqrt{2} \sqrt{b^{2}} x\right ) e^{c} - 8 \,{\left (\pi b^{3} x^{2} - \pi b\right )} \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{16 \, \pi b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b^2*x^2+c)*x^3*erfi(b*x),x, algorithm="fricas")

[Out]

-1/16*(4*sqrt(pi)*b^2*x*e^(2*b^2*x^2 + c) - 5*sqrt(2)*pi*sqrt(b^2)*erfi(sqrt(2)*sqrt(b^2)*x)*e^c - 8*(pi*b^3*x
^2 - pi*b)*erfi(b*x)*e^(b^2*x^2 + c))/(pi*b^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{c} \int x^{3} e^{b^{2} x^{2}} \operatorname{erfi}{\left (b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b**2*x**2+c)*x**3*erfi(b*x),x)

[Out]

exp(c)*Integral(x**3*exp(b**2*x**2)*erfi(b*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b^2*x^2+c)*x^3*erfi(b*x),x, algorithm="giac")

[Out]

integrate(x^3*erfi(b*x)*e^(b^2*x^2 + c), x)