3.261 \(\int e^{c+d x^2} x \text {erfi}(b x) \, dx\)

Optimal. Leaf size=53 \[ \frac {\text {erfi}(b x) e^{c+d x^2}}{2 d}-\frac {b e^c \text {erfi}\left (x \sqrt {b^2+d}\right )}{2 d \sqrt {b^2+d}} \]

[Out]

1/2*exp(d*x^2+c)*erfi(b*x)/d-1/2*b*exp(c)*erfi(x*(b^2+d)^(1/2))/d/(b^2+d)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6384, 2204} \[ \frac {\text {Erfi}(b x) e^{c+d x^2}}{2 d}-\frac {b e^c \text {Erfi}\left (x \sqrt {b^2+d}\right )}{2 d \sqrt {b^2+d}} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x^2)*x*Erfi[b*x],x]

[Out]

(E^(c + d*x^2)*Erfi[b*x])/(2*d) - (b*E^c*Erfi[Sqrt[b^2 + d]*x])/(2*d*Sqrt[b^2 + d])

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

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Mathematica [A]  time = 0.03, size = 47, normalized size = 0.89 \[ \frac {e^c \left (e^{d x^2} \text {erfi}(b x)-\frac {b \text {erfi}\left (x \sqrt {b^2+d}\right )}{\sqrt {b^2+d}}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^c*(E^(d*x^2)*Erfi[b*x] - (b*Erfi[Sqrt[b^2 + d]*x])/Sqrt[b^2 + d]))/(2*d)

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fricas [A]  time = 0.45, size = 61, normalized size = 1.15 \[ \frac {\sqrt {-b^{2} - d} b \operatorname {erf}\left (\sqrt {-b^{2} - d} x\right ) e^{c} + {\left (b^{2} + d\right )} \operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{2 \, {\left (b^{2} d + d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(-b^2 - d)*b*erf(sqrt(-b^2 - d)*x)*e^c + (b^2 + d)*erfi(b*x)*e^(d*x^2 + c))/(b^2*d + d^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{d \,x^{2}+c} x \erfi \left (b x \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.17, size = 51, normalized size = 0.96 \[ \frac {{\mathrm {e}}^{d\,x^2}\,{\mathrm {e}}^c\,\mathrm {erfi}\left (b\,x\right )}{2\,d}-\frac {b\,{\mathrm {e}}^c\,\mathrm {erf}\left (x\,\sqrt {-b^2-d}\right )}{2\,d\,\sqrt {-b^2-d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*exp(c + d*x^2)*erfi(b*x),x)

[Out]

(exp(d*x^2)*exp(c)*erfi(b*x))/(2*d) - (b*exp(c)*erf(x*(- d - b^2)^(1/2)))/(2*d*(- d - b^2)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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