3.260 \(\int e^{c+d x^2} x^3 \text {erfi}(b x) \, dx\)

Optimal. Leaf size=142 \[ \frac {b e^c \text {erfi}\left (x \sqrt {b^2+d}\right )}{2 d^2 \sqrt {b^2+d}}+\frac {b e^c \text {erfi}\left (x \sqrt {b^2+d}\right )}{4 d \left (b^2+d\right )^{3/2}}-\frac {b x e^{x^2 \left (b^2+d\right )+c}}{2 \sqrt {\pi } d \left (b^2+d\right )}-\frac {\text {erfi}(b x) e^{c+d x^2}}{2 d^2}+\frac {x^2 \text {erfi}(b x) e^{c+d x^2}}{2 d} \]

[Out]

-1/2*exp(d*x^2+c)*erfi(b*x)/d^2+1/2*exp(d*x^2+c)*x^2*erfi(b*x)/d+1/4*b*exp(c)*erfi(x*(b^2+d)^(1/2))/d/(b^2+d)^
(3/2)+1/2*b*exp(c)*erfi(x*(b^2+d)^(1/2))/d^2/(b^2+d)^(1/2)-1/2*b*exp(c+(b^2+d)*x^2)*x/d/(b^2+d)/Pi^(1/2)

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Rubi [A]  time = 0.15, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6387, 6384, 2204, 2212} \[ \frac {b e^c \text {Erfi}\left (x \sqrt {b^2+d}\right )}{2 d^2 \sqrt {b^2+d}}+\frac {b e^c \text {Erfi}\left (x \sqrt {b^2+d}\right )}{4 d \left (b^2+d\right )^{3/2}}-\frac {b x e^{x^2 \left (b^2+d\right )+c}}{2 \sqrt {\pi } d \left (b^2+d\right )}-\frac {\text {Erfi}(b x) e^{c+d x^2}}{2 d^2}+\frac {x^2 \text {Erfi}(b x) e^{c+d x^2}}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x^2)*x^3*Erfi[b*x],x]

[Out]

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

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 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]

Rubi steps

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

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Mathematica [A]  time = 0.22, size = 91, normalized size = 0.64 \[ \frac {e^c \left (-\frac {2 b d x e^{x^2 \left (b^2+d\right )}}{\sqrt {\pi } \left (b^2+d\right )}+\frac {\left (2 b^3+3 b d\right ) \text {erfi}\left (x \sqrt {b^2+d}\right )}{\left (b^2+d\right )^{3/2}}+2 e^{d x^2} \left (d x^2-1\right ) \text {erfi}(b x)\right )}{4 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^c*((-2*b*d*E^((b^2 + d)*x^2)*x)/((b^2 + d)*Sqrt[Pi]) + 2*E^(d*x^2)*(-1 + d*x^2)*Erfi[b*x] + ((2*b^3 + 3*b*d
)*Erfi[Sqrt[b^2 + d]*x])/(b^2 + d)^(3/2)))/(4*d^2)

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fricas [A]  time = 0.41, size = 151, normalized size = 1.06 \[ -\frac {\pi {\left (2 \, b^{3} + 3 \, b d\right )} \sqrt {-b^{2} - d} \operatorname {erf}\left (\sqrt {-b^{2} - d} x\right ) e^{c} + 2 \, \sqrt {\pi } {\left (b^{3} d + b d^{2}\right )} x e^{\left (b^{2} x^{2} + d x^{2} + c\right )} - 2 \, {\left (\pi {\left (b^{4} d + 2 \, b^{2} d^{2} + d^{3}\right )} x^{2} - \pi {\left (b^{4} + 2 \, b^{2} d + d^{2}\right )}\right )} \operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{4 \, \pi {\left (b^{4} d^{2} + 2 \, b^{2} d^{3} + d^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(pi*(2*b^3 + 3*b*d)*sqrt(-b^2 - d)*erf(sqrt(-b^2 - d)*x)*e^c + 2*sqrt(pi)*(b^3*d + b*d^2)*x*e^(b^2*x^2 +
d*x^2 + c) - 2*(pi*(b^4*d + 2*b^2*d^2 + d^3)*x^2 - pi*(b^4 + 2*b^2*d + d^2))*erfi(b*x)*e^(d*x^2 + c))/(pi*(b^4
*d^2 + 2*b^2*d^3 + d^4))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.22, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{d \,x^{2}+c} x^{3} \erfi \left (b x \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.59, size = 128, normalized size = 0.90 \[ \frac {b\,\mathrm {erfi}\left (x\,\sqrt {b^2+d}\right )\,{\mathrm {e}}^c}{4\,d\,{\left (b^2+d\right )}^{3/2}}-\mathrm {erfi}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{d\,x^2+c}}{2\,d^2}-\frac {x^2\,{\mathrm {e}}^{d\,x^2+c}}{2\,d}\right )-\frac {b\,x\,{\mathrm {e}}^{b^2\,x^2+d\,x^2+c}}{2\,\sqrt {\pi }\,\left (b^2\,d+d^2\right )}+\frac {b\,{\mathrm {e}}^c\,\mathrm {erf}\left (x\,\sqrt {-b^2-d}\right )}{2\,d^2\,\sqrt {-b^2-d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*exp(c + d*x^2)*erfi(b*x),x)

[Out]

(b*erfi(x*(d + b^2)^(1/2))*exp(c))/(4*d*(d + b^2)^(3/2)) - erfi(b*x)*(exp(c + d*x^2)/(2*d^2) - (x^2*exp(c + d*
x^2))/(2*d)) - (b*x*exp(c + d*x^2 + b^2*x^2))/(2*pi^(1/2)*(b^2*d + d^2)) + (b*exp(c)*erf(x*(- d - b^2)^(1/2)))
/(2*d^2*(- d - b^2)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{c} \int x^{3} e^{d x^{2}} \operatorname {erfi}{\left (b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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