3.174 \(\int e^{c+b^2 x^2} x^2 \text {erfc}(b x) \, dx\)

Optimal. Leaf size=95 \[ \frac {e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi } b}-\frac {\sqrt {\pi } e^c \text {erfi}(b x)}{4 b^3}+\frac {x e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^c x^2}{2 \sqrt {\pi } b} \]

[Out]

1/2*exp(b^2*x^2+c)*x*erfc(b*x)/b^2+1/2*exp(c)*x^2/b/Pi^(1/2)+1/2*exp(c)*x^2*HypergeometricPFQ([1, 1],[3/2, 2],
b^2*x^2)/b/Pi^(1/2)-1/4*exp(c)*erfi(b*x)*Pi^(1/2)/b^3

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Rubi [A]  time = 0.08, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6386, 6377, 2204, 6376, 12, 30} \[ \frac {e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi } b}+\frac {x e^{b^2 x^2+c} \text {Erfc}(b x)}{2 b^2}-\frac {\sqrt {\pi } e^c \text {Erfi}(b x)}{4 b^3}+\frac {e^c x^2}{2 \sqrt {\pi } b} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + b^2*x^2)*x^2*Erfc[b*x],x]

[Out]

(E^c*x^2)/(2*b*Sqrt[Pi]) + (E^(c + b^2*x^2)*x*Erfc[b*x])/(2*b^2) - (E^c*Sqrt[Pi]*Erfi[b*x])/(4*b^3) + (E^c*x^2
*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(2*b*Sqrt[Pi])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 6376

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)], x_Symbol] :> Simp[(b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2},
b^2*x^2])/Sqrt[Pi], x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]

Rule 6377

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)], x_Symbol] :> Int[E^(c + d*x^2), x] - Int[E^(c + d*x^2)*Erf[b*x]
, x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]

Rule 6386

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m - 1)*E^(c + d*x^2)*Er
fc[a + b*x])/(2*d), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erfc[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+b^2 x^2} x^2 \text {erfc}(b x) \, dx &=\frac {e^{c+b^2 x^2} x \text {erfc}(b x)}{2 b^2}-\frac {\int e^{c+b^2 x^2} \text {erfc}(b x) \, dx}{2 b^2}+\frac {\int e^c x \, dx}{b \sqrt {\pi }}\\ &=\frac {e^{c+b^2 x^2} x \text {erfc}(b x)}{2 b^2}-\frac {\int e^{c+b^2 x^2} \, dx}{2 b^2}+\frac {\int e^{c+b^2 x^2} \text {erf}(b x) \, dx}{2 b^2}+\frac {e^c \int x \, dx}{b \sqrt {\pi }}\\ &=\frac {e^c x^2}{2 b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} x \text {erfc}(b x)}{2 b^2}-\frac {e^c \sqrt {\pi } \text {erfi}(b x)}{4 b^3}+\frac {e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 b \sqrt {\pi }}\\ \end {align*}

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Mathematica [A]  time = 0.34, size = 104, normalized size = 1.09 \[ -\frac {e^c \left (2 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )+\text {erf}(b x) \left (2 \sqrt {\pi } b x e^{b^2 x^2}-\pi \text {erfi}(b x)\right )-2 b^2 x^2-2 \sqrt {\pi } b x e^{b^2 x^2}+\pi \text {erfi}(b x)\right )}{4 \sqrt {\pi } b^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/4*(E^c*(-2*b*E^(b^2*x^2)*Sqrt[Pi]*x - 2*b^2*x^2 + Pi*Erfi[b*x] + Erf[b*x]*(2*b*E^(b^2*x^2)*Sqrt[Pi]*x - Pi*
Erfi[b*x]) + 2*b^2*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, -(b^2*x^2)]))/(b^3*Sqrt[Pi])

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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (x^{2} \operatorname {erf}\left (b x\right ) - x^{2}\right )} e^{\left (b^{2} x^{2} + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(x^2*erf(b*x) - x^2)*e^(b^2*x^2 + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.18, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{b^{2} x^{2}+c} x^{2} \mathrm {erfc}\left (b x \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*exp(c + b^2*x^2)*erfc(b*x),x)

[Out]

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

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b**2*x**2+c)*x**2*erfc(b*x),x)

[Out]

Exception raised: AttributeError

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