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

Optimal. Leaf size=50 \[ \frac{\sqrt{\pi } e^c \text{Erfi}(b x)}{2 b}-\frac{b e^c x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},b^2 x^2\right )}{\sqrt{\pi }} \]

[Out]

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

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Rubi [A]  time = 0.0346488, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {6377, 2204, 6376} \[ \frac{\sqrt{\pi } e^c \text{Erfi}(b x)}{2 b}-\frac{b e^c x^2 \, _2F_2\left (1,1;\frac{3}{2},2;b^2 x^2\right )}{\sqrt{\pi }} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + b^2*x^2)*Erfc[b*x],x]

[Out]

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

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 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]

Rubi steps

\begin{align*} \int e^{c+b^2 x^2} \text{erfc}(b x) \, dx &=\int e^{c+b^2 x^2} \, dx-\int e^{c+b^2 x^2} \text{erf}(b x) \, dx\\ &=\frac{e^c \sqrt{\pi } \text{erfi}(b x)}{2 b}-\frac{b e^c x^2 \, _2F_2\left (1,1;\frac{3}{2},2;b^2 x^2\right )}{\sqrt{\pi }}\\ \end{align*}

Mathematica [F]  time = 0.0810612, size = 0, normalized size = 0. \[ \int e^{c+b^2 x^2} \text{Erfc}(b x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.111, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{b}^{2}{x}^{2}+c}}{\it erfc} \left ( bx \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erfc}\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)*erfc(b*x),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (\operatorname{erf}\left (b x\right ) - 1\right )} e^{\left (b^{2} x^{2} + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(erf(b*x) - 1)*e^(b^2*x^2 + c), x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erfc}\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)*erfc(b*x),x, algorithm="giac")

[Out]

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