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

Optimal. Leaf size=50 \[ \frac {\sqrt {\pi } e^c \text {erfi}(b x)}{2 b}-\frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 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]

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*}

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Mathematica [F]  time = 0.11, size = 0, normalized size = 0.00 \[ \int e^{c+b^2 x^2} \text {erfc}(b x) \, dx \]

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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fricas [F]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (\operatorname {erf}\left (b x\right ) - 1\right )} e^{\left (b^{2} x^{2} + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{b^{2} x^{2}+c} \mathrm {erfc}\left (b x \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfc}\left (b\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(c + b^2*x^2)*erfc(b*x),x)

[Out]

int(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)*erfc(b*x),x)

[Out]

Exception raised: AttributeError

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