3.2.75 \(\int e^{c+b^2 x^2} \text {Erfc}(b x) \, dx\) [175]

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

[Out]

-b*exp(c)*x^2*hypergeom([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 = {6512, 2235, 6511} \begin {gather*} \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 }} \end {gather*}

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 2235

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 6511

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

Rule 6512

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 = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int e^{c+b^2 x^2} \text {Erfc}(b x) \, dx \end {gather*}

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 = 0.07, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 [C] Result contains complex when optimal does not.
time = 4.61, size = 44, normalized size = 0.88 \begin {gather*} - \frac {b x^{2} e^{c} {{}_{2}F_{2}\left (\begin {matrix} 1, 1 \\ \frac {3}{2}, 2 \end {matrix}\middle | {b^{2} x^{2}} \right )}}{\sqrt {\pi }} - \frac {i \sqrt {\pi } e^{c} \operatorname {erf}{\left (i b x \right )}}{2 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*x**2*exp(c)*hyper((1, 1), (3/2, 2), b**2*x**2)/sqrt(pi) - I*sqrt(pi)*exp(c)*erf(I*b*x)/(2*b)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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