3.67 \(\int \frac{e^{c+b^2 x^2} \text{Erf}(b x)}{x} \, dx\)

Optimal. Leaf size=32 \[ \frac{2 b e^c x \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},1\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},b^2 x^2\right )}{\sqrt{\pi }} \]

[Out]

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

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Rubi [A]  time = 0.0434965, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {6388} \[ \frac{2 b e^c x \, _2F_2\left (\frac{1}{2},1;\frac{3}{2},\frac{3}{2};b^2 x^2\right )}{\sqrt{\pi }} \]

Antiderivative was successfully verified.

[In]

Int[(E^(c + b^2*x^2)*Erf[b*x])/x,x]

[Out]

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

Rule 6388

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

Rubi steps

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

Mathematica [A]  time = 0.0860145, size = 32, normalized size = 1. \[ \frac{2 b e^c x \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},1\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},b^2 x^2\right )}{\sqrt{\pi }} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{{b}^{2}{x}^{2}+c}}{\it Erf} \left ( bx \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: AttributeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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