3.74 \(\int \frac{e^{c+b^2 x^2} \text{Erf}(b x)}{x^4} \, dx\)

Optimal. Leaf size=115 \[ \frac{4 b^5 e^c x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},b^2 x^2\right )}{3 \sqrt{\pi }}-\frac{2 b^2 e^{b^2 x^2+c} \text{Erf}(b x)}{3 x}-\frac{e^{b^2 x^2+c} \text{Erf}(b x)}{3 x^3}+\frac{4 b^3 e^c \log (x)}{3 \sqrt{\pi }}-\frac{b e^c}{3 \sqrt{\pi } x^2} \]

[Out]

-(b*E^c)/(3*Sqrt[Pi]*x^2) - (E^(c + b^2*x^2)*Erf[b*x])/(3*x^3) - (2*b^2*E^(c + b^2*x^2)*Erf[b*x])/(3*x) + (4*b
^5*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(3*Sqrt[Pi]) + (4*b^3*E^c*Log[x])/(3*Sqrt[Pi])

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Rubi [A]  time = 0.10828, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {6391, 6376, 12, 29, 30} \[ \frac{4 b^5 e^c x^2 \, _2F_2\left (1,1;\frac{3}{2},2;b^2 x^2\right )}{3 \sqrt{\pi }}-\frac{2 b^2 e^{b^2 x^2+c} \text{Erf}(b x)}{3 x}-\frac{e^{b^2 x^2+c} \text{Erf}(b x)}{3 x^3}+\frac{4 b^3 e^c \log (x)}{3 \sqrt{\pi }}-\frac{b e^c}{3 \sqrt{\pi } x^2} \]

Antiderivative was successfully verified.

[In]

Int[(E^(c + b^2*x^2)*Erf[b*x])/x^4,x]

[Out]

-(b*E^c)/(3*Sqrt[Pi]*x^2) - (E^(c + b^2*x^2)*Erf[b*x])/(3*x^3) - (2*b^2*E^(c + b^2*x^2)*Erf[b*x])/(3*x) + (4*b
^5*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(3*Sqrt[Pi]) + (4*b^3*E^c*Log[x])/(3*Sqrt[Pi])

Rule 6391

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m + 1)*E^(c + d*x^2)*Erf
[a + b*x])/(m + 1), x] + (-Dist[(2*d)/(m + 1), Int[x^(m + 2)*E^(c + d*x^2)*Erf[a + b*x], x], x] - Dist[(2*b)/(
(m + 1)*Sqrt[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] &&
ILtQ[m, -1]

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 12

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 30

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

Rubi steps

\begin{align*} \int \frac{e^{c+b^2 x^2} \text{erf}(b x)}{x^4} \, dx &=-\frac{e^{c+b^2 x^2} \text{erf}(b x)}{3 x^3}+\frac{1}{3} \left (2 b^2\right ) \int \frac{e^{c+b^2 x^2} \text{erf}(b x)}{x^2} \, dx+\frac{(2 b) \int \frac{e^c}{x^3} \, dx}{3 \sqrt{\pi }}\\ &=-\frac{e^{c+b^2 x^2} \text{erf}(b x)}{3 x^3}-\frac{2 b^2 e^{c+b^2 x^2} \text{erf}(b x)}{3 x}+\frac{1}{3} \left (4 b^4\right ) \int e^{c+b^2 x^2} \text{erf}(b x) \, dx+\frac{\left (4 b^3\right ) \int \frac{e^c}{x} \, dx}{3 \sqrt{\pi }}+\frac{\left (2 b e^c\right ) \int \frac{1}{x^3} \, dx}{3 \sqrt{\pi }}\\ &=-\frac{b e^c}{3 \sqrt{\pi } x^2}-\frac{e^{c+b^2 x^2} \text{erf}(b x)}{3 x^3}-\frac{2 b^2 e^{c+b^2 x^2} \text{erf}(b x)}{3 x}+\frac{4 b^5 e^c x^2 \, _2F_2\left (1,1;\frac{3}{2},2;b^2 x^2\right )}{3 \sqrt{\pi }}+\frac{\left (4 b^3 e^c\right ) \int \frac{1}{x} \, dx}{3 \sqrt{\pi }}\\ &=-\frac{b e^c}{3 \sqrt{\pi } x^2}-\frac{e^{c+b^2 x^2} \text{erf}(b x)}{3 x^3}-\frac{2 b^2 e^{c+b^2 x^2} \text{erf}(b x)}{3 x}+\frac{4 b^5 e^c x^2 \, _2F_2\left (1,1;\frac{3}{2},2;b^2 x^2\right )}{3 \sqrt{\pi }}+\frac{4 b^3 e^c \log (x)}{3 \sqrt{\pi }}\\ \end{align*}

Mathematica [A]  time = 0.340162, size = 100, normalized size = 0.87 \[ -\frac{e^c \left (4 b^5 x^5 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-b^2 x^2\right )-2 \pi b^3 x^3 \text{Erf}(b x) \text{Erfi}(b x)+\sqrt{\pi } e^{b^2 x^2} \left (2 b^2 x^2+1\right ) \text{Erf}(b x)-4 b^3 x^3 \log (x)+b x\right )}{3 \sqrt{\pi } x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(c + b^2*x^2)*Erf[b*x])/x^4,x]

[Out]

-(E^c*(b*x + E^(b^2*x^2)*Sqrt[Pi]*(1 + 2*b^2*x^2)*Erf[b*x] - 2*b^3*Pi*x^3*Erf[b*x]*Erfi[b*x] + 4*b^5*x^5*Hyper
geometricPFQ[{1, 1}, {3/2, 2}, -(b^2*x^2)] - 4*b^3*x^3*Log[x]))/(3*Sqrt[Pi]*x^3)

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Maple [F]  time = 0.273, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{{b}^{2}{x}^{2}+c}}{\it Erf} \left ( bx \right ) }{{x}^{4}}}\, 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^4,x)

[Out]

int(exp(b^2*x^2+c)*erf(b*x)/x^4,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^{4}}\,{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^4,x, algorithm="maxima")

[Out]

integrate(erf(b*x)*e^(b^2*x^2 + c)/x^4, 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^{4}}, 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^4,x, algorithm="fricas")

[Out]

integral(erf(b*x)*e^(b^2*x^2 + c)/x^4, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b**2*x**2+c)*erf(b*x)/x**4,x)

[Out]

Timed out

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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^{4}}\,{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^4,x, algorithm="giac")

[Out]

integrate(erf(b*x)*e^(b^2*x^2 + c)/x^4, x)