3.220 \(\int \frac{\text{Erfi}(b x)}{x^6} \, dx\)

Optimal. Leaf size=78 \[ \frac{b^5 \text{ExpIntegralEi}\left (b^2 x^2\right )}{10 \sqrt{\pi }}-\frac{b^3 e^{b^2 x^2}}{10 \sqrt{\pi } x^2}-\frac{b e^{b^2 x^2}}{10 \sqrt{\pi } x^4}-\frac{\text{Erfi}(b x)}{5 x^5} \]

[Out]

-(b*E^(b^2*x^2))/(10*Sqrt[Pi]*x^4) - (b^3*E^(b^2*x^2))/(10*Sqrt[Pi]*x^2) - Erfi[b*x]/(5*x^5) + (b^5*ExpIntegra
lEi[b^2*x^2])/(10*Sqrt[Pi])

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Rubi [A]  time = 0.0701197, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6363, 2214, 2210} \[ \frac{b^5 \text{Ei}\left (b^2 x^2\right )}{10 \sqrt{\pi }}-\frac{b^3 e^{b^2 x^2}}{10 \sqrt{\pi } x^2}-\frac{b e^{b^2 x^2}}{10 \sqrt{\pi } x^4}-\frac{\text{Erfi}(b x)}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[Erfi[b*x]/x^6,x]

[Out]

-(b*E^(b^2*x^2))/(10*Sqrt[Pi]*x^4) - (b^3*E^(b^2*x^2))/(10*Sqrt[Pi]*x^2) - Erfi[b*x]/(5*x^5) + (b^5*ExpIntegra
lEi[b^2*x^2])/(10*Sqrt[Pi])

Rule 6363

Int[Erfi[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*Erfi[a + b*x])/(
d*(m + 1)), x] - Dist[(2*b)/(Sqrt[Pi]*d*(m + 1)), Int[(c + d*x)^(m + 1)*E^(a + b*x)^2, x], x] /; FreeQ[{a, b,
c, d, m}, x] && NeQ[m, -1]

Rule 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{\text{erfi}(b x)}{x^6} \, dx &=-\frac{\text{erfi}(b x)}{5 x^5}+\frac{(2 b) \int \frac{e^{b^2 x^2}}{x^5} \, dx}{5 \sqrt{\pi }}\\ &=-\frac{b e^{b^2 x^2}}{10 \sqrt{\pi } x^4}-\frac{\text{erfi}(b x)}{5 x^5}+\frac{b^3 \int \frac{e^{b^2 x^2}}{x^3} \, dx}{5 \sqrt{\pi }}\\ &=-\frac{b e^{b^2 x^2}}{10 \sqrt{\pi } x^4}-\frac{b^3 e^{b^2 x^2}}{10 \sqrt{\pi } x^2}-\frac{\text{erfi}(b x)}{5 x^5}+\frac{b^5 \int \frac{e^{b^2 x^2}}{x} \, dx}{5 \sqrt{\pi }}\\ &=-\frac{b e^{b^2 x^2}}{10 \sqrt{\pi } x^4}-\frac{b^3 e^{b^2 x^2}}{10 \sqrt{\pi } x^2}-\frac{\text{erfi}(b x)}{5 x^5}+\frac{b^5 \text{Ei}\left (b^2 x^2\right )}{10 \sqrt{\pi }}\\ \end{align*}

Mathematica [A]  time = 0.0273478, size = 61, normalized size = 0.78 \[ \frac{b^5 x^5 \text{ExpIntegralEi}\left (b^2 x^2\right )-b x e^{b^2 x^2} \left (b^2 x^2+1\right )-2 \sqrt{\pi } \text{Erfi}(b x)}{10 \sqrt{\pi } x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Erfi[b*x]/x^6,x]

[Out]

(-(b*E^(b^2*x^2)*x*(1 + b^2*x^2)) - 2*Sqrt[Pi]*Erfi[b*x] + b^5*x^5*ExpIntegralEi[b^2*x^2])/(10*Sqrt[Pi]*x^5)

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Maple [A]  time = 0.043, size = 68, normalized size = 0.9 \begin{align*}{b}^{5} \left ( -{\frac{{\it erfi} \left ( bx \right ) }{5\,{b}^{5}{x}^{5}}}+{\frac{2}{5\,\sqrt{\pi }} \left ( -{\frac{{{\rm e}^{{b}^{2}{x}^{2}}}}{4\,{x}^{4}{b}^{4}}}-{\frac{{{\rm e}^{{b}^{2}{x}^{2}}}}{4\,{b}^{2}{x}^{2}}}-{\frac{{\it Ei} \left ( 1,-{b}^{2}{x}^{2} \right ) }{4}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(b*x)/x^6,x)

[Out]

b^5*(-1/5/b^5/x^5*erfi(b*x)+2/5/Pi^(1/2)*(-1/4*exp(b^2*x^2)/b^4/x^4-1/4*exp(b^2*x^2)/b^2/x^2-1/4*Ei(1,-b^2*x^2
)))

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Maxima [A]  time = 1.12932, size = 38, normalized size = 0.49 \begin{align*} -\frac{b^{5} \Gamma \left (-2, -b^{2} x^{2}\right )}{5 \, \sqrt{\pi }} - \frac{\operatorname{erfi}\left (b x\right )}{5 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/x^6,x, algorithm="maxima")

[Out]

-1/5*b^5*gamma(-2, -b^2*x^2)/sqrt(pi) - 1/5*erfi(b*x)/x^5

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Fricas [A]  time = 2.39239, size = 130, normalized size = 1.67 \begin{align*} -\frac{2 \, \pi \operatorname{erfi}\left (b x\right ) - \sqrt{\pi }{\left (b^{5} x^{5}{\rm Ei}\left (b^{2} x^{2}\right ) -{\left (b^{3} x^{3} + b x\right )} e^{\left (b^{2} x^{2}\right )}\right )}}{10 \, \pi x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/x^6,x, algorithm="fricas")

[Out]

-1/10*(2*pi*erfi(b*x) - sqrt(pi)*(b^5*x^5*Ei(b^2*x^2) - (b^3*x^3 + b*x)*e^(b^2*x^2)))/(pi*x^5)

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Sympy [C]  time = 5.01015, size = 85, normalized size = 1.09 \begin{align*} - \frac{b^{5} \operatorname{E}_{1}\left (b^{2} x^{2} e^{i \pi }\right )}{10 \sqrt{\pi }} - \frac{b^{3} e^{b^{2} x^{2}}}{10 \sqrt{\pi } x^{2}} - \frac{b e^{b^{2} x^{2}}}{10 \sqrt{\pi } x^{4}} - \frac{i \operatorname{erfc}{\left (i b x \right )}}{5 x^{5}} + \frac{i}{5 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/x**6,x)

[Out]

-b**5*expint(1, b**2*x**2*exp_polar(I*pi))/(10*sqrt(pi)) - b**3*exp(b**2*x**2)/(10*sqrt(pi)*x**2) - b*exp(b**2
*x**2)/(10*sqrt(pi)*x**4) - I*erfc(I*b*x)/(5*x**5) + I/(5*x**5)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}\left (b x\right )}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/x^6,x, algorithm="giac")

[Out]

integrate(erfi(b*x)/x^6, x)