3.213 \(\int \frac{\text{Erfi}(b x)}{x^7} \, dx\)

Optimal. Leaf size=93 \[ \frac{4}{45} b^6 \text{Erfi}(b x)-\frac{4 b^5 e^{b^2 x^2}}{45 \sqrt{\pi } x}-\frac{2 b^3 e^{b^2 x^2}}{45 \sqrt{\pi } x^3}-\frac{b e^{b^2 x^2}}{15 \sqrt{\pi } x^5}-\frac{\text{Erfi}(b x)}{6 x^6} \]

[Out]

-(b*E^(b^2*x^2))/(15*Sqrt[Pi]*x^5) - (2*b^3*E^(b^2*x^2))/(45*Sqrt[Pi]*x^3) - (4*b^5*E^(b^2*x^2))/(45*Sqrt[Pi]*
x) + (4*b^6*Erfi[b*x])/45 - Erfi[b*x]/(6*x^6)

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Rubi [A]  time = 0.0785057, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6363, 2214, 2204} \[ \frac{4}{45} b^6 \text{Erfi}(b x)-\frac{4 b^5 e^{b^2 x^2}}{45 \sqrt{\pi } x}-\frac{2 b^3 e^{b^2 x^2}}{45 \sqrt{\pi } x^3}-\frac{b e^{b^2 x^2}}{15 \sqrt{\pi } x^5}-\frac{\text{Erfi}(b x)}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[Erfi[b*x]/x^7,x]

[Out]

-(b*E^(b^2*x^2))/(15*Sqrt[Pi]*x^5) - (2*b^3*E^(b^2*x^2))/(45*Sqrt[Pi]*x^3) - (4*b^5*E^(b^2*x^2))/(45*Sqrt[Pi]*
x) + (4*b^6*Erfi[b*x])/45 - Erfi[b*x]/(6*x^6)

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 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]

Rubi steps

\begin{align*} \int \frac{\text{erfi}(b x)}{x^7} \, dx &=-\frac{\text{erfi}(b x)}{6 x^6}+\frac{b \int \frac{e^{b^2 x^2}}{x^6} \, dx}{3 \sqrt{\pi }}\\ &=-\frac{b e^{b^2 x^2}}{15 \sqrt{\pi } x^5}-\frac{\text{erfi}(b x)}{6 x^6}+\frac{\left (2 b^3\right ) \int \frac{e^{b^2 x^2}}{x^4} \, dx}{15 \sqrt{\pi }}\\ &=-\frac{b e^{b^2 x^2}}{15 \sqrt{\pi } x^5}-\frac{2 b^3 e^{b^2 x^2}}{45 \sqrt{\pi } x^3}-\frac{\text{erfi}(b x)}{6 x^6}+\frac{\left (4 b^5\right ) \int \frac{e^{b^2 x^2}}{x^2} \, dx}{45 \sqrt{\pi }}\\ &=-\frac{b e^{b^2 x^2}}{15 \sqrt{\pi } x^5}-\frac{2 b^3 e^{b^2 x^2}}{45 \sqrt{\pi } x^3}-\frac{4 b^5 e^{b^2 x^2}}{45 \sqrt{\pi } x}-\frac{\text{erfi}(b x)}{6 x^6}+\frac{\left (8 b^7\right ) \int e^{b^2 x^2} \, dx}{45 \sqrt{\pi }}\\ &=-\frac{b e^{b^2 x^2}}{15 \sqrt{\pi } x^5}-\frac{2 b^3 e^{b^2 x^2}}{45 \sqrt{\pi } x^3}-\frac{4 b^5 e^{b^2 x^2}}{45 \sqrt{\pi } x}+\frac{4}{45} b^6 \text{erfi}(b x)-\frac{\text{erfi}(b x)}{6 x^6}\\ \end{align*}

Mathematica [A]  time = 0.0239837, size = 64, normalized size = 0.69 \[ \frac{\sqrt{\pi } \left (8 b^6 x^6-15\right ) \text{Erfi}(b x)-2 b x e^{b^2 x^2} \left (4 b^4 x^4+2 b^2 x^2+3\right )}{90 \sqrt{\pi } x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Erfi[b*x]/x^7,x]

[Out]

(-2*b*E^(b^2*x^2)*x*(3 + 2*b^2*x^2 + 4*b^4*x^4) + Sqrt[Pi]*(-15 + 8*b^6*x^6)*Erfi[b*x])/(90*Sqrt[Pi]*x^6)

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Maple [A]  time = 0.043, size = 81, normalized size = 0.9 \begin{align*}{b}^{6} \left ( -{\frac{{\it erfi} \left ( bx \right ) }{6\,{b}^{6}{x}^{6}}}+{\frac{1}{3\,\sqrt{\pi }} \left ( -{\frac{{{\rm e}^{{b}^{2}{x}^{2}}}}{5\,{b}^{5}{x}^{5}}}-{\frac{2\,{{\rm e}^{{b}^{2}{x}^{2}}}}{15\,{x}^{3}{b}^{3}}}-{\frac{4\,{{\rm e}^{{b}^{2}{x}^{2}}}}{15\,bx}}+{\frac{4\,\sqrt{\pi }{\it erfi} \left ( bx \right ) }{15}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(b*x)/x^7,x)

[Out]

b^6*(-1/6/b^6/x^6*erfi(b*x)+1/3/Pi^(1/2)*(-1/5*exp(b^2*x^2)/b^5/x^5-2/15*exp(b^2*x^2)/b^3/x^3-4/15*exp(b^2*x^2
)/b/x+4/15*Pi^(1/2)*erfi(b*x)))

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Maxima [A]  time = 1.07059, size = 53, normalized size = 0.57 \begin{align*} -\frac{\left (-b^{2} x^{2}\right )^{\frac{5}{2}} b \Gamma \left (-\frac{5}{2}, -b^{2} x^{2}\right )}{6 \, \sqrt{\pi } x^{5}} - \frac{\operatorname{erfi}\left (b x\right )}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/x^7,x, algorithm="maxima")

[Out]

-1/6*(-b^2*x^2)^(5/2)*b*gamma(-5/2, -b^2*x^2)/(sqrt(pi)*x^5) - 1/6*erfi(b*x)/x^6

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Fricas [A]  time = 2.4588, size = 146, normalized size = 1.57 \begin{align*} -\frac{2 \, \sqrt{\pi }{\left (4 \, b^{5} x^{5} + 2 \, b^{3} x^{3} + 3 \, b x\right )} e^{\left (b^{2} x^{2}\right )} +{\left (15 \, \pi - 8 \, \pi b^{6} x^{6}\right )} \operatorname{erfi}\left (b x\right )}{90 \, \pi x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/x^7,x, algorithm="fricas")

[Out]

-1/90*(2*sqrt(pi)*(4*b^5*x^5 + 2*b^3*x^3 + 3*b*x)*e^(b^2*x^2) + (15*pi - 8*pi*b^6*x^6)*erfi(b*x))/(pi*x^6)

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Sympy [A]  time = 4.45284, size = 87, normalized size = 0.94 \begin{align*} \frac{4 b^{6} \operatorname{erfi}{\left (b x \right )}}{45} - \frac{4 b^{5} e^{b^{2} x^{2}}}{45 \sqrt{\pi } x} - \frac{2 b^{3} e^{b^{2} x^{2}}}{45 \sqrt{\pi } x^{3}} - \frac{b e^{b^{2} x^{2}}}{15 \sqrt{\pi } x^{5}} - \frac{\operatorname{erfi}{\left (b x \right )}}{6 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/x**7,x)

[Out]

4*b**6*erfi(b*x)/45 - 4*b**5*exp(b**2*x**2)/(45*sqrt(pi)*x) - 2*b**3*exp(b**2*x**2)/(45*sqrt(pi)*x**3) - b*exp
(b**2*x**2)/(15*sqrt(pi)*x**5) - erfi(b*x)/(6*x**6)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}\left (b x\right )}{x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/x^7,x, algorithm="giac")

[Out]

integrate(erfi(b*x)/x^7, x)