3.211 \(\int \frac{\text{Erfi}(b x)}{x^3} \, dx\)

Optimal. Leaf size=40 \[ b^2 \text{Erfi}(b x)-\frac{b e^{b^2 x^2}}{\sqrt{\pi } x}-\frac{\text{Erfi}(b x)}{2 x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0356932, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6363, 2214, 2204} \[ b^2 \text{Erfi}(b x)-\frac{b e^{b^2 x^2}}{\sqrt{\pi } x}-\frac{\text{Erfi}(b x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[Erfi[b*x]/x^3,x]

[Out]

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

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

Mathematica [A]  time = 0.0224858, size = 37, normalized size = 0.92 \[ \left (b^2-\frac{1}{2 x^2}\right ) \text{Erfi}(b x)-\frac{b e^{b^2 x^2}}{\sqrt{\pi } x} \]

Antiderivative was successfully verified.

[In]

Integrate[Erfi[b*x]/x^3,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.042, size = 47, normalized size = 1.2 \begin{align*}{b}^{2} \left ( -{\frac{{\it erfi} \left ( bx \right ) }{2\,{b}^{2}{x}^{2}}}+{\frac{1}{\sqrt{\pi }} \left ( -{\frac{{{\rm e}^{{b}^{2}{x}^{2}}}}{bx}}+\sqrt{\pi }{\it erfi} \left ( bx \right ) \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(b*x)/x^3,x)

[Out]

b^2*(-1/2/b^2/x^2*erfi(b*x)+1/Pi^(1/2)*(-exp(b^2*x^2)/b/x+Pi^(1/2)*erfi(b*x)))

________________________________________________________________________________________

Maxima [A]  time = 1.08743, size = 53, normalized size = 1.32 \begin{align*} -\frac{\sqrt{-b^{2} x^{2}} b \Gamma \left (-\frac{1}{2}, -b^{2} x^{2}\right )}{2 \, \sqrt{\pi } x} - \frac{\operatorname{erfi}\left (b x\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/x^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.24377, size = 103, normalized size = 2.58 \begin{align*} -\frac{2 \, \sqrt{\pi } b x e^{\left (b^{2} x^{2}\right )} +{\left (\pi - 2 \, \pi b^{2} x^{2}\right )} \operatorname{erfi}\left (b x\right )}{2 \, \pi x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/x^3,x, algorithm="fricas")

[Out]

-1/2*(2*sqrt(pi)*b*x*e^(b^2*x^2) + (pi - 2*pi*b^2*x^2)*erfi(b*x))/(pi*x^2)

________________________________________________________________________________________

Sympy [A]  time = 0.557861, size = 34, normalized size = 0.85 \begin{align*} b^{2} \operatorname{erfi}{\left (b x \right )} - \frac{b e^{b^{2} x^{2}}}{\sqrt{\pi } x} - \frac{\operatorname{erfi}{\left (b x \right )}}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/x**3,x)

[Out]

b**2*erfi(b*x) - b*exp(b**2*x**2)/(sqrt(pi)*x) - erfi(b*x)/(2*x**2)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}\left (b x\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/x^3,x, algorithm="giac")

[Out]

integrate(erfi(b*x)/x^3, x)