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

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

[Out]

-1/5*erfi(b*x)/x^5-1/10*b*exp(b^2*x^2)/x^4/Pi^(1/2)-1/10*b^3*exp(b^2*x^2)/x^2/Pi^(1/2)+1/10*b^5*Ei(b^2*x^2)/Pi
^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 78, normalized size of antiderivative = 1.00, 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 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]

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 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]

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 = 0.03, size = 61, normalized size = 0.78 \[ \frac {-b x e^{b^2 x^2} \left (b^2 x^2+1\right )+b^5 x^5 \text {Ei}\left (b^2 x^2\right )-2 \sqrt {\pi } \text {erfi}(b x)}{10 \sqrt {\pi } x^5} \]

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)

________________________________________________________________________________________

fricas [A]  time = 1.26, size = 58, normalized size = 0.74 \[ -\frac {2 \, \pi \operatorname {erfi}\left (b x\right ) - \sqrt {\pi } {\left (b^{5} x^{5} {\rm Ei}\left (b^{2} x^{2}\right ) - {\left (b^{3} x^{3} + b x\right )} e^{\left (b^{2} x^{2}\right )}\right )}}{10 \, \pi x^{5}} \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfi}\left (b x\right )}{x^{6}}\,{d x} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 68, normalized size = 0.87 \[ b^{5} \left (-\frac {\erfi \left (b x \right )}{5 b^{5} x^{5}}+\frac {-\frac {{\mathrm e}^{b^{2} x^{2}}}{10 b^{4} x^{4}}-\frac {{\mathrm e}^{b^{2} x^{2}}}{10 b^{2} x^{2}}-\frac {\Ei \left (1, -b^{2} x^{2}\right )}{10}}{\sqrt {\pi }}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^5*(-1/5*erfi(b*x)/b^5/x^5+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
)))

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 28, normalized size = 0.36 \[ -\frac {b^{5} \Gamma \left (-2, -b^{2} x^{2}\right )}{5 \, \sqrt {\pi }} - \frac {\operatorname {erfi}\left (b x\right )}{5 \, x^{5}} \]

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

________________________________________________________________________________________

mupad [B]  time = 0.22, size = 62, normalized size = 0.79 \[ \frac {b^5\,\mathrm {ei}\left (b^2\,x^2\right )}{10\,\sqrt {\pi }}-\frac {\mathrm {erfi}\left (b\,x\right )}{5\,x^5}-\frac {\frac {b\,{\mathrm {e}}^{b^2\,x^2}}{2}+\frac {b^3\,x^2\,{\mathrm {e}}^{b^2\,x^2}}{2}}{5\,x^4\,\sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^5*ei(b^2*x^2))/(10*pi^(1/2)) - erfi(b*x)/(5*x^5) - ((b*exp(b^2*x^2))/2 + (b^3*x^2*exp(b^2*x^2))/2)/(5*x^4*p
i^(1/2))

________________________________________________________________________________________

sympy [C]  time = 2.91, size = 85, normalized size = 1.09 \[ - \frac {b^{5} \operatorname {E}_{1}\left (b^{2} x^{2} e^{i \pi }\right )}{10 \sqrt {\pi }} - \frac {b^{3} e^{b^{2} x^{2}}}{10 \sqrt {\pi } x^{2}} - \frac {b e^{b^{2} x^{2}}}{10 \sqrt {\pi } x^{4}} - \frac {i \operatorname {erfc}{\left (i b x \right )}}{5 x^{5}} + \frac {i}{5 x^{5}} \]

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)

________________________________________________________________________________________