\(\int \frac {\text {erf}(b x)}{x^4} \, dx\) [13]

Optimal result

Integrand size = 8, antiderivative size = 56 \[ \int \frac {\text {erf}(b x)}{x^4} \, dx=-\frac {b e^{-b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {\text {erf}(b x)}{3 x^3}-\frac {b^3 \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{3 \sqrt {\pi }} \]

[Out]

-1/3*erf(b*x)/x^3-1/3*b/exp(b^2*x^2)/x^2/Pi^(1/2)-1/3*b^3*Ei(-b^2*x^2)/Pi^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6496, 2245, 2241} \[ \int \frac {\text {erf}(b x)}{x^4} \, dx=-\frac {b e^{-b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {b^3 \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{3 \sqrt {\pi }}-\frac {\text {erf}(b x)}{3 x^3} \]

[In]

Int[Erf[b*x]/x^4,x]

[Out]

-1/3*b/(E^(b^2*x^2)*Sqrt[Pi]*x^2) - Erf[b*x]/(3*x^3) - (b^3*ExpIntegralEi[-(b^2*x^2)])/(3*Sqrt[Pi])

Rule 2241

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 2245

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 6496

Int[Erf[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Erf[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*} \text {integral}& = -\frac {\text {erf}(b x)}{3 x^3}+\frac {(2 b) \int \frac {e^{-b^2 x^2}}{x^3} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b e^{-b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {\text {erf}(b x)}{3 x^3}-\frac {\left (2 b^3\right ) \int \frac {e^{-b^2 x^2}}{x} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b e^{-b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {\text {erf}(b x)}{3 x^3}-\frac {b^3 \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{3 \sqrt {\pi }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.84 \[ \int \frac {\text {erf}(b x)}{x^4} \, dx=-\frac {\text {erf}(b x)+\frac {b x \left (e^{-b^2 x^2}+b^2 x^2 \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )\right )}{\sqrt {\pi }}}{3 x^3} \]

[In]

Integrate[Erf[b*x]/x^4,x]

[Out]

-1/3*(Erf[b*x] + (b*x*(E^(-(b^2*x^2)) + b^2*x^2*ExpIntegralEi[-(b^2*x^2)]))/Sqrt[Pi])/x^3

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82

method	result	size
parts	\(-\frac {\operatorname {erf}\left (b x \right )}{3 x^{3}}+\frac {2 b \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2 x^{2}}+\frac {b^{2} \operatorname {Ei}_{1}\left (b^{2} x^{2}\right )}{2}\right )}{3 \sqrt {\pi }}\)	\(46\)
derivativedivides	\(b^{3} \left (-\frac {\operatorname {erf}\left (b x \right )}{3 b^{3} x^{3}}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{3 x^{2} b^{2}}+\frac {\operatorname {Ei}_{1}\left (b^{2} x^{2}\right )}{3}}{\sqrt {\pi }}\right )\)	\(53\)
default	\(b^{3} \left (-\frac {\operatorname {erf}\left (b x \right )}{3 b^{3} x^{3}}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{3 x^{2} b^{2}}+\frac {\operatorname {Ei}_{1}\left (b^{2} x^{2}\right )}{3}}{\sqrt {\pi }}\right )\)	\(53\)
meijerg	\(\frac {b^{3} \left (\frac {b^{2} x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{2}\right ], \left [2, 3, \frac {7}{2}\right ], -b^{2} x^{2}\right )}{5}+\frac {10}{9}-\frac {2 \gamma }{3}-\frac {4 \ln \left (x \right )}{3}-\frac {4 \ln \left (b \right )}{3}-\frac {2}{b^{2} x^{2}}\right )}{2 \sqrt {\pi }}\)	\(55\)

[In]

int(erf(b*x)/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*erf(b*x)/x^3+2/3/Pi^(1/2)*b*(-1/2/x^2*exp(-b^2*x^2)+1/2*b^2*Ei(1,b^2*x^2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int \frac {\text {erf}(b x)}{x^4} \, dx=-\frac {\pi \operatorname {erf}\left (b x\right ) + \sqrt {\pi } {\left (b^{3} x^{3} {\rm Ei}\left (-b^{2} x^{2}\right ) + b x e^{\left (-b^{2} x^{2}\right )}\right )}}{3 \, \pi x^{3}} \]

[In]

integrate(erf(b*x)/x^4,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.96 \[ \int \frac {\text {erf}(b x)}{x^4} \, dx=\frac {b^{3} \operatorname {E}_{1}\left (b^{2} x^{2}\right )}{3 \sqrt {\pi }} - \frac {b e^{- b^{2} x^{2}}}{3 \sqrt {\pi } x^{2}} + \frac {\operatorname {erfc}{\left (b x \right )}}{3 x^{3}} - \frac {1}{3 x^{3}} \]

[In]

integrate(erf(b*x)/x**4,x)

[Out]

b**3*expint(1, b**2*x**2)/(3*sqrt(pi)) - b*exp(-b**2*x**2)/(3*sqrt(pi)*x**2) + erfc(b*x)/(3*x**3) - 1/(3*x**3)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.48 \[ \int \frac {\text {erf}(b x)}{x^4} \, dx=-\frac {b^{3} \Gamma \left (-1, b^{2} x^{2}\right )}{3 \, \sqrt {\pi }} - \frac {\operatorname {erf}\left (b x\right )}{3 \, x^{3}} \]

[In]

integrate(erf(b*x)/x^4,x, algorithm="maxima")

[Out]

-1/3*b^3*gamma(-1, b^2*x^2)/sqrt(pi) - 1/3*erf(b*x)/x^3

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int \frac {\text {erf}(b x)}{x^4} \, dx=-\frac {\operatorname {erf}\left (b x\right )}{3 \, x^{3}} - \frac {b^{6} x^{2} {\rm Ei}\left (-b^{2} x^{2}\right ) + b^{4} e^{\left (-b^{2} x^{2}\right )}}{3 \, \sqrt {\pi } b^{3} x^{2}} \]

[In]

integrate(erf(b*x)/x^4,x, algorithm="giac")

[Out]

-1/3*erf(b*x)/x^3 - 1/3*(b^6*x^2*Ei(-b^2*x^2) + b^4*e^(-b^2*x^2))/(sqrt(pi)*b^3*x^2)

Mupad [B] (verification not implemented)

Time = 5.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.80 \[ \int \frac {\text {erf}(b x)}{x^4} \, dx=-\frac {\mathrm {erf}\left (b\,x\right )}{3\,x^3}-\frac {b^3\,\mathrm {ei}\left (-b^2\,x^2\right )}{3\,\sqrt {\pi }}-\frac {b\,{\mathrm {e}}^{-b^2\,x^2}}{3\,x^2\,\sqrt {\pi }} \]

[In]

int(erf(b*x)/x^4,x)

[Out]

- erf(b*x)/(3*x^3) - (b^3*ei(-b^2*x^2))/(3*pi^(1/2)) - (b*exp(-b^2*x^2))/(3*x^2*pi^(1/2))