\(\int \frac {\text {erf}(b x)}{x^5} \, dx\) [6]

Optimal result

Integrand size = 8, antiderivative size = 71 \[ \int \frac {\text {erf}(b x)}{x^5} \, dx=-\frac {b e^{-b^2 x^2}}{6 \sqrt {\pi } x^3}+\frac {b^3 e^{-b^2 x^2}}{3 \sqrt {\pi } x}+\frac {1}{3} b^4 \text {erf}(b x)-\frac {\text {erf}(b x)}{4 x^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6496, 2245, 2236} \[ \int \frac {\text {erf}(b x)}{x^5} \, dx=\frac {1}{3} b^4 \text {erf}(b x)-\frac {b e^{-b^2 x^2}}{6 \sqrt {\pi } x^3}+\frac {b^3 e^{-b^2 x^2}}{3 \sqrt {\pi } x}-\frac {\text {erf}(b x)}{4 x^4} \]

[In]

Int[Erf[b*x]/x^5,x]

[Out]

-1/6*b/(E^(b^2*x^2)*Sqrt[Pi]*x^3) + b^3/(3*E^(b^2*x^2)*Sqrt[Pi]*x) + (b^4*Erf[b*x])/3 - Erf[b*x]/(4*x^4)

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

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)}{4 x^4}+\frac {b \int \frac {e^{-b^2 x^2}}{x^4} \, dx}{2 \sqrt {\pi }} \\ & = -\frac {b e^{-b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {\text {erf}(b x)}{4 x^4}-\frac {b^3 \int \frac {e^{-b^2 x^2}}{x^2} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b e^{-b^2 x^2}}{6 \sqrt {\pi } x^3}+\frac {b^3 e^{-b^2 x^2}}{3 \sqrt {\pi } x}-\frac {\text {erf}(b x)}{4 x^4}+\frac {\left (2 b^5\right ) \int e^{-b^2 x^2} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b e^{-b^2 x^2}}{6 \sqrt {\pi } x^3}+\frac {b^3 e^{-b^2 x^2}}{3 \sqrt {\pi } x}+\frac {1}{3} b^4 \text {erf}(b x)-\frac {\text {erf}(b x)}{4 x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89 \[ \int \frac {\text {erf}(b x)}{x^5} \, dx=e^{-b^2 x^2} \left (-\frac {b}{6 \sqrt {\pi } x^3}+\frac {b^3}{3 \sqrt {\pi } x}\right )+\frac {1}{3} b^4 \text {erf}(b x)-\frac {\text {erf}(b x)}{4 x^4} \]

[In]

Integrate[Erf[b*x]/x^5,x]

[Out]

(-1/6*b/(Sqrt[Pi]*x^3) + b^3/(3*Sqrt[Pi]*x))/E^(b^2*x^2) + (b^4*Erf[b*x])/3 - Erf[b*x]/(4*x^4)

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.87

method	result	size
meijerg	\(\frac {b^{4} \left (-\frac {4 \left (-\frac {b^{2} x^{2}}{2}+\frac {1}{4}\right ) {\mathrm e}^{-b^{2} x^{2}}}{3 x^{3} b^{3}}-\frac {\left (-4 b^{4} x^{4}+3\right ) \operatorname {erf}\left (b x \right ) \sqrt {\pi }}{6 x^{4} b^{4}}\right )}{2 \sqrt {\pi }}\)	\(62\)
parts	\(-\frac {\operatorname {erf}\left (b x \right )}{4 x^{4}}+\frac {b \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{3 x^{3}}-\frac {2 b^{2} \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{x}-b \sqrt {\pi }\, \operatorname {erf}\left (b x \right )\right )}{3}\right )}{2 \sqrt {\pi }}\)	\(62\)
parallelrisch	\(\frac {4 \,\operatorname {erf}\left (b x \right ) x^{4} b^{4} \sqrt {\pi }+4 x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}-2 \,{\mathrm e}^{-b^{2} x^{2}} b x -3 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{12 \sqrt {\pi }\, x^{4}}\)	\(64\)
derivativedivides	\(b^{4} \left (-\frac {\operatorname {erf}\left (b x \right )}{4 b^{4} x^{4}}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{3 b^{3} x^{3}}+\frac {2 \,{\mathrm e}^{-b^{2} x^{2}}}{3 b x}+\frac {2 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{3}}{2 \sqrt {\pi }}\right )\)	\(69\)
default	\(b^{4} \left (-\frac {\operatorname {erf}\left (b x \right )}{4 b^{4} x^{4}}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{3 b^{3} x^{3}}+\frac {2 \,{\mathrm e}^{-b^{2} x^{2}}}{3 b x}+\frac {2 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{3}}{2 \sqrt {\pi }}\right )\)	\(69\)

[In]

int(erf(b*x)/x^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.77 \[ \int \frac {\text {erf}(b x)}{x^5} \, dx=\frac {2 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} - b x\right )} e^{\left (-b^{2} x^{2}\right )} - {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right )}{12 \, \pi x^{4}} \]

[In]

integrate(erf(b*x)/x^5,x, algorithm="fricas")

[Out]

1/12*(2*sqrt(pi)*(2*b^3*x^3 - b*x)*e^(-b^2*x^2) - (3*pi - 4*pi*b^4*x^4)*erf(b*x))/(pi*x^4)

Sympy [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.85 \[ \int \frac {\text {erf}(b x)}{x^5} \, dx=\frac {b^{4} \operatorname {erf}{\left (b x \right )}}{3} + \frac {b^{3} e^{- b^{2} x^{2}}}{3 \sqrt {\pi } x} - \frac {b e^{- b^{2} x^{2}}}{6 \sqrt {\pi } x^{3}} - \frac {\operatorname {erf}{\left (b x \right )}}{4 x^{4}} \]

[In]

integrate(erf(b*x)/x**5,x)

[Out]

b**4*erf(b*x)/3 + b**3*exp(-b**2*x**2)/(3*sqrt(pi)*x) - b*exp(-b**2*x**2)/(6*sqrt(pi)*x**3) - erf(b*x)/(4*x**4
)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.49 \[ \int \frac {\text {erf}(b x)}{x^5} \, dx=-\frac {b^{4} {\left (x^{2}\right )}^{\frac {3}{2}} \Gamma \left (-\frac {3}{2}, b^{2} x^{2}\right )}{4 \, \sqrt {\pi } x^{3}} - \frac {\operatorname {erf}\left (b x\right )}{4 \, x^{4}} \]

[In]

integrate(erf(b*x)/x^5,x, algorithm="maxima")

[Out]

-1/4*b^4*(x^2)^(3/2)*gamma(-3/2, b^2*x^2)/(sqrt(pi)*x^3) - 1/4*erf(b*x)/x^4

Giac [F]

\[ \int \frac {\text {erf}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erf}\left (b x\right )}{x^{5}} \,d x } \]

[In]

integrate(erf(b*x)/x^5,x, algorithm="giac")

[Out]

integrate(erf(b*x)/x^5, x)

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.24 \[ \int \frac {\text {erf}(b x)}{x^5} \, dx=\frac {b\,{\left (b^2\,x^2\right )}^{3/2}}{3\,x^3}-\frac {\mathrm {erf}\left (b\,x\right )}{4\,x^4}+\frac {b^3\,{\mathrm {e}}^{-b^2\,x^2}}{3\,x\,\sqrt {\pi }}-\frac {b\,{\mathrm {e}}^{-b^2\,x^2}}{6\,x^3\,\sqrt {\pi }}-\frac {b\,\mathrm {erfc}\left (\sqrt {b^2\,x^2}\right )\,{\left (b^2\,x^2\right )}^{3/2}}{3\,x^3} \]

[In]

int(erf(b*x)/x^5,x)

[Out]

(b*(b^2*x^2)^(3/2))/(3*x^3) - erf(b*x)/(4*x^4) + (b^3*exp(-b^2*x^2))/(3*x*pi^(1/2)) - (b*exp(-b^2*x^2))/(6*x^3
*pi^(1/2)) - (b*erfc((b^2*x^2)^(1/2))*(b^2*x^2)^(3/2))/(3*x^3)