\(\int \frac {\text {erf}(b x)}{x^7} \, dx\) [7]

Optimal result

Integrand size = 8, antiderivative size = 96 \[ \int \frac {\text {erf}(b x)}{x^7} \, dx=-\frac {b e^{-b^2 x^2}}{15 \sqrt {\pi } x^5}+\frac {2 b^3 e^{-b^2 x^2}}{45 \sqrt {\pi } x^3}-\frac {4 b^5 e^{-b^2 x^2}}{45 \sqrt {\pi } x}-\frac {4}{45} b^6 \text {erf}(b x)-\frac {\text {erf}(b x)}{6 x^6} \]

[Out]

-4/45*b^6*erf(b*x)-1/6*erf(b*x)/x^6-1/15*b/exp(b^2*x^2)/x^5/Pi^(1/2)+2/45*b^3/exp(b^2*x^2)/x^3/Pi^(1/2)-4/45*b
^5/exp(b^2*x^2)/x/Pi^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6496, 2245, 2236} \[ \int \frac {\text {erf}(b x)}{x^7} \, dx=-\frac {4}{45} b^6 \text {erf}(b x)-\frac {b e^{-b^2 x^2}}{15 \sqrt {\pi } x^5}-\frac {4 b^5 e^{-b^2 x^2}}{45 \sqrt {\pi } x}+\frac {2 b^3 e^{-b^2 x^2}}{45 \sqrt {\pi } x^3}-\frac {\text {erf}(b x)}{6 x^6} \]

[In]

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

[Out]

-1/15*b/(E^(b^2*x^2)*Sqrt[Pi]*x^5) + (2*b^3)/(45*E^(b^2*x^2)*Sqrt[Pi]*x^3) - (4*b^5)/(45*E^(b^2*x^2)*Sqrt[Pi]*
x) - (4*b^6*Erf[b*x])/45 - Erf[b*x]/(6*x^6)

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)}{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 {erf}(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 {erf}(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 {erf}(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 {erf}(b x)-\frac {\text {erf}(b x)}{6 x^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.76 \[ \int \frac {\text {erf}(b x)}{x^7} \, dx=\frac {e^{-b^2 x^2} \left (-6 b x+4 b^3 x^3-8 b^5 x^5-e^{b^2 x^2} \sqrt {\pi } \left (15+8 b^6 x^6\right ) \text {erf}(b x)\right )}{90 \sqrt {\pi } x^6} \]

[In]

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

[Out]

(-6*b*x + 4*b^3*x^3 - 8*b^5*x^5 - E^(b^2*x^2)*Sqrt[Pi]*(15 + 8*b^6*x^6)*Erf[b*x])/(90*E^(b^2*x^2)*Sqrt[Pi]*x^6
)

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.73

method	result	size
meijerg	\(\frac {b^{6} \left (-\frac {4 \left (\frac {2}{9} b^{4} x^{4}-\frac {1}{9} b^{2} x^{2}+\frac {1}{6}\right ) {\mathrm e}^{-b^{2} x^{2}}}{5 x^{5} b^{5}}-\frac {\left (8 b^{6} x^{6}+15\right ) \operatorname {erf}\left (b x \right ) \sqrt {\pi }}{45 x^{6} b^{6}}\right )}{2 \sqrt {\pi }}\)	\(70\)
parallelrisch	\(-\frac {8 \,\operatorname {erf}\left (b x \right ) x^{6} b^{6} \sqrt {\pi }+8 \,{\mathrm e}^{-b^{2} x^{2}} x^{5} b^{5}-4 x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}+6 \,{\mathrm e}^{-b^{2} x^{2}} b x +15 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{90 \sqrt {\pi }\, x^{6}}\)	\(81\)
parts	\(-\frac {\operatorname {erf}\left (b x \right )}{6 x^{6}}+\frac {b \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{5 x^{5}}-\frac {2 b^{2} \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{3 x^{3}}-\frac {2 b^{2} \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{x}-b \sqrt {\pi }\, \operatorname {erf}\left (b x \right )\right )}{3}\right )}{5}\right )}{3 \sqrt {\pi }}\)	\(82\)
derivativedivides	\(b^{6} \left (-\frac {\operatorname {erf}\left (b x \right )}{6 b^{6} x^{6}}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{5 b^{5} x^{5}}+\frac {2 \,{\mathrm e}^{-b^{2} x^{2}}}{15 b^{3} x^{3}}-\frac {4 \,{\mathrm e}^{-b^{2} x^{2}}}{15 b x}-\frac {4 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{15}}{3 \sqrt {\pi }}\right )\)	\(87\)
default	\(b^{6} \left (-\frac {\operatorname {erf}\left (b x \right )}{6 b^{6} x^{6}}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{5 b^{5} x^{5}}+\frac {2 \,{\mathrm e}^{-b^{2} x^{2}}}{15 b^{3} x^{3}}-\frac {4 \,{\mathrm e}^{-b^{2} x^{2}}}{15 b x}-\frac {4 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{15}}{3 \sqrt {\pi }}\right )\)	\(87\)

[In]

int(erf(b*x)/x^7,x,method=_RETURNVERBOSE)

[Out]

1/2/Pi^(1/2)*b^6*(-4/5/x^5/b^5*(2/9*b^4*x^4-1/9*b^2*x^2+1/6)*exp(-b^2*x^2)-1/45/x^6/b^6*(8*b^6*x^6+15)*erf(b*x
)*Pi^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.65 \[ \int \frac {\text {erf}(b x)}{x^7} \, dx=-\frac {2 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} - 2 \, b^{3} x^{3} + 3 \, b x\right )} e^{\left (-b^{2} x^{2}\right )} + {\left (15 \, \pi + 8 \, \pi b^{6} x^{6}\right )} \operatorname {erf}\left (b x\right )}{90 \, \pi x^{6}} \]

[In]

integrate(erf(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)*erf(b*x))/(pi*x^6)

Sympy [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.91 \[ \int \frac {\text {erf}(b x)}{x^7} \, dx=- \frac {4 b^{6} \operatorname {erf}{\left (b x \right )}}{45} - \frac {4 b^{5} e^{- b^{2} x^{2}}}{45 \sqrt {\pi } x} + \frac {2 b^{3} e^{- b^{2} x^{2}}}{45 \sqrt {\pi } x^{3}} - \frac {b e^{- b^{2} x^{2}}}{15 \sqrt {\pi } x^{5}} - \frac {\operatorname {erf}{\left (b x \right )}}{6 x^{6}} \]

[In]

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

[Out]

-4*b**6*erf(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*e
xp(-b**2*x**2)/(15*sqrt(pi)*x**5) - erf(b*x)/(6*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.36 \[ \int \frac {\text {erf}(b x)}{x^7} \, dx=-\frac {b^{6} {\left (x^{2}\right )}^{\frac {5}{2}} \Gamma \left (-\frac {5}{2}, b^{2} x^{2}\right )}{6 \, \sqrt {\pi } x^{5}} - \frac {\operatorname {erf}\left (b x\right )}{6 \, x^{6}} \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\text {erf}(b x)}{x^7} \, dx=\int { \frac {\operatorname {erf}\left (b x\right )}{x^{7}} \,d x } \]

[In]

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

[Out]

integrate(erf(b*x)/x^7, x)

Mupad [B] (verification not implemented)

Time = 5.13 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.18 \[ \int \frac {\text {erf}(b x)}{x^7} \, dx=-\frac {\mathrm {erf}\left (b\,x\right )}{6\,x^6}-\frac {3\,b\,{\mathrm {e}}^{-b^2\,x^2}-2\,b^3\,x^2\,{\mathrm {e}}^{-b^2\,x^2}+4\,b^5\,x^4\,{\mathrm {e}}^{-b^2\,x^2}+4\,b^5\,\sqrt {\pi }\,\sqrt {b^2}\,{\left (x^2\right )}^{5/2}-4\,b^5\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {b^2}\,\sqrt {x^2}\right )\,\sqrt {b^2}\,{\left (x^2\right )}^{5/2}}{45\,x^5\,\sqrt {\pi }} \]

[In]

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

[Out]

- erf(b*x)/(6*x^6) - (3*b*exp(-b^2*x^2) - 2*b^3*x^2*exp(-b^2*x^2) + 4*b^5*x^4*exp(-b^2*x^2) + 4*b^5*pi^(1/2)*(
b^2)^(1/2)*(x^2)^(5/2) - 4*b^5*pi^(1/2)*erfc((b^2)^(1/2)*(x^2)^(1/2))*(b^2)^(1/2)*(x^2)^(5/2))/(45*x^5*pi^(1/2
))