\(\int \frac {\text {erfi}(b x)}{x^7} \, dx\) [213]

Optimal result

Integrand size = 8, antiderivative size = 93 \[ \int \frac {\text {erfi}(b x)}{x^7} \, dx=-\frac {b e^{b^2 x^2}}{15 \sqrt {\pi } x^5}-\frac {2 b^3 e^{b^2 x^2}}{45 \sqrt {\pi } x^3}-\frac {4 b^5 e^{b^2 x^2}}{45 \sqrt {\pi } x}+\frac {4}{45} b^6 \text {erfi}(b x)-\frac {\text {erfi}(b x)}{6 x^6} \]

[Out]

4/45*b^6*erfi(b*x)-1/6*erfi(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 = 0.05 (sec) , antiderivative size = 93, 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 = {6498, 2245, 2235} \[ \int \frac {\text {erfi}(b x)}{x^7} \, dx=\frac {4}{45} b^6 \text {erfi}(b x)-\frac {b e^{b^2 x^2}}{15 \sqrt {\pi } x^5}-\frac {4 b^5 e^{b^2 x^2}}{45 \sqrt {\pi } x}-\frac {2 b^3 e^{b^2 x^2}}{45 \sqrt {\pi } x^3}-\frac {\text {erfi}(b x)}{6 x^6} \]

[In]

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

[Out]

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

Rule 2235

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]

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 6498

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*} \text {integral}& = -\frac {\text {erfi}(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 {erfi}(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 {erfi}(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 {erfi}(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 {erfi}(b x)-\frac {\text {erfi}(b x)}{6 x^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.69 \[ \int \frac {\text {erfi}(b x)}{x^7} \, dx=\frac {-2 b e^{b^2 x^2} x \left (3+2 b^2 x^2+4 b^4 x^4\right )+\sqrt {\pi } \left (-15+8 b^6 x^6\right ) \text {erfi}(b x)}{90 \sqrt {\pi } x^6} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.54 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.77

method	result	size
meijerg	\(\frac {i b^{6} \left (\frac {4 i \left (\frac {2}{9} b^{4} x^{4}+\frac {1}{9} b^{2} x^{2}+\frac {1}{6}\right ) {\mathrm e}^{b^{2} x^{2}}}{5 x^{5} b^{5}}+\frac {i \left (-8 b^{6} x^{6}+15\right ) \operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{45 x^{6} b^{6}}\right )}{2 \sqrt {\pi }}\)	\(72\)
parallelrisch	\(\frac {8 \,\operatorname {erfi}\left (b x \right ) x^{6} b^{6} \sqrt {\pi }-8 \,{\mathrm e}^{b^{2} x^{2}} b^{5} x^{5}-4 \,{\mathrm e}^{b^{2} x^{2}} b^{3} x^{3}-6 \,{\mathrm e}^{b^{2} x^{2}} b x -15 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{90 \sqrt {\pi }\, x^{6}}\)	\(78\)
derivativedivides	\(b^{6} \left (-\frac {\operatorname {erfi}\left (b x \right )}{6 b^{6} x^{6}}+\frac {-\frac {{\mathrm e}^{b^{2} x^{2}}}{5 b^{5} x^{5}}-\frac {2 \,{\mathrm e}^{b^{2} x^{2}}}{15 b^{3} x^{3}}-\frac {4 \,{\mathrm e}^{b^{2} x^{2}}}{15 b x}+\frac {4 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{15}}{3 \sqrt {\pi }}\right )\)	\(81\)
default	\(b^{6} \left (-\frac {\operatorname {erfi}\left (b x \right )}{6 b^{6} x^{6}}+\frac {-\frac {{\mathrm e}^{b^{2} x^{2}}}{5 b^{5} x^{5}}-\frac {2 \,{\mathrm e}^{b^{2} x^{2}}}{15 b^{3} x^{3}}-\frac {4 \,{\mathrm e}^{b^{2} x^{2}}}{15 b x}+\frac {4 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{15}}{3 \sqrt {\pi }}\right )\)	\(81\)
parts	\(-\frac {\operatorname {erfi}\left (b x \right )}{6 x^{6}}+\frac {b \left (-\frac {{\mathrm e}^{b^{2} x^{2}}}{5 x^{5}}+\frac {2 b^{2} \left (-\frac {{\mathrm e}^{b^{2} x^{2}}}{3 x^{3}}+\frac {2 b^{2} \left (-\frac {{\mathrm e}^{b^{2} x^{2}}}{x}-i b \sqrt {\pi }\, \operatorname {erf}\left (i b x \right )\right )}{3}\right )}{5}\right )}{3 \sqrt {\pi }}\)	\(82\)

[In]

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

[Out]

1/2*I/Pi^(1/2)*b^6*(4/5*I/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*I/x^6/b^6*(-8*b^6*x^6+15)*er
fi(b*x)*Pi^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.66 \[ \int \frac {\text {erfi}(b x)}{x^7} \, dx=-\frac {2 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} + 2 \, b^{3} x^{3} + 3 \, b x\right )} e^{\left (b^{2} x^{2}\right )} + {\left (15 \, \pi - 8 \, \pi b^{6} x^{6}\right )} \operatorname {erfi}\left (b x\right )}{90 \, \pi x^{6}} \]

[In]

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

Sympy [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.94 \[ \int \frac {\text {erfi}(b x)}{x^7} \, dx=\frac {4 b^{6} \operatorname {erfi}{\left (b x \right )}}{45} - \frac {4 b^{5} e^{b^{2} x^{2}}}{45 \sqrt {\pi } x} - \frac {2 b^{3} e^{b^{2} x^{2}}}{45 \sqrt {\pi } x^{3}} - \frac {b e^{b^{2} x^{2}}}{15 \sqrt {\pi } x^{5}} - \frac {\operatorname {erfi}{\left (b x \right )}}{6 x^{6}} \]

[In]

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

[Out]

4*b**6*erfi(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*exp
(b**2*x**2)/(15*sqrt(pi)*x**5) - erfi(b*x)/(6*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.42 \[ \int \frac {\text {erfi}(b x)}{x^7} \, dx=-\frac {\left (-b^{2} x^{2}\right )^{\frac {5}{2}} b \Gamma \left (-\frac {5}{2}, -b^{2} x^{2}\right )}{6 \, \sqrt {\pi } x^{5}} - \frac {\operatorname {erfi}\left (b x\right )}{6 \, x^{6}} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(erfi(b*x)/x^7, x)

Mupad [B] (verification not implemented)

Time = 4.85 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.16 \[ \int \frac {\text {erfi}(b x)}{x^7} \, dx=-\frac {\mathrm {erfi}\left (b\,x\right )}{6\,x^6}-\frac {3\,b\,{\mathrm {e}}^{b^2\,x^2}+2\,b^3\,x^2\,{\mathrm {e}}^{b^2\,x^2}+4\,b^5\,x^4\,{\mathrm {e}}^{b^2\,x^2}+4\,b\,\sqrt {\pi }\,{\left (-b^2\,x^2\right )}^{5/2}-4\,b\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-b^2}\,\sqrt {x^2}\right )\,{\left (-b^2\,x^2\right )}^{5/2}}{45\,x^5\,\sqrt {\pi }} \]

[In]

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

[Out]

- erfi(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*pi^(1/2)*(-b^2
*x^2)^(5/2) - 4*b*pi^(1/2)*erfc((-b^2)^(1/2)*(x^2)^(1/2))*(-b^2*x^2)^(5/2))/(45*x^5*pi^(1/2))