\(\int \frac {\text {erfi}(b x)}{x^5} \, dx\) [212]

Optimal result

Integrand size = 8, antiderivative size = 69 \[ \int \frac {\text {erfi}(b x)}{x^5} \, dx=-\frac {b e^{b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {b^3 e^{b^2 x^2}}{3 \sqrt {\pi } x}+\frac {1}{3} b^4 \text {erfi}(b x)-\frac {\text {erfi}(b x)}{4 x^4} \]

[Out]

1/3*b^4*erfi(b*x)-1/4*erfi(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 = 0.03 (sec) , antiderivative size = 69, 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 = {6498, 2245, 2235} \[ \int \frac {\text {erfi}(b x)}{x^5} \, dx=\frac {1}{3} b^4 \text {erfi}(b x)-\frac {b e^{b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {b^3 e^{b^2 x^2}}{3 \sqrt {\pi } x}-\frac {\text {erfi}(b x)}{4 x^4} \]

[In]

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

[Out]

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

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)}{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 {erfi}(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 {erfi}(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 {erfi}(b x)-\frac {\text {erfi}(b x)}{4 x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.74 \[ \int \frac {\text {erfi}(b x)}{x^5} \, dx=\frac {-\frac {2 b e^{b^2 x^2} x \left (1+2 b^2 x^2\right )}{\sqrt {\pi }}+\left (-3+4 b^4 x^4\right ) \text {erfi}(b x)}{12 x^4} \]

[In]

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

[Out]

((-2*b*E^(b^2*x^2)*x*(1 + 2*b^2*x^2))/Sqrt[Pi] + (-3 + 4*b^4*x^4)*Erfi[b*x])/(12*x^4)

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.90

method	result	size
parallelrisch	\(\frac {4 \,\operatorname {erfi}\left (b x \right ) x^{4} \sqrt {\pi }\, b^{4}-4 \,{\mathrm e}^{b^{2} x^{2}} b^{3} x^{3}-2 \,{\mathrm e}^{b^{2} x^{2}} b x -3 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{12 \sqrt {\pi }\, x^{4}}\)	\(62\)
parts	\(-\frac {\operatorname {erfi}\left (b x \right )}{4 x^{4}}+\frac {b \left (-\frac {{\mathrm e}^{b^{2} x^{2}}}{3 x^{3}}+\frac {2 b^{2} \left (-\frac {{\mathrm e}^{b^{2} x^{2}}}{x}-i b \sqrt {\pi }\, \operatorname {erf}\left (i b x \right )\right )}{3}\right )}{2 \sqrt {\pi }}\)	\(63\)
meijerg	\(-\frac {i b^{4} \left (-\frac {4 i \left (\frac {b^{2} x^{2}}{2}+\frac {1}{4}\right ) {\mathrm e}^{b^{2} x^{2}}}{3 x^{3} b^{3}}-\frac {i \left (-4 b^{4} x^{4}+3\right ) \operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{6 x^{4} b^{4}}\right )}{2 \sqrt {\pi }}\)	\(64\)
derivativedivides	\(b^{4} \left (-\frac {\operatorname {erfi}\left (b x \right )}{4 b^{4} x^{4}}+\frac {-\frac {{\mathrm e}^{b^{2} x^{2}}}{3 b^{3} x^{3}}-\frac {2 \,{\mathrm e}^{b^{2} x^{2}}}{3 b x}+\frac {2 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{3}}{2 \sqrt {\pi }}\right )\)	\(65\)
default	\(b^{4} \left (-\frac {\operatorname {erfi}\left (b x \right )}{4 b^{4} x^{4}}+\frac {-\frac {{\mathrm e}^{b^{2} x^{2}}}{3 b^{3} x^{3}}-\frac {2 \,{\mathrm e}^{b^{2} x^{2}}}{3 b x}+\frac {2 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{3}}{2 \sqrt {\pi }}\right )\)	\(65\)

[In]

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

[Out]

1/12*(4*erfi(b*x)*x^4*Pi^(1/2)*b^4-4*exp(b^2*x^2)*b^3*x^3-2*exp(b^2*x^2)*b*x-3*erfi(b*x)*Pi^(1/2))/Pi^(1/2)/x^
4

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.75 \[ \int \frac {\text {erfi}(b x)}{x^5} \, dx=-\frac {2 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} + b x\right )} e^{\left (b^{2} x^{2}\right )} + {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erfi}\left (b x\right )}{12 \, \pi x^{4}} \]

[In]

integrate(erfi(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)*erfi(b*x))/(pi*x^4)

Sympy [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.87 \[ \int \frac {\text {erfi}(b x)}{x^5} \, dx=\frac {b^{4} \operatorname {erfi}{\left (b x \right )}}{3} - \frac {b^{3} e^{b^{2} x^{2}}}{3 \sqrt {\pi } x} - \frac {b e^{b^{2} x^{2}}}{6 \sqrt {\pi } x^{3}} - \frac {\operatorname {erfi}{\left (b x \right )}}{4 x^{4}} \]

[In]

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

[Out]

b**4*erfi(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) - erfi(b*x)/(4*x**4
)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.57 \[ \int \frac {\text {erfi}(b x)}{x^5} \, dx=-\frac {\left (-b^{2} x^{2}\right )^{\frac {3}{2}} b \Gamma \left (-\frac {3}{2}, -b^{2} x^{2}\right )}{4 \, \sqrt {\pi } x^{3}} - \frac {\operatorname {erfi}\left (b x\right )}{4 \, x^{4}} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(erfi(b*x)/x^5, x)

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.29 \[ \int \frac {\text {erfi}(b x)}{x^5} \, dx=\frac {b\,{\left (-b^2\,x^2\right )}^{3/2}}{3\,x^3}-\frac {\mathrm {erfi}\left (b\,x\right )}{4\,x^4}-\frac {b^3\,{\mathrm {e}}^{b^2\,x^2}}{3\,x\,\sqrt {\pi }}-\frac {b\,{\mathrm {e}}^{b^2\,x^2}}{6\,x^3\,\sqrt {\pi }}-\frac {b\,\mathrm {erfc}\left (\sqrt {-b^2\,x^2}\right )\,{\left (-b^2\,x^2\right )}^{3/2}}{3\,x^3} \]

[In]

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

[Out]

(b*(-b^2*x^2)^(3/2))/(3*x^3) - erfi(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)