\(\int \frac {\text {erfi}(b x)}{x^6} \, dx\) [220]

Optimal result

Integrand size = 8, antiderivative size = 78 \[ \int \frac {\text {erfi}(b x)}{x^6} \, dx=-\frac {b e^{b^2 x^2}}{10 \sqrt {\pi } x^4}-\frac {b^3 e^{b^2 x^2}}{10 \sqrt {\pi } x^2}-\frac {\text {erfi}(b x)}{5 x^5}+\frac {b^5 \operatorname {ExpIntegralEi}\left (b^2 x^2\right )}{10 \sqrt {\pi }} \]

[Out]

-1/5*erfi(b*x)/x^5-1/10*b*exp(b^2*x^2)/x^4/Pi^(1/2)-1/10*b^3*exp(b^2*x^2)/x^2/Pi^(1/2)+1/10*b^5*Ei(b^2*x^2)/Pi
^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 78, 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, 2241} \[ \int \frac {\text {erfi}(b x)}{x^6} \, dx=-\frac {b e^{b^2 x^2}}{10 \sqrt {\pi } x^4}+\frac {b^5 \operatorname {ExpIntegralEi}\left (b^2 x^2\right )}{10 \sqrt {\pi }}-\frac {b^3 e^{b^2 x^2}}{10 \sqrt {\pi } x^2}-\frac {\text {erfi}(b x)}{5 x^5} \]

[In]

Int[Erfi[b*x]/x^6,x]

[Out]

-1/10*(b*E^(b^2*x^2))/(Sqrt[Pi]*x^4) - (b^3*E^(b^2*x^2))/(10*Sqrt[Pi]*x^2) - Erfi[b*x]/(5*x^5) + (b^5*ExpInteg
ralEi[b^2*x^2])/(10*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 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)}{5 x^5}+\frac {(2 b) \int \frac {e^{b^2 x^2}}{x^5} \, dx}{5 \sqrt {\pi }} \\ & = -\frac {b e^{b^2 x^2}}{10 \sqrt {\pi } x^4}-\frac {\text {erfi}(b x)}{5 x^5}+\frac {b^3 \int \frac {e^{b^2 x^2}}{x^3} \, dx}{5 \sqrt {\pi }} \\ & = -\frac {b e^{b^2 x^2}}{10 \sqrt {\pi } x^4}-\frac {b^3 e^{b^2 x^2}}{10 \sqrt {\pi } x^2}-\frac {\text {erfi}(b x)}{5 x^5}+\frac {b^5 \int \frac {e^{b^2 x^2}}{x} \, dx}{5 \sqrt {\pi }} \\ & = -\frac {b e^{b^2 x^2}}{10 \sqrt {\pi } x^4}-\frac {b^3 e^{b^2 x^2}}{10 \sqrt {\pi } x^2}-\frac {\text {erfi}(b x)}{5 x^5}+\frac {b^5 \operatorname {ExpIntegralEi}\left (b^2 x^2\right )}{10 \sqrt {\pi }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.78 \[ \int \frac {\text {erfi}(b x)}{x^6} \, dx=\frac {-b e^{b^2 x^2} x \left (1+b^2 x^2\right )-2 \sqrt {\pi } \text {erfi}(b x)+b^5 x^5 \operatorname {ExpIntegralEi}\left (b^2 x^2\right )}{10 \sqrt {\pi } x^5} \]

[In]

Integrate[Erfi[b*x]/x^6,x]

[Out]

(-(b*E^(b^2*x^2)*x*(1 + b^2*x^2)) - 2*Sqrt[Pi]*Erfi[b*x] + b^5*x^5*ExpIntegralEi[b^2*x^2])/(10*Sqrt[Pi]*x^5)

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.83

method	result	size
parts	\(-\frac {\operatorname {erfi}\left (b x \right )}{5 x^{5}}+\frac {2 b \left (-\frac {{\mathrm e}^{b^{2} x^{2}}}{4 x^{4}}+\frac {b^{2} \left (-\frac {{\mathrm e}^{b^{2} x^{2}}}{2 x^{2}}-\frac {b^{2} \operatorname {Ei}_{1}\left (-b^{2} x^{2}\right )}{2}\right )}{2}\right )}{5 \sqrt {\pi }}\)	\(65\)
derivativedivides	\(b^{5} \left (-\frac {\operatorname {erfi}\left (b x \right )}{5 b^{5} x^{5}}+\frac {-\frac {{\mathrm e}^{b^{2} x^{2}}}{10 b^{4} x^{4}}-\frac {{\mathrm e}^{b^{2} x^{2}}}{10 x^{2} b^{2}}-\frac {\operatorname {Ei}_{1}\left (-b^{2} x^{2}\right )}{10}}{\sqrt {\pi }}\right )\)	\(68\)
default	\(b^{5} \left (-\frac {\operatorname {erfi}\left (b x \right )}{5 b^{5} x^{5}}+\frac {-\frac {{\mathrm e}^{b^{2} x^{2}}}{10 b^{4} x^{4}}-\frac {{\mathrm e}^{b^{2} x^{2}}}{10 x^{2} b^{2}}-\frac {\operatorname {Ei}_{1}\left (-b^{2} x^{2}\right )}{10}}{\sqrt {\pi }}\right )\)	\(68\)
meijerg	\(\frac {b^{5} \left (\frac {399 b^{4} x^{4}+700 b^{2} x^{2}+1050}{1050 b^{4} x^{4}}-\frac {\left (21 b^{2} x^{2}+21\right ) {\mathrm e}^{b^{2} x^{2}}}{105 b^{4} x^{4}}-\frac {2 \sqrt {\pi }\, \operatorname {erfi}\left (b x \right )}{5 b^{5} x^{5}}-\frac {\ln \left (-b^{2} x^{2}\right )}{5}-\frac {\operatorname {Ei}_{1}\left (-b^{2} x^{2}\right )}{5}-\frac {19}{50}+\frac {2 \ln \left (x \right )}{5}+\frac {2 \ln \left (i b \right )}{5}-\frac {1}{b^{4} x^{4}}-\frac {2}{3 b^{2} x^{2}}\right )}{2 \sqrt {\pi }}\)	\(128\)

[In]

int(erfi(b*x)/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/5*erfi(b*x)/x^5+2/5/Pi^(1/2)*b*(-1/4*exp(b^2*x^2)/x^4+1/2*b^2*(-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 = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.74 \[ \int \frac {\text {erfi}(b x)}{x^6} \, dx=-\frac {2 \, \pi \operatorname {erfi}\left (b x\right ) - \sqrt {\pi } {\left (b^{5} x^{5} {\rm Ei}\left (b^{2} x^{2}\right ) - {\left (b^{3} x^{3} + b x\right )} e^{\left (b^{2} x^{2}\right )}\right )}}{10 \, \pi x^{5}} \]

[In]

integrate(erfi(b*x)/x^6,x, algorithm="fricas")

[Out]

-1/10*(2*pi*erfi(b*x) - sqrt(pi)*(b^5*x^5*Ei(b^2*x^2) - (b^3*x^3 + b*x)*e^(b^2*x^2)))/(pi*x^5)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.81 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.09 \[ \int \frac {\text {erfi}(b x)}{x^6} \, dx=- \frac {b^{5} \operatorname {E}_{1}\left (b^{2} x^{2} e^{i \pi }\right )}{10 \sqrt {\pi }} - \frac {b^{3} e^{b^{2} x^{2}}}{10 \sqrt {\pi } x^{2}} - \frac {b e^{b^{2} x^{2}}}{10 \sqrt {\pi } x^{4}} - \frac {i \operatorname {erfc}{\left (i b x \right )}}{5 x^{5}} + \frac {i}{5 x^{5}} \]

[In]

integrate(erfi(b*x)/x**6,x)

[Out]

-b**5*expint(1, b**2*x**2*exp_polar(I*pi))/(10*sqrt(pi)) - b**3*exp(b**2*x**2)/(10*sqrt(pi)*x**2) - b*exp(b**2
*x**2)/(10*sqrt(pi)*x**4) - I*erfc(I*b*x)/(5*x**5) + I/(5*x**5)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.36 \[ \int \frac {\text {erfi}(b x)}{x^6} \, dx=-\frac {b^{5} \Gamma \left (-2, -b^{2} x^{2}\right )}{5 \, \sqrt {\pi }} - \frac {\operatorname {erfi}\left (b x\right )}{5 \, x^{5}} \]

[In]

integrate(erfi(b*x)/x^6,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\text {erfi}(b x)}{x^6} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right )}{x^{6}} \,d x } \]

[In]

integrate(erfi(b*x)/x^6,x, algorithm="giac")

[Out]

integrate(erfi(b*x)/x^6, x)

Mupad [B] (verification not implemented)

Time = 4.94 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.79 \[ \int \frac {\text {erfi}(b x)}{x^6} \, dx=\frac {b^5\,\mathrm {ei}\left (b^2\,x^2\right )}{10\,\sqrt {\pi }}-\frac {\mathrm {erfi}\left (b\,x\right )}{5\,x^5}-\frac {\frac {b\,{\mathrm {e}}^{b^2\,x^2}}{2}+\frac {b^3\,x^2\,{\mathrm {e}}^{b^2\,x^2}}{2}}{5\,x^4\,\sqrt {\pi }} \]

[In]

int(erfi(b*x)/x^6,x)

[Out]

(b^5*ei(b^2*x^2))/(10*pi^(1/2)) - erfi(b*x)/(5*x^5) - ((b*exp(b^2*x^2))/2 + (b^3*x^2*exp(b^2*x^2))/2)/(5*x^4*p
i^(1/2))