\(\int \frac {\text {erfi}(b x)}{x^4} \, dx\) [219]

Optimal result

Integrand size = 8, antiderivative size = 54 \[ \int \frac {\text {erfi}(b x)}{x^4} \, dx=-\frac {b e^{b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {\text {erfi}(b x)}{3 x^3}+\frac {b^3 \operatorname {ExpIntegralEi}\left (b^2 x^2\right )}{3 \sqrt {\pi }} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6498, 2245, 2241} \[ \int \frac {\text {erfi}(b x)}{x^4} \, dx=-\frac {b e^{b^2 x^2}}{3 \sqrt {\pi } x^2}+\frac {b^3 \operatorname {ExpIntegralEi}\left (b^2 x^2\right )}{3 \sqrt {\pi }}-\frac {\text {erfi}(b x)}{3 x^3} \]

[In]

Int[Erfi[b*x]/x^4,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.93 \[ \int \frac {\text {erfi}(b x)}{x^4} \, dx=-\frac {\frac {b e^{b^2 x^2} x}{\sqrt {\pi }}+\text {erfi}(b x)-\frac {b^3 x^3 \operatorname {ExpIntegralEi}\left (b^2 x^2\right )}{\sqrt {\pi }}}{3 x^3} \]

[In]

Integrate[Erfi[b*x]/x^4,x]

[Out]

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

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85

method	result	size
parts	\(-\frac {\operatorname {erfi}\left (b x \right )}{3 x^{3}}+\frac {2 b \left (-\frac {{\mathrm e}^{b^{2} x^{2}}}{2 x^{2}}-\frac {b^{2} \operatorname {Ei}_{1}\left (-b^{2} x^{2}\right )}{2}\right )}{3 \sqrt {\pi }}\)	\(46\)
derivativedivides	\(b^{3} \left (-\frac {\operatorname {erfi}\left (b x \right )}{3 b^{3} x^{3}}+\frac {-\frac {{\mathrm e}^{b^{2} x^{2}}}{3 x^{2} b^{2}}-\frac {\operatorname {Ei}_{1}\left (-b^{2} x^{2}\right )}{3}}{\sqrt {\pi }}\right )\)	\(52\)
default	\(b^{3} \left (-\frac {\operatorname {erfi}\left (b x \right )}{3 b^{3} x^{3}}+\frac {-\frac {{\mathrm e}^{b^{2} x^{2}}}{3 x^{2} b^{2}}-\frac {\operatorname {Ei}_{1}\left (-b^{2} x^{2}\right )}{3}}{\sqrt {\pi }}\right )\)	\(52\)
meijerg	\(-\frac {b^{3} \left (-\frac {50 b^{2} x^{2}+90}{45 b^{2} x^{2}}+\frac {2 \,{\mathrm e}^{b^{2} x^{2}}}{3 x^{2} b^{2}}+\frac {2 \sqrt {\pi }\, \operatorname {erfi}\left (b x \right )}{3 b^{3} x^{3}}+\frac {2 \ln \left (-b^{2} x^{2}\right )}{3}+\frac {2 \,\operatorname {Ei}_{1}\left (-b^{2} x^{2}\right )}{3}+\frac {10}{9}-\frac {4 \ln \left (x \right )}{3}-\frac {4 \ln \left (i b \right )}{3}+\frac {2}{b^{2} x^{2}}\right )}{2 \sqrt {\pi }}\)	\(102\)

[In]

int(erfi(b*x)/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*erfi(b*x)/x^3+2/3/Pi^(1/2)*b*(-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 = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89 \[ \int \frac {\text {erfi}(b x)}{x^4} \, dx=-\frac {\pi \operatorname {erfi}\left (b x\right ) - \sqrt {\pi } {\left (b^{3} x^{3} {\rm Ei}\left (b^{2} x^{2}\right ) - b x e^{\left (b^{2} x^{2}\right )}\right )}}{3 \, \pi x^{3}} \]

[In]

integrate(erfi(b*x)/x^4,x, algorithm="fricas")

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.17 \[ \int \frac {\text {erfi}(b x)}{x^4} \, dx=- \frac {b^{3} \operatorname {E}_{1}\left (b^{2} x^{2} e^{i \pi }\right )}{3 \sqrt {\pi }} - \frac {b e^{b^{2} x^{2}}}{3 \sqrt {\pi } x^{2}} - \frac {i \operatorname {erfc}{\left (i b x \right )}}{3 x^{3}} + \frac {i}{3 x^{3}} \]

[In]

integrate(erfi(b*x)/x**4,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.52 \[ \int \frac {\text {erfi}(b x)}{x^4} \, dx=\frac {b^{3} \Gamma \left (-1, -b^{2} x^{2}\right )}{3 \, \sqrt {\pi }} - \frac {\operatorname {erfi}\left (b x\right )}{3 \, x^{3}} \]

[In]

integrate(erfi(b*x)/x^4,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\text {erfi}(b x)}{x^4} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right )}{x^{4}} \,d x } \]

[In]

integrate(erfi(b*x)/x^4,x, algorithm="giac")

[Out]

integrate(erfi(b*x)/x^4, x)

Mupad [B] (verification not implemented)

Time = 4.91 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80 \[ \int \frac {\text {erfi}(b x)}{x^4} \, dx=\frac {b^3\,\mathrm {ei}\left (b^2\,x^2\right )}{3\,\sqrt {\pi }}-\frac {\mathrm {erfi}\left (b\,x\right )}{3\,x^3}-\frac {b\,{\mathrm {e}}^{b^2\,x^2}}{3\,x^2\,\sqrt {\pi }} \]

[In]

int(erfi(b*x)/x^4,x)

[Out]

(b^3*ei(b^2*x^2))/(3*pi^(1/2)) - erfi(b*x)/(3*x^3) - (b*exp(b^2*x^2))/(3*x^2*pi^(1/2))