\(\int \frac {\text {erfi}(b x)^2}{x^5} \, dx\) [233]

Optimal result

Integrand size = 10, antiderivative size = 123 \[ \int \frac {\text {erfi}(b x)^2}{x^5} \, dx=-\frac {b^2 e^{2 b^2 x^2}}{3 \pi x^2}-\frac {b e^{b^2 x^2} \text {erfi}(b x)}{3 \sqrt {\pi } x^3}-\frac {2 b^3 e^{b^2 x^2} \text {erfi}(b x)}{3 \sqrt {\pi } x}+\frac {1}{3} b^4 \text {erfi}(b x)^2-\frac {\text {erfi}(b x)^2}{4 x^4}+\frac {4 b^4 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{3 \pi } \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6501, 6528, 6510, 30, 2241, 2245} \[ \int \frac {\text {erfi}(b x)^2}{x^5} \, dx=\frac {1}{3} b^4 \text {erfi}(b x)^2-\frac {b e^{b^2 x^2} \text {erfi}(b x)}{3 \sqrt {\pi } x^3}-\frac {b^2 e^{2 b^2 x^2}}{3 \pi x^2}+\frac {4 b^4 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{3 \pi }-\frac {2 b^3 e^{b^2 x^2} \text {erfi}(b x)}{3 \sqrt {\pi } x}-\frac {\text {erfi}(b x)^2}{4 x^4} \]

[In]

Int[Erfi[b*x]^2/x^5,x]

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 6501

Int[Erfi[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erfi[b*x]^2/(m + 1)), x] - Dist[4*(b/(Sqrt[Pi]
*(m + 1))), Int[x^(m + 1)*E^(b^2*x^2)*Erfi[b*x], x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])

Rule 6510

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[E^c*(Sqrt[Pi]/(2*b)), Subst[Int[x^n, x]
, x, Erfi[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, b^2]

Rule 6528

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^(m + 1)*E^(c + d*x^2)*(Er
fi[a + b*x]/(m + 1)), x] + (-Dist[2*(d/(m + 1)), Int[x^(m + 2)*E^(c + d*x^2)*Erfi[a + b*x], x], x] - Dist[2*(b
/((m + 1)*Sqrt[Pi])), Int[x^(m + 1)*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x] &
& ILtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {erfi}(b x)^2}{4 x^4}+\frac {b \int \frac {e^{b^2 x^2} \text {erfi}(b x)}{x^4} \, dx}{\sqrt {\pi }} \\ & = -\frac {b e^{b^2 x^2} \text {erfi}(b x)}{3 \sqrt {\pi } x^3}-\frac {\text {erfi}(b x)^2}{4 x^4}+\frac {\left (2 b^2\right ) \int \frac {e^{2 b^2 x^2}}{x^3} \, dx}{3 \pi }+\frac {\left (2 b^3\right ) \int \frac {e^{b^2 x^2} \text {erfi}(b x)}{x^2} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b^2 e^{2 b^2 x^2}}{3 \pi x^2}-\frac {b e^{b^2 x^2} \text {erfi}(b x)}{3 \sqrt {\pi } x^3}-\frac {2 b^3 e^{b^2 x^2} \text {erfi}(b x)}{3 \sqrt {\pi } x}-\frac {\text {erfi}(b x)^2}{4 x^4}+2 \frac {\left (4 b^4\right ) \int \frac {e^{2 b^2 x^2}}{x} \, dx}{3 \pi }+\frac {\left (4 b^5\right ) \int e^{b^2 x^2} \text {erfi}(b x) \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b^2 e^{2 b^2 x^2}}{3 \pi x^2}-\frac {b e^{b^2 x^2} \text {erfi}(b x)}{3 \sqrt {\pi } x^3}-\frac {2 b^3 e^{b^2 x^2} \text {erfi}(b x)}{3 \sqrt {\pi } x}-\frac {\text {erfi}(b x)^2}{4 x^4}+\frac {4 b^4 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{3 \pi }+\frac {1}{3} \left (2 b^4\right ) \text {Subst}(\int x \, dx,x,\text {erfi}(b x)) \\ & = -\frac {b^2 e^{2 b^2 x^2}}{3 \pi x^2}-\frac {b e^{b^2 x^2} \text {erfi}(b x)}{3 \sqrt {\pi } x^3}-\frac {2 b^3 e^{b^2 x^2} \text {erfi}(b x)}{3 \sqrt {\pi } x}+\frac {1}{3} b^4 \text {erfi}(b x)^2-\frac {\text {erfi}(b x)^2}{4 x^4}+\frac {4 b^4 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{3 \pi } \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.79 \[ \int \frac {\text {erfi}(b x)^2}{x^5} \, dx=\frac {-4 b e^{b^2 x^2} \sqrt {\pi } x \left (1+2 b^2 x^2\right ) \text {erfi}(b x)+\pi \left (-3+4 b^4 x^4\right ) \text {erfi}(b x)^2-4 b^2 x^2 \left (e^{2 b^2 x^2}-4 b^2 x^2 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )\right )}{12 \pi x^4} \]

[In]

Integrate[Erfi[b*x]^2/x^5,x]

[Out]

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

Maple [F]

\[\int \frac {\operatorname {erfi}\left (b x \right )^{2}}{x^{5}}d x\]

[In]

int(erfi(b*x)^2/x^5,x)

[Out]

int(erfi(b*x)^2/x^5,x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.76 \[ \int \frac {\text {erfi}(b x)^2}{x^5} \, dx=\frac {16 \, b^{4} x^{4} {\rm Ei}\left (2 \, b^{2} x^{2}\right ) - 4 \, b^{2} x^{2} e^{\left (2 \, b^{2} x^{2}\right )} - 4 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} + b x\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erfi}\left (b x\right )^{2}}{12 \, \pi x^{4}} \]

[In]

integrate(erfi(b*x)^2/x^5,x, algorithm="fricas")

[Out]

1/12*(16*b^4*x^4*Ei(2*b^2*x^2) - 4*b^2*x^2*e^(2*b^2*x^2) - 4*sqrt(pi)*(2*b^3*x^3 + b*x)*erfi(b*x)*e^(b^2*x^2)
- (3*pi - 4*pi*b^4*x^4)*erfi(b*x)^2)/(pi*x^4)

Sympy [F]

\[ \int \frac {\text {erfi}(b x)^2}{x^5} \, dx=\int \frac {\operatorname {erfi}^{2}{\left (b x \right )}}{x^{5}}\, dx \]

[In]

integrate(erfi(b*x)**2/x**5,x)

[Out]

Integral(erfi(b*x)**2/x**5, x)

Maxima [F]

\[ \int \frac {\text {erfi}(b x)^2}{x^5} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right )^{2}}{x^{5}} \,d x } \]

[In]

integrate(erfi(b*x)^2/x^5,x, algorithm="maxima")

[Out]

integrate(erfi(b*x)^2/x^5, x)

Giac [F]

\[ \int \frac {\text {erfi}(b x)^2}{x^5} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right )^{2}}{x^{5}} \,d x } \]

[In]

integrate(erfi(b*x)^2/x^5,x, algorithm="giac")

[Out]

integrate(erfi(b*x)^2/x^5, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {erfi}(b x)^2}{x^5} \, dx=\int \frac {{\mathrm {erfi}\left (b\,x\right )}^2}{x^5} \,d x \]

[In]

int(erfi(b*x)^2/x^5,x)

[Out]

int(erfi(b*x)^2/x^5, x)