Integrand size = 10, antiderivative size = 123 \[ \int \frac {\text {erfi}(b x)^2}{x^5} \, dx=-\frac {b^2 e^{2 b^2 x^2}}{3 \pi x^2}-\frac {b e^{b^2 x^2} \text {erfi}(b x)}{3 \sqrt {\pi } x^3}-\frac {2 b^3 e^{b^2 x^2} \text {erfi}(b x)}{3 \sqrt {\pi } x}+\frac {1}{3} b^4 \text {erfi}(b x)^2-\frac {\text {erfi}(b x)^2}{4 x^4}+\frac {4 b^4 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{3 \pi } \]
-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*ex p(b^2*x^2)*erfi(b*x)/x/Pi^(1/2)
Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.79 \[ \int \frac {\text {erfi}(b x)^2}{x^5} \, dx=\frac {-4 b e^{b^2 x^2} \sqrt {\pi } x \left (1+2 b^2 x^2\right ) \text {erfi}(b x)+\pi \left (-3+4 b^4 x^4\right ) \text {erfi}(b x)^2-4 b^2 x^2 \left (e^{2 b^2 x^2}-4 b^2 x^2 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )\right )}{12 \pi x^4} \]
(-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)
Time = 0.71 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6920, 6947, 2643, 2639, 6947, 2639, 6929, 15}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\text {erfi}(b x)^2}{x^5} \, dx\) |
\(\Big \downarrow \) 6920 |
\(\displaystyle \frac {b \int \frac {e^{b^2 x^2} \text {erfi}(b x)}{x^4}dx}{\sqrt {\pi }}-\frac {\text {erfi}(b x)^2}{4 x^4}\) |
\(\Big \downarrow \) 6947 |
\(\displaystyle \frac {b \left (\frac {2}{3} b^2 \int \frac {e^{b^2 x^2} \text {erfi}(b x)}{x^2}dx+\frac {2 b \int \frac {e^{2 b^2 x^2}}{x^3}dx}{3 \sqrt {\pi }}-\frac {e^{b^2 x^2} \text {erfi}(b x)}{3 x^3}\right )}{\sqrt {\pi }}-\frac {\text {erfi}(b x)^2}{4 x^4}\) |
\(\Big \downarrow \) 2643 |
\(\displaystyle \frac {b \left (\frac {2}{3} b^2 \int \frac {e^{b^2 x^2} \text {erfi}(b x)}{x^2}dx+\frac {2 b \left (2 b^2 \int \frac {e^{2 b^2 x^2}}{x}dx-\frac {e^{2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}-\frac {e^{b^2 x^2} \text {erfi}(b x)}{3 x^3}\right )}{\sqrt {\pi }}-\frac {\text {erfi}(b x)^2}{4 x^4}\) |
\(\Big \downarrow \) 2639 |
\(\displaystyle \frac {b \left (\frac {2}{3} b^2 \int \frac {e^{b^2 x^2} \text {erfi}(b x)}{x^2}dx-\frac {e^{b^2 x^2} \text {erfi}(b x)}{3 x^3}+\frac {2 b \left (b^2 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )-\frac {e^{2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\right )}{\sqrt {\pi }}-\frac {\text {erfi}(b x)^2}{4 x^4}\) |
\(\Big \downarrow \) 6947 |
\(\displaystyle \frac {b \left (\frac {2}{3} b^2 \left (2 b^2 \int e^{b^2 x^2} \text {erfi}(b x)dx+\frac {2 b \int \frac {e^{2 b^2 x^2}}{x}dx}{\sqrt {\pi }}-\frac {e^{b^2 x^2} \text {erfi}(b x)}{x}\right )-\frac {e^{b^2 x^2} \text {erfi}(b x)}{3 x^3}+\frac {2 b \left (b^2 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )-\frac {e^{2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\right )}{\sqrt {\pi }}-\frac {\text {erfi}(b x)^2}{4 x^4}\) |
\(\Big \downarrow \) 2639 |
\(\displaystyle \frac {b \left (\frac {2}{3} b^2 \left (2 b^2 \int e^{b^2 x^2} \text {erfi}(b x)dx-\frac {e^{b^2 x^2} \text {erfi}(b x)}{x}+\frac {b \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{\sqrt {\pi }}\right )-\frac {e^{b^2 x^2} \text {erfi}(b x)}{3 x^3}+\frac {2 b \left (b^2 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )-\frac {e^{2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\right )}{\sqrt {\pi }}-\frac {\text {erfi}(b x)^2}{4 x^4}\) |
\(\Big \downarrow \) 6929 |
\(\displaystyle \frac {b \left (\frac {2}{3} b^2 \left (\sqrt {\pi } b \int \text {erfi}(b x)d\text {erfi}(b x)-\frac {e^{b^2 x^2} \text {erfi}(b x)}{x}+\frac {b \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{\sqrt {\pi }}\right )-\frac {e^{b^2 x^2} \text {erfi}(b x)}{3 x^3}+\frac {2 b \left (b^2 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )-\frac {e^{2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\right )}{\sqrt {\pi }}-\frac {\text {erfi}(b x)^2}{4 x^4}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {b \left (\frac {2}{3} b^2 \left (-\frac {e^{b^2 x^2} \text {erfi}(b x)}{x}+\frac {b \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{\sqrt {\pi }}+\frac {1}{2} \sqrt {\pi } b \text {erfi}(b x)^2\right )-\frac {e^{b^2 x^2} \text {erfi}(b x)}{3 x^3}+\frac {2 b \left (b^2 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )-\frac {e^{2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\right )}{\sqrt {\pi }}-\frac {\text {erfi}(b x)^2}{4 x^4}\) |
-1/4*Erfi[b*x]^2/x^4 + (b*(-1/3*(E^(b^2*x^2)*Erfi[b*x])/x^3 + (2*b*(-1/2*E ^(2*b^2*x^2)/x^2 + b^2*ExpIntegralEi[2*b^2*x^2]))/(3*Sqrt[Pi]) + (2*b^2*(- ((E^(b^2*x^2)*Erfi[b*x])/x) + (b*Sqrt[Pi]*Erfi[b*x]^2)/2 + (b*ExpIntegralE i[2*b^2*x^2])/Sqrt[Pi]))/3))/Sqrt[Pi]
3.3.33.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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] /; Free Q[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]
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] - Simp[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]))
Int[Erfi[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erfi[b*x]^2 /(m + 1)), x] - Simp[4*(b/(Sqrt[Pi]*(m + 1))) Int[x^(m + 1)*E^(b^2*x^2)*E rfi[b*x], x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])
Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[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]
Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^(m + 1)*E^(c + d*x^2)*(Erfi[a + b*x]/(m + 1)), x] + (-Simp[2*(d/( m + 1)) Int[x^(m + 2)*E^(c + d*x^2)*Erfi[a + b*x], x], x] - Simp[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]
\[\int \frac {\operatorname {erfi}\left (b x \right )^{2}}{x^{5}}d x\]
Time = 0.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.76 \[ \int \frac {\text {erfi}(b x)^2}{x^5} \, dx=\frac {16 \, b^{4} x^{4} {\rm Ei}\left (2 \, b^{2} x^{2}\right ) - 4 \, b^{2} x^{2} e^{\left (2 \, b^{2} x^{2}\right )} - 4 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} + b x\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erfi}\left (b x\right )^{2}}{12 \, \pi x^{4}} \]
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)
\[ \int \frac {\text {erfi}(b x)^2}{x^5} \, dx=\int \frac {\operatorname {erfi}^{2}{\left (b x \right )}}{x^{5}}\, dx \]
\[ \int \frac {\text {erfi}(b x)^2}{x^5} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right )^{2}}{x^{5}} \,d x } \]
\[ \int \frac {\text {erfi}(b x)^2}{x^5} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right )^{2}}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {\text {erfi}(b x)^2}{x^5} \, dx=\int \frac {{\mathrm {erfi}\left (b\,x\right )}^2}{x^5} \,d x \]