Integrand size = 10, antiderivative size = 67 \[ \int \frac {\text {erf}(b x)^2}{x^3} \, dx=-\frac {2 b e^{-b^2 x^2} \text {erf}(b x)}{\sqrt {\pi } x}-b^2 \text {erf}(b x)^2-\frac {\text {erf}(b x)^2}{2 x^2}+\frac {2 b^2 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\pi } \] Output:
-2*b*erf(b*x)/exp(b^2*x^2)/Pi^(1/2)/x-b^2*erf(b*x)^2-1/2*erf(b*x)^2/x^2+2* b^2*Ei(-2*b^2*x^2)/Pi
Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.94 \[ \int \frac {\text {erf}(b x)^2}{x^3} \, dx=-\frac {2 b e^{-b^2 x^2} \text {erf}(b x)}{\sqrt {\pi } x}+\left (-b^2-\frac {1}{2 x^2}\right ) \text {erf}(b x)^2+\frac {2 b^2 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\pi } \] Input:
Integrate[Erf[b*x]^2/x^3,x]
Output:
(-2*b*Erf[b*x])/(E^(b^2*x^2)*Sqrt[Pi]*x) + (-b^2 - 1/(2*x^2))*Erf[b*x]^2 + (2*b^2*ExpIntegralEi[-2*b^2*x^2])/Pi
Time = 0.42 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6918, 6945, 2639, 6927, 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.
\(\displaystyle \int \frac {\text {erf}(b x)^2}{x^3} \, dx\) |
\(\Big \downarrow \) 6918 |
\(\displaystyle \frac {2 b \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^2}dx}{\sqrt {\pi }}-\frac {\text {erf}(b x)^2}{2 x^2}\) |
\(\Big \downarrow \) 6945 |
\(\displaystyle \frac {2 b \left (-2 b^2 \int e^{-b^2 x^2} \text {erf}(b x)dx+\frac {2 b \int \frac {e^{-2 b^2 x^2}}{x}dx}{\sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{x}\right )}{\sqrt {\pi }}-\frac {\text {erf}(b x)^2}{2 x^2}\) |
\(\Big \downarrow \) 2639 |
\(\displaystyle \frac {2 b \left (-2 b^2 \int e^{-b^2 x^2} \text {erf}(b x)dx-\frac {e^{-b^2 x^2} \text {erf}(b x)}{x}+\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}\right )}{\sqrt {\pi }}-\frac {\text {erf}(b x)^2}{2 x^2}\) |
\(\Big \downarrow \) 6927 |
\(\displaystyle \frac {2 b \left (-\sqrt {\pi } b \int \text {erf}(b x)d\text {erf}(b x)-\frac {e^{-b^2 x^2} \text {erf}(b x)}{x}+\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}\right )}{\sqrt {\pi }}-\frac {\text {erf}(b x)^2}{2 x^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {2 b \left (-\frac {e^{-b^2 x^2} \text {erf}(b x)}{x}+\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}-\frac {1}{2} \sqrt {\pi } b \text {erf}(b x)^2\right )}{\sqrt {\pi }}-\frac {\text {erf}(b x)^2}{2 x^2}\) |
Input:
Int[Erf[b*x]^2/x^3,x]
Output:
-1/2*Erf[b*x]^2/x^2 + (2*b*(-(Erf[b*x]/(E^(b^2*x^2)*x)) - (b*Sqrt[Pi]*Erf[ b*x]^2)/2 + (b*ExpIntegralEi[-2*b^2*x^2])/Sqrt[Pi]))/Sqrt[Pi]
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[Erf[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erf[b*x]^2/( m + 1)), x] - Simp[4*(b/(Sqrt[Pi]*(m + 1))) Int[(x^(m + 1)*Erf[b*x])/E^(b ^2*x^2), x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])
Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[E^c*( Sqrt[Pi]/(2*b)) Subst[Int[x^n, x], x, Erf[b*x]], x] /; FreeQ[{b, c, d, n} , x] && EqQ[d, -b^2]
Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] : > Simp[x^(m + 1)*E^(c + d*x^2)*(Erf[a + b*x]/(m + 1)), x] + (-Simp[2*(d/(m + 1)) Int[x^(m + 2)*E^(c + d*x^2)*Erf[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]
\[\int \frac {\operatorname {erf}\left (b x \right )^{2}}{x^{3}}d x\]
Input:
int(erf(b*x)^2/x^3,x)
Output:
int(erf(b*x)^2/x^3,x)
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.97 \[ \int \frac {\text {erf}(b x)^2}{x^3} \, dx=\frac {4 \, b^{2} x^{2} {\rm Ei}\left (-2 \, b^{2} x^{2}\right ) - 4 \, \sqrt {\pi } b x \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - {\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right )^{2}}{2 \, \pi x^{2}} \] Input:
integrate(erf(b*x)^2/x^3,x, algorithm="fricas")
Output:
1/2*(4*b^2*x^2*Ei(-2*b^2*x^2) - 4*sqrt(pi)*b*x*erf(b*x)*e^(-b^2*x^2) - (pi + 2*pi*b^2*x^2)*erf(b*x)^2)/(pi*x^2)
\[ \int \frac {\text {erf}(b x)^2}{x^3} \, dx=\int \frac {\operatorname {erf}^{2}{\left (b x \right )}}{x^{3}}\, dx \] Input:
integrate(erf(b*x)**2/x**3,x)
Output:
Integral(erf(b*x)**2/x**3, x)
\[ \int \frac {\text {erf}(b x)^2}{x^3} \, dx=\int { \frac {\operatorname {erf}\left (b x\right )^{2}}{x^{3}} \,d x } \] Input:
integrate(erf(b*x)^2/x^3,x, algorithm="maxima")
Output:
2*b*integrate(erf(b*x)*e^(-b^2*x^2)/x^2, x)/sqrt(pi) - 1/2*erf(b*x)^2/x^2
\[ \int \frac {\text {erf}(b x)^2}{x^3} \, dx=\int { \frac {\operatorname {erf}\left (b x\right )^{2}}{x^{3}} \,d x } \] Input:
integrate(erf(b*x)^2/x^3,x, algorithm="giac")
Output:
integrate(erf(b*x)^2/x^3, x)
Timed out. \[ \int \frac {\text {erf}(b x)^2}{x^3} \, dx=\int \frac {{\mathrm {erf}\left (b\,x\right )}^2}{x^3} \,d x \] Input:
int(erf(b*x)^2/x^3,x)
Output:
int(erf(b*x)^2/x^3, x)
\[ \int \frac {\text {erf}(b x)^2}{x^3} \, dx=\frac {-\mathrm {erf}\left (b x \right )^{2} \pi +4 \sqrt {\pi }\, \left (\int \frac {\mathrm {erf}\left (b x \right )}{e^{b^{2} x^{2}} x^{2}}d x \right ) b \,x^{2}}{2 \pi \,x^{2}} \] Input:
int(erf(b*x)^2/x^3,x)
Output:
( - erf(b*x)**2*pi + 4*sqrt(pi)*int(erf(b*x)/(e**(b**2*x**2)*x**2),x)*b*x* *2)/(2*pi*x**2)