\(\int \frac {\text {erf}(b x)^2}{x^5} \, dx\) [27]

Optimal result

Integrand size = 10, antiderivative size = 125 \[ \int \frac {\text {erf}(b x)^2}{x^5} \, dx=-\frac {b^2 e^{-2 b^2 x^2}}{3 \pi x^2}-\frac {b e^{-b^2 x^2} \text {erf}(b x)}{3 \sqrt {\pi } x^3}+\frac {2 b^3 e^{-b^2 x^2} \text {erf}(b x)}{3 \sqrt {\pi } x}+\frac {1}{3} b^4 \text {erf}(b x)^2-\frac {\text {erf}(b x)^2}{4 x^4}-\frac {4 b^4 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{3 \pi } \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.78 \[ \int \frac {\text {erf}(b x)^2}{x^5} \, dx=\frac {\frac {4 b e^{-b^2 x^2} x \left (-1+2 b^2 x^2\right ) \text {erf}(b x)}{\sqrt {\pi }}+\left (-3+4 b^4 x^4\right ) \text {erf}(b x)^2-\frac {4 b^2 x^2 \left (e^{-2 b^2 x^2}+4 b^2 x^2 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )}{\pi }}{12 x^4} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 144, 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 = {6918, 6945, 2643, 2639, 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.

\(\displaystyle \int \frac {\text {erf}(b x)^2}{x^5} \, dx\)

\(\Big \downarrow \) 6918

\(\displaystyle \frac {b \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^4}dx}{\sqrt {\pi }}-\frac {\text {erf}(b x)^2}{4 x^4}\)

\(\Big \downarrow \) 6945

\(\displaystyle \frac {b \left (-\frac {2}{3} b^2 \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^2}dx+\frac {2 b \int \frac {e^{-2 b^2 x^2}}{x^3}dx}{3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}\right )}{\sqrt {\pi }}-\frac {\text {erf}(b x)^2}{4 x^4}\)

\(\Big \downarrow \) 2643

\(\displaystyle \frac {b \left (-\frac {2}{3} b^2 \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^2}dx+\frac {2 b \left (-2 b^2 \int \frac {e^{-2 b^2 x^2}}{x}dx-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}\right )}{\sqrt {\pi }}-\frac {\text {erf}(b x)^2}{4 x^4}\)

\(\Big \downarrow \) 2639

\(\displaystyle \frac {b \left (-\frac {2}{3} b^2 \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^2}dx-\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}+\frac {2 b \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\right )}{\sqrt {\pi }}-\frac {\text {erf}(b x)^2}{4 x^4}\)

\(\Big \downarrow \) 6945

\(\displaystyle \frac {b \left (-\frac {2}{3} b^2 \left (-2 b^2 \int e^{-b^2 x^2} \text {erf}(b x)dx+\frac {2 b \int \frac {e^{-2 b^2 x^2}}{x}dx}{\sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{x}\right )-\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}+\frac {2 b \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\right )}{\sqrt {\pi }}-\frac {\text {erf}(b x)^2}{4 x^4}\)

\(\Big \downarrow \) 2639

\(\displaystyle \frac {b \left (-\frac {2}{3} b^2 \left (-2 b^2 \int e^{-b^2 x^2} \text {erf}(b x)dx-\frac {e^{-b^2 x^2} \text {erf}(b x)}{x}+\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}\right )-\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}+\frac {2 b \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\right )}{\sqrt {\pi }}-\frac {\text {erf}(b x)^2}{4 x^4}\)

\(\Big \downarrow \) 6927

\(\displaystyle \frac {b \left (-\frac {2}{3} b^2 \left (-\sqrt {\pi } b \int \text {erf}(b x)d\text {erf}(b x)-\frac {e^{-b^2 x^2} \text {erf}(b x)}{x}+\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}\right )-\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}+\frac {2 b \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\right )}{\sqrt {\pi }}-\frac {\text {erf}(b x)^2}{4 x^4}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {b \left (-\frac {2}{3} b^2 \left (-\frac {e^{-b^2 x^2} \text {erf}(b x)}{x}+\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}-\frac {1}{2} \sqrt {\pi } b \text {erf}(b x)^2\right )-\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}+\frac {2 b \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\right )}{\sqrt {\pi }}-\frac {\text {erf}(b x)^2}{4 x^4}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 15

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

rule 2639

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]
 

rule 2643

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]))
 

rule 6918

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])
 

rule 6927

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]
 

rule 6945

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.75 \[ \int \frac {\text {erf}(b x)^2}{x^5} \, dx=-\frac {16 \, b^{4} x^{4} {\rm Ei}\left (-2 \, b^{2} x^{2}\right ) + 4 \, b^{2} x^{2} e^{\left (-2 \, b^{2} x^{2}\right )} - 4 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} - b x\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} + {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right )^{2}}{12 \, \pi x^{4}} \] Input:

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

Output:

-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)*erf(b*x)*e^(-b^2*x^2) + (3*pi - 4*pi*b^4*x^4)*erf(b*x)^2) 
/(pi*x^4)
 

Sympy [F]

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

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

Output:

Integral(erf(b*x)**2/x**5, x)
 

Maxima [F]

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

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

Output:

b*integrate(erf(b*x)*e^(-b^2*x^2)/x^4, x)/sqrt(pi) - 1/4*erf(b*x)^2/x^4
 

Giac [F]

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

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

Output:

integrate(erf(b*x)^2/x^5, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\text {erf}(b x)^2}{x^5} \, dx=\frac {-\mathrm {erf}\left (b x \right )^{2} \pi +4 \sqrt {\pi }\, \left (\int \frac {\mathrm {erf}\left (b x \right )}{e^{b^{2} x^{2}} x^{4}}d x \right ) b \,x^{4}}{4 \pi \,x^{4}} \] Input:

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

Output:

( - erf(b*x)**2*pi + 4*sqrt(pi)*int(erf(b*x)/(e**(b**2*x**2)*x**4),x)*b*x* 
*4)/(4*pi*x**4)