3.1.28 \(\int \frac {\text {erf}(b x)^2}{x^7} \, dx\) [28]

3.1.28.1 Optimal result

Integrand size = 10, antiderivative size = 177 \[ \int \frac {\text {erf}(b x)^2}{x^7} \, dx=-\frac {b^2 e^{-2 b^2 x^2}}{15 \pi x^4}+\frac {2 b^4 e^{-2 b^2 x^2}}{9 \pi x^2}-\frac {2 b e^{-b^2 x^2} \text {erf}(b x)}{15 \sqrt {\pi } x^5}+\frac {4 b^3 e^{-b^2 x^2} \text {erf}(b x)}{45 \sqrt {\pi } x^3}-\frac {8 b^5 e^{-b^2 x^2} \text {erf}(b x)}{45 \sqrt {\pi } x}-\frac {4}{45} b^6 \text {erf}(b x)^2-\frac {\text {erf}(b x)^2}{6 x^6}+\frac {28 b^6 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{45 \pi } \]

output

-1/15*b^2/exp(2*b^2*x^2)/Pi/x^4+2/9*b^4/exp(2*b^2*x^2)/Pi/x^2+28/45*b^6*Ei 
(-2*b^2*x^2)/Pi-4/45*b^6*erf(b*x)^2-1/6*erf(b*x)^2/x^6-2/15*b*erf(b*x)/exp 
(b^2*x^2)/x^5/Pi^(1/2)+4/45*b^3*erf(b*x)/exp(b^2*x^2)/x^3/Pi^(1/2)-8/45*b^ 
5*erf(b*x)/exp(b^2*x^2)/x/Pi^(1/2)
 

3.1.28.2 Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.75 \[ \int \frac {\text {erf}(b x)^2}{x^7} \, dx=\frac {e^{-2 b^2 x^2} \left (-6 b^2 x^2+20 b^4 x^4-4 b e^{b^2 x^2} \sqrt {\pi } x \left (3-2 b^2 x^2+4 b^4 x^4\right ) \text {erf}(b x)-e^{2 b^2 x^2} \pi \left (15+8 b^6 x^6\right ) \text {erf}(b x)^2+56 b^6 e^{2 b^2 x^2} x^6 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )}{90 \pi x^6} \]

input

Integrate[Erf[b*x]^2/x^7,x]
 

output

(-6*b^2*x^2 + 20*b^4*x^4 - 4*b*E^(b^2*x^2)*Sqrt[Pi]*x*(3 - 2*b^2*x^2 + 4*b 
^4*x^4)*Erf[b*x] - E^(2*b^2*x^2)*Pi*(15 + 8*b^6*x^6)*Erf[b*x]^2 + 56*b^6*E 
^(2*b^2*x^2)*x^6*ExpIntegralEi[-2*b^2*x^2])/(90*E^(2*b^2*x^2)*Pi*x^6)
 

3.1.28.3 Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.36, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {6918, 6945, 2643, 2643, 2639, 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.

\(\displaystyle \int \frac {\text {erf}(b x)^2}{x^7} \, dx\)

\(\Big \downarrow \) 6918

\(\displaystyle \frac {2 b \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^6}dx}{3 \sqrt {\pi }}-\frac {\text {erf}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 6945

\(\displaystyle \frac {2 b \left (-\frac {2}{5} b^2 \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^4}dx+\frac {2 b \int \frac {e^{-2 b^2 x^2}}{x^5}dx}{5 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{5 x^5}\right )}{3 \sqrt {\pi }}-\frac {\text {erf}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 2643

\(\displaystyle \frac {2 b \left (-\frac {2}{5} b^2 \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^4}dx+\frac {2 b \left (b^2 \left (-\int \frac {e^{-2 b^2 x^2}}{x^3}dx\right )-\frac {e^{-2 b^2 x^2}}{4 x^4}\right )}{5 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{5 x^5}\right )}{3 \sqrt {\pi }}-\frac {\text {erf}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 2643

\(\displaystyle \frac {2 b \left (-\frac {2}{5} b^2 \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^4}dx+\frac {2 b \left (-\left (b^2 \left (-2 b^2 \int \frac {e^{-2 b^2 x^2}}{x}dx-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )\right )-\frac {e^{-2 b^2 x^2}}{4 x^4}\right )}{5 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{5 x^5}\right )}{3 \sqrt {\pi }}-\frac {\text {erf}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 2639

\(\displaystyle \frac {2 b \left (-\frac {2}{5} b^2 \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^4}dx-\frac {e^{-b^2 x^2} \text {erf}(b x)}{5 x^5}+\frac {2 b \left (-\left (b^2 \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )\right )-\frac {e^{-2 b^2 x^2}}{4 x^4}\right )}{5 \sqrt {\pi }}\right )}{3 \sqrt {\pi }}-\frac {\text {erf}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 6945

\(\displaystyle \frac {2 b \left (-\frac {2}{5} b^2 \left (-\frac {2}{3} b^2 \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^2}dx+\frac {2 b \int \frac {e^{-2 b^2 x^2}}{x^3}dx}{3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}\right )-\frac {e^{-b^2 x^2} \text {erf}(b x)}{5 x^5}+\frac {2 b \left (-\left (b^2 \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )\right )-\frac {e^{-2 b^2 x^2}}{4 x^4}\right )}{5 \sqrt {\pi }}\right )}{3 \sqrt {\pi }}-\frac {\text {erf}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 2643

\(\displaystyle \frac {2 b \left (-\frac {2}{5} b^2 \left (-\frac {2}{3} b^2 \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^2}dx+\frac {2 b \left (-2 b^2 \int \frac {e^{-2 b^2 x^2}}{x}dx-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}\right )-\frac {e^{-b^2 x^2} \text {erf}(b x)}{5 x^5}+\frac {2 b \left (-\left (b^2 \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )\right )-\frac {e^{-2 b^2 x^2}}{4 x^4}\right )}{5 \sqrt {\pi }}\right )}{3 \sqrt {\pi }}-\frac {\text {erf}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 2639

\(\displaystyle \frac {2 b \left (-\frac {2}{5} b^2 \left (-\frac {2}{3} b^2 \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^2}dx-\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}+\frac {2 b \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\right )-\frac {e^{-b^2 x^2} \text {erf}(b x)}{5 x^5}+\frac {2 b \left (-\left (b^2 \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )\right )-\frac {e^{-2 b^2 x^2}}{4 x^4}\right )}{5 \sqrt {\pi }}\right )}{3 \sqrt {\pi }}-\frac {\text {erf}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 6945

\(\displaystyle \frac {2 b \left (-\frac {2}{5} b^2 \left (-\frac {2}{3} b^2 \left (-2 b^2 \int e^{-b^2 x^2} \text {erf}(b x)dx+\frac {2 b \int \frac {e^{-2 b^2 x^2}}{x}dx}{\sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{x}\right )-\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}+\frac {2 b \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\right )-\frac {e^{-b^2 x^2} \text {erf}(b x)}{5 x^5}+\frac {2 b \left (-\left (b^2 \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )\right )-\frac {e^{-2 b^2 x^2}}{4 x^4}\right )}{5 \sqrt {\pi }}\right )}{3 \sqrt {\pi }}-\frac {\text {erf}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 2639

\(\displaystyle \frac {2 b \left (-\frac {2}{5} b^2 \left (-\frac {2}{3} b^2 \left (-2 b^2 \int e^{-b^2 x^2} \text {erf}(b x)dx-\frac {e^{-b^2 x^2} \text {erf}(b x)}{x}+\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}\right )-\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}+\frac {2 b \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\right )-\frac {e^{-b^2 x^2} \text {erf}(b x)}{5 x^5}+\frac {2 b \left (-\left (b^2 \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )\right )-\frac {e^{-2 b^2 x^2}}{4 x^4}\right )}{5 \sqrt {\pi }}\right )}{3 \sqrt {\pi }}-\frac {\text {erf}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 6927

\(\displaystyle \frac {2 b \left (-\frac {2}{5} b^2 \left (-\frac {2}{3} b^2 \left (-\sqrt {\pi } b \int \text {erf}(b x)d\text {erf}(b x)-\frac {e^{-b^2 x^2} \text {erf}(b x)}{x}+\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}\right )-\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}+\frac {2 b \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\right )-\frac {e^{-b^2 x^2} \text {erf}(b x)}{5 x^5}+\frac {2 b \left (-\left (b^2 \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )\right )-\frac {e^{-2 b^2 x^2}}{4 x^4}\right )}{5 \sqrt {\pi }}\right )}{3 \sqrt {\pi }}-\frac {\text {erf}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {2 b \left (-\frac {2}{5} b^2 \left (-\frac {2}{3} b^2 \left (-\frac {e^{-b^2 x^2} \text {erf}(b x)}{x}+\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}-\frac {1}{2} \sqrt {\pi } b \text {erf}(b x)^2\right )-\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}+\frac {2 b \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\right )-\frac {e^{-b^2 x^2} \text {erf}(b x)}{5 x^5}+\frac {2 b \left (-\left (b^2 \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )\right )-\frac {e^{-2 b^2 x^2}}{4 x^4}\right )}{5 \sqrt {\pi }}\right )}{3 \sqrt {\pi }}-\frac {\text {erf}(b x)^2}{6 x^6}\)

input

Int[Erf[b*x]^2/x^7,x]
 

output

-1/6*Erf[b*x]^2/x^6 + (2*b*(-1/5*Erf[b*x]/(E^(b^2*x^2)*x^5) + (2*b*(-1/4*1 
/(E^(2*b^2*x^2)*x^4) - b^2*(-1/2*1/(E^(2*b^2*x^2)*x^2) - b^2*ExpIntegralEi 
[-2*b^2*x^2])))/(5*Sqrt[Pi]) - (2*b^2*(-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*ExpIntegralEi[-2*b^2*x^2])/Sqrt[Pi]))/3))/5))/(3*Sqrt[Pi])
 

3.1.28.3.1 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]
 

3.1.28.4 Maple [F]

\[\int \frac {\operatorname {erf}\left (b x \right )^{2}}{x^{7}}d x\]

input

int(erf(b*x)^2/x^7,x)
 

output

int(erf(b*x)^2/x^7,x)
 

3.1.28.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.64 \[ \int \frac {\text {erf}(b x)^2}{x^7} \, dx=\frac {56 \, b^{6} x^{6} {\rm Ei}\left (-2 \, b^{2} x^{2}\right ) - 4 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} - 2 \, b^{3} x^{3} + 3 \, b x\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - {\left (15 \, \pi + 8 \, \pi b^{6} x^{6}\right )} \operatorname {erf}\left (b x\right )^{2} + 2 \, {\left (10 \, b^{4} x^{4} - 3 \, b^{2} x^{2}\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{90 \, \pi x^{6}} \]

input

integrate(erf(b*x)^2/x^7,x, algorithm="fricas")
 

output

1/90*(56*b^6*x^6*Ei(-2*b^2*x^2) - 4*sqrt(pi)*(4*b^5*x^5 - 2*b^3*x^3 + 3*b* 
x)*erf(b*x)*e^(-b^2*x^2) - (15*pi + 8*pi*b^6*x^6)*erf(b*x)^2 + 2*(10*b^4*x 
^4 - 3*b^2*x^2)*e^(-2*b^2*x^2))/(pi*x^6)
 

3.1.28.6 Sympy [F]

\[ \int \frac {\text {erf}(b x)^2}{x^7} \, dx=\int \frac {\operatorname {erf}^{2}{\left (b x \right )}}{x^{7}}\, dx \]

input

integrate(erf(b*x)**2/x**7,x)
 

output

Integral(erf(b*x)**2/x**7, x)
 

3.1.28.7 Maxima [F]

\[ \int \frac {\text {erf}(b x)^2}{x^7} \, dx=\int { \frac {\operatorname {erf}\left (b x\right )^{2}}{x^{7}} \,d x } \]

input

integrate(erf(b*x)^2/x^7,x, algorithm="maxima")
 

output

2/3*b*integrate(erf(b*x)*e^(-b^2*x^2)/x^6, x)/sqrt(pi) - 1/6*erf(b*x)^2/x^ 
6
 

3.1.28.8 Giac [F]

\[ \int \frac {\text {erf}(b x)^2}{x^7} \, dx=\int { \frac {\operatorname {erf}\left (b x\right )^{2}}{x^{7}} \,d x } \]

input

integrate(erf(b*x)^2/x^7,x, algorithm="giac")
 

output

integrate(erf(b*x)^2/x^7, x)
 

3.1.28.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\text {erf}(b x)^2}{x^7} \, dx=\int \frac {{\mathrm {erf}\left (b\,x\right )}^2}{x^7} \,d x \]

input

int(erf(b*x)^2/x^7,x)
 

output

int(erf(b*x)^2/x^7, x)