\(\int \frac {\text {erfi}(b x)^2}{x^7} \, dx\) [234]

Optimal result

Integrand size = 10, antiderivative size = 174 \[ \int \frac {\text {erfi}(b x)^2}{x^7} \, dx=-\frac {b^2 e^{2 b^2 x^2}}{15 \pi x^4}-\frac {2 b^4 e^{2 b^2 x^2}}{9 \pi x^2}-\frac {2 b e^{b^2 x^2} \text {erfi}(b x)}{15 \sqrt {\pi } x^5}-\frac {4 b^3 e^{b^2 x^2} \text {erfi}(b x)}{45 \sqrt {\pi } x^3}-\frac {8 b^5 e^{b^2 x^2} \text {erfi}(b x)}{45 \sqrt {\pi } x}+\frac {4}{45} b^6 \text {erfi}(b x)^2-\frac {\text {erfi}(b x)^2}{6 x^6}+\frac {28 b^6 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{45 \pi } \] 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^2-2/15*b*exp(b 
^2*x^2)*erfi(b*x)/Pi^(1/2)/x^5-4/45*b^3*exp(b^2*x^2)*erfi(b*x)/Pi^(1/2)/x^ 
3-8/45*b^5*exp(b^2*x^2)*erfi(b*x)/Pi^(1/2)/x+4/45*b^6*erfi(b*x)^2-1/6*erfi 
(b*x)^2/x^6+28/45*b^6*Ei(2*b^2*x^2)/Pi
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.66 \[ \int \frac {\text {erfi}(b x)^2}{x^7} \, dx=\frac {-2 b^2 e^{2 b^2 x^2} x^2 \left (3+10 b^2 x^2\right )-4 b e^{b^2 x^2} \sqrt {\pi } x \left (3+2 b^2 x^2+4 b^4 x^4\right ) \text {erfi}(b x)+\pi \left (-15+8 b^6 x^6\right ) \text {erfi}(b x)^2+56 b^6 x^6 \operatorname {ExpIntegralEi}\left (2 b^2 x^2\right )}{90 \pi x^6} \] Input:

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

Output:

(-2*b^2*E^(2*b^2*x^2)*x^2*(3 + 10*b^2*x^2) - 4*b*E^(b^2*x^2)*Sqrt[Pi]*x*(3 
 + 2*b^2*x^2 + 4*b^4*x^4)*Erfi[b*x] + Pi*(-15 + 8*b^6*x^6)*Erfi[b*x]^2 + 5 
6*b^6*x^6*ExpIntegralEi[2*b^2*x^2])/(90*Pi*x^6)
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.35, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {6920, 6947, 2643, 2643, 2639, 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.

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

\(\Big \downarrow \) 6920

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

\(\Big \downarrow \) 6947

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

\(\Big \downarrow \) 2643

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

\(\Big \downarrow \) 2643

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

\(\Big \downarrow \) 2639

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

\(\Big \downarrow \) 6947

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

\(\Big \downarrow \) 2643

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

\(\Big \downarrow \) 2639

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

\(\Big \downarrow \) 6947

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

\(\Big \downarrow \) 2639

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

\(\Big \downarrow \) 6929

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

\(\Big \downarrow \) 15

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

Input:

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

Output:

-1/6*Erfi[b*x]^2/x^6 + (2*b*(-1/5*(E^(b^2*x^2)*Erfi[b*x])/x^5 + (2*b*(-1/4 
*E^(2*b^2*x^2)/x^4 + b^2*(-1/2*E^(2*b^2*x^2)/x^2 + b^2*ExpIntegralEi[2*b^2 
*x^2])))/(5*Sqrt[Pi]) + (2*b^2*(-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*ExpInte 
gralEi[2*b^2*x^2])/Sqrt[Pi]))/3))/5))/(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 6920

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])
 

rule 6929

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]
 

rule 6947

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.65 \[ \int \frac {\text {erfi}(b x)^2}{x^7} \, dx=\frac {56 \, b^{6} x^{6} {\rm Ei}\left (2 \, b^{2} x^{2}\right ) - 4 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} + 2 \, b^{3} x^{3} + 3 \, b x\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - {\left (15 \, \pi - 8 \, \pi b^{6} x^{6}\right )} \operatorname {erfi}\left (b x\right )^{2} - 2 \, {\left (10 \, b^{4} x^{4} + 3 \, b^{2} x^{2}\right )} e^{\left (2 \, b^{2} x^{2}\right )}}{90 \, \pi x^{6}} \] Input:

integrate(erfi(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 
)*erfi(b*x)*e^(b^2*x^2) - (15*pi - 8*pi*b^6*x^6)*erfi(b*x)^2 - 2*(10*b^4*x 
^4 + 3*b^2*x^2)*e^(2*b^2*x^2))/(pi*x^6)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\text {erfi}(b x)^2}{x^7} \, dx=-\left (\int \frac {\mathrm {erf}\left (b i x \right )^{2}}{x^{7}}d x \right ) \] Input:

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

Output:

 - int(erf(b*i*x)**2/x**7,x)