\(\int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^4} \, dx\) [85]

Optimal result

Integrand size = 18, antiderivative size = 108 \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^4} \, dx=-\frac {b e^{-2 b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}+\frac {2 b^2 e^{-b^2 x^2} \text {erf}(b x)}{3 x}+\frac {1}{3} b^3 \sqrt {\pi } \text {erf}(b x)^2-\frac {4 b^3 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{3 \sqrt {\pi }} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6526, 6508, 30, 2241, 2245} \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^4} \, dx=\frac {1}{3} \sqrt {\pi } b^3 \text {erf}(b x)^2+\frac {2 b^2 e^{-b^2 x^2} \text {erf}(b x)}{3 x}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}-\frac {b e^{-2 b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {4 b^3 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{3 \sqrt {\pi }} \]

[In]

Int[Erf[b*x]/(E^(b^2*x^2)*x^4),x]

[Out]

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

Rule 30

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

Rule 2241

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] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rule 2245

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] - Dist[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 6508

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[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 6526

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] + (-Dist[2*(d/(m + 1)), Int[x^(m + 2)*E^(c + d*x^2)*Erf[a + b*x], x], x] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}-\frac {1}{3} \left (2 b^2\right ) \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 {b e^{-2 b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}+\frac {2 b^2 e^{-b^2 x^2} \text {erf}(b x)}{3 x}+\frac {1}{3} \left (4 b^4\right ) \int e^{-b^2 x^2} \text {erf}(b x) \, dx-2 \frac {\left (4 b^3\right ) \int \frac {e^{-2 b^2 x^2}}{x} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b e^{-2 b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}+\frac {2 b^2 e^{-b^2 x^2} \text {erf}(b x)}{3 x}-\frac {4 b^3 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{3 \sqrt {\pi }}+\frac {1}{3} \left (2 b^3 \sqrt {\pi }\right ) \text {Subst}(\int x \, dx,x,\text {erf}(b x)) \\ & = -\frac {b e^{-2 b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{3 x^3}+\frac {2 b^2 e^{-b^2 x^2} \text {erf}(b x)}{3 x}+\frac {1}{3} b^3 \sqrt {\pi } \text {erf}(b x)^2-\frac {4 b^3 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{3 \sqrt {\pi }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.79 \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^4} \, dx=\frac {1}{3} \left (\frac {e^{-b^2 x^2} \left (-1+2 b^2 x^2\right ) \text {erf}(b x)}{x^3}+b^3 \sqrt {\pi } \text {erf}(b x)^2+\frac {b \left (-\frac {e^{-2 b^2 x^2}}{x^2}-4 b^2 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )}{\sqrt {\pi }}\right ) \]

[In]

Integrate[Erf[b*x]/(E^(b^2*x^2)*x^4),x]

[Out]

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

Maple [F]

\[\int \frac {\operatorname {erf}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x^{4}}d x\]

[In]

int(erf(b*x)/exp(b^2*x^2)/x^4,x)

[Out]

int(erf(b*x)/exp(b^2*x^2)/x^4,x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.78 \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^4} \, dx=-\frac {{\left (\pi - 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - \sqrt {\pi } {\left (\pi b^{3} x^{3} \operatorname {erf}\left (b x\right )^{2} - 4 \, b^{3} x^{3} {\rm Ei}\left (-2 \, b^{2} x^{2}\right ) - b x e^{\left (-2 \, b^{2} x^{2}\right )}\right )}}{3 \, \pi x^{3}} \]

[In]

integrate(erf(b*x)/exp(b^2*x^2)/x^4,x, algorithm="fricas")

[Out]

-1/3*((pi - 2*pi*b^2*x^2)*erf(b*x)*e^(-b^2*x^2) - sqrt(pi)*(pi*b^3*x^3*erf(b*x)^2 - 4*b^3*x^3*Ei(-2*b^2*x^2) -
b*x*e^(-2*b^2*x^2)))/(pi*x^3)

Sympy [F]

\[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^4} \, dx=\int \frac {e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{x^{4}}\, dx \]

[In]

integrate(erf(b*x)/exp(b**2*x**2)/x**4,x)

[Out]

Integral(exp(-b**2*x**2)*erf(b*x)/x**4, x)

Maxima [F]

\[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^4} \, dx=\int { \frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{4}} \,d x } \]

[In]

integrate(erf(b*x)/exp(b^2*x^2)/x^4,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^4} \, dx=\int { \frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{4}} \,d x } \]

[In]

integrate(erf(b*x)/exp(b^2*x^2)/x^4,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^4} \, dx=\int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{x^4} \,d x \]

[In]

int((exp(-b^2*x^2)*erf(b*x))/x^4,x)

[Out]

int((exp(-b^2*x^2)*erf(b*x))/x^4, x)