\(\int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^5} \, dx\) [275]

Optimal result

Integrand size = 18, antiderivative size = 105 \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=-\frac {b}{6 \sqrt {\pi } x^3}+\frac {b^3}{2 \sqrt {\pi } x}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{4 x^4}+\frac {b^2 e^{-b^2 x^2} \text {erfi}(b x)}{4 x^2}+\frac {b^5 x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }} \]

[Out]

-1/4*erfi(b*x)/exp(b^2*x^2)/x^4+1/4*b^2*erfi(b*x)/exp(b^2*x^2)/x^2-1/6*b/x^3/Pi^(1/2)+1/2*b^3/x/Pi^(1/2)+b^5*x
*hypergeom([1/2, 1],[3/2, 3/2],-b^2*x^2)/Pi^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6528, 6525, 30} \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=\frac {b^5 x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }}+\frac {b^3}{2 \sqrt {\pi } x}+\frac {b^2 e^{-b^2 x^2} \text {erfi}(b x)}{4 x^2}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{4 x^4}-\frac {b}{6 \sqrt {\pi } x^3} \]

[In]

Int[Erfi[b*x]/(E^(b^2*x^2)*x^5),x]

[Out]

-1/6*b/(Sqrt[Pi]*x^3) + b^3/(2*Sqrt[Pi]*x) - Erfi[b*x]/(4*E^(b^2*x^2)*x^4) + (b^2*Erfi[b*x])/(4*E^(b^2*x^2)*x^
2) + (b^5*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt[Pi]

Rule 30

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

Rule 6525

Int[(E^((c_.) + (d_.)*(x_)^2)*Erfi[(b_.)*(x_)])/(x_), x_Symbol] :> Simp[2*b*E^c*(x/Sqrt[Pi])*HypergeometricPFQ
[{1/2, 1}, {3/2, 3/2}, (-b^2)*x^2], x] /; FreeQ[{b, c, d}, x] && EqQ[d, -b^2]

Rule 6528

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^(m + 1)*E^(c + d*x^2)*(Er
fi[a + b*x]/(m + 1)), x] + (-Dist[2*(d/(m + 1)), Int[x^(m + 2)*E^(c + d*x^2)*Erfi[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 {erfi}(b x)}{4 x^4}-\frac {1}{2} b^2 \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3} \, dx+\frac {b \int \frac {1}{x^4} \, dx}{2 \sqrt {\pi }} \\ & = -\frac {b}{6 \sqrt {\pi } x^3}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{4 x^4}+\frac {b^2 e^{-b^2 x^2} \text {erfi}(b x)}{4 x^2}+\frac {1}{2} b^4 \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x} \, dx-\frac {b^3 \int \frac {1}{x^2} \, dx}{2 \sqrt {\pi }} \\ & = -\frac {b}{6 \sqrt {\pi } x^3}+\frac {b^3}{2 \sqrt {\pi } x}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{4 x^4}+\frac {b^2 e^{-b^2 x^2} \text {erfi}(b x)}{4 x^2}+\frac {b^5 x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.32 \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=-\frac {2 b \, _2F_2\left (-\frac {3}{2},1;-\frac {1}{2},\frac {3}{2};-b^2 x^2\right )}{3 \sqrt {\pi } x^3} \]

[In]

Integrate[Erfi[b*x]/(E^(b^2*x^2)*x^5),x]

[Out]

(-2*b*HypergeometricPFQ[{-3/2, 1}, {-1/2, 3/2}, -(b^2*x^2)])/(3*Sqrt[Pi]*x^3)

Maple [F]

\[\int \frac {\operatorname {erfi}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x^{5}}d x\]

[In]

int(erfi(b*x)/exp(b^2*x^2)/x^5,x)

[Out]

int(erfi(b*x)/exp(b^2*x^2)/x^5,x)

Fricas [F]

\[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{5}} \,d x } \]

[In]

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

[Out]

integral(erfi(b*x)*e^(-b^2*x^2)/x^5, x)

Sympy [A] (verification not implemented)

Time = 49.39 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.30 \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=- \frac {2 b {{}_{2}F_{2}\left (\begin {matrix} - \frac {3}{2}, 1 \\ - \frac {1}{2}, \frac {3}{2} \end {matrix}\middle | {- b^{2} x^{2}} \right )}}{3 \sqrt {\pi } x^{3}} \]

[In]

integrate(erfi(b*x)/exp(b**2*x**2)/x**5,x)

[Out]

-2*b*hyper((-3/2, 1), (-1/2, 3/2), -b**2*x**2)/(3*sqrt(pi)*x**3)

Maxima [F]

\[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{5}} \,d x } \]

[In]

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

[Out]

integrate(erfi(b*x)*e^(-b^2*x^2)/x^5, x)

Giac [F]

\[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{5}} \,d x } \]

[In]

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

[Out]

integrate(erfi(b*x)*e^(-b^2*x^2)/x^5, x)

Mupad [F(-1)]

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

[In]

int((exp(-b^2*x^2)*erfi(b*x))/x^5,x)

[Out]

int((exp(-b^2*x^2)*erfi(b*x))/x^5, x)