\(\int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^4} \, dx\) [281]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6528, 6513, 29, 30} \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^4} \, dx=\frac {4 b^5 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{3 \sqrt {\pi }}-\frac {4 b^3 \log (x)}{3 \sqrt {\pi }}+\frac {2 b^2 e^{-b^2 x^2} \text {erfi}(b x)}{3 x}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{3 x^3}-\frac {b}{3 \sqrt {\pi } x^2} \]

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 30

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

Rule 6513

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(b_.)*(x_)], x_Symbol] :> Simp[b*E^c*(x^2/Sqrt[Pi])*HypergeometricPFQ[{1, 1}
, {3/2, 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)}{3 x^3}-\frac {1}{3} \left (2 b^2\right ) \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^2} \, dx+\frac {(2 b) \int \frac {1}{x^3} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b}{3 \sqrt {\pi } x^2}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{3 x^3}+\frac {2 b^2 e^{-b^2 x^2} \text {erfi}(b x)}{3 x}+\frac {1}{3} \left (4 b^4\right ) \int e^{-b^2 x^2} \text {erfi}(b x) \, dx-\frac {\left (4 b^3\right ) \int \frac {1}{x} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b}{3 \sqrt {\pi } x^2}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{3 x^3}+\frac {2 b^2 e^{-b^2 x^2} \text {erfi}(b x)}{3 x}+\frac {4 b^5 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{3 \sqrt {\pi }}-\frac {4 b^3 \log (x)}{3 \sqrt {\pi }} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.28 \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^4} \, dx=-\frac {b G_{2,3}^{2,1}\left (b^2 x^2|\begin {array}{c} 0,2 \\ 0,1,-\frac {1}{2} \\\end {array}\right )}{2 x^2} \]

[In]

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

[Out]

-1/2*(b*MeijerG[{{0}, {2}}, {{0, 1}, {-1/2}}, b^2*x^2])/x^2

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(erfi(b*x)*e^(-b^2*x^2)/x^4, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 24.92 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.23 \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^4} \, dx=- \frac {b^{3} {G_{3, 2}^{1, 2}\left (\begin {matrix} 2, 1 & \frac {5}{2} \\2 & 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{2} \]

[In]

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

[Out]

-b**3*meijerg(((2, 1), (5/2,)), ((2,), (0,)), exp_polar(-2*I*pi)/(b**2*x**2))/2

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

int((exp(-b^2*x^2)*erfi(b*x))/x^4, x)