\(\int \frac {\text {erfi}(b x)}{x} \, dx\) [210]

Optimal result

Integrand size = 8, antiderivative size = 31 \[ \int \frac {\text {erfi}(b x)}{x} \, dx=\frac {2 b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }} \]

[Out]

2*b*x*hypergeom([1/2, 1/2],[3/2, 3/2],b^2*x^2)/Pi^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6495} \[ \int \frac {\text {erfi}(b x)}{x} \, dx=\frac {2 b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }} \]

[In]

Int[Erfi[b*x]/x,x]

[Out]

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

Rule 6495

Int[Erfi[(b_.)*(x_)]/(x_), x_Symbol] :> Simp[2*b*(x/Sqrt[Pi])*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, b^2*x^
2], x] /; FreeQ[b, x]

Rubi steps \begin{align*} \text {integral}& = \frac {2 b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {\text {erfi}(b x)}{x} \, dx=\frac {2 b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }} \]

[In]

Integrate[Erfi[b*x]/x,x]

[Out]

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

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71

method	result	size
meijerg	\(\frac {2 b x \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {1}{2}\right ], \left [\frac {3}{2}, \frac {3}{2}\right ], b^{2} x^{2}\right )}{\sqrt {\pi }}\)	\(22\)

[In]

int(erfi(b*x)/x,x,method=_RETURNVERBOSE)

[Out]

2*b*x*hypergeom([1/2,1/2],[3/2,3/2],b^2*x^2)/Pi^(1/2)

Fricas [F]

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

[In]

integrate(erfi(b*x)/x,x, algorithm="fricas")

[Out]

integral(erfi(b*x)/x, x)

Sympy [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {\text {erfi}(b x)}{x} \, dx=\frac {2 b x {{}_{2}F_{2}\left (\begin {matrix} \frac {1}{2}, \frac {1}{2} \\ \frac {3}{2}, \frac {3}{2} \end {matrix}\middle | {b^{2} x^{2}} \right )}}{\sqrt {\pi }} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(erfi(b*x)/x, x)

Giac [F]

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

[In]

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

[Out]

integrate(erfi(b*x)/x, x)

Mupad [F(-1)]

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

[In]

int(erfi(b*x)/x,x)

[Out]

int(erfi(b*x)/x, x)