\(\int \text {erfi}(b x) \, dx\) [217]

Optimal result

Integrand size = 4, antiderivative size = 26 \[ \int \text {erfi}(b x) \, dx=-\frac {e^{b^2 x^2}}{b \sqrt {\pi }}+x \text {erfi}(b x) \]

[Out]

x*erfi(b*x)-exp(b^2*x^2)/b/Pi^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6486} \[ \int \text {erfi}(b x) \, dx=x \text {erfi}(b x)-\frac {e^{b^2 x^2}}{\sqrt {\pi } b} \]

[In]

Int[Erfi[b*x],x]

[Out]

-(E^(b^2*x^2)/(b*Sqrt[Pi])) + x*Erfi[b*x]

Rule 6486

Int[Erfi[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(Erfi[a + b*x]/b), x] - Simp[E^(a + b*x)^2/(b*Sqrt[P
i]), x] /; FreeQ[{a, b}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{b^2 x^2}}{b \sqrt {\pi }}+x \text {erfi}(b x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \text {erfi}(b x) \, dx=-\frac {e^{b^2 x^2}}{b \sqrt {\pi }}+x \text {erfi}(b x) \]

[In]

Integrate[Erfi[b*x],x]

[Out]

-(E^(b^2*x^2)/(b*Sqrt[Pi])) + x*Erfi[b*x]

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92

method	result	size
parts	\(x \,\operatorname {erfi}\left (b x \right )-\frac {{\mathrm e}^{b^{2} x^{2}}}{b \sqrt {\pi }}\)	\(24\)
derivativedivides	\(\frac {b x \,\operatorname {erfi}\left (b x \right )-\frac {{\mathrm e}^{b^{2} x^{2}}}{\sqrt {\pi }}}{b}\)	\(26\)
default	\(\frac {b x \,\operatorname {erfi}\left (b x \right )-\frac {{\mathrm e}^{b^{2} x^{2}}}{\sqrt {\pi }}}{b}\)	\(26\)
parallelrisch	\(\frac {b x \sqrt {\pi }\, \operatorname {erfi}\left (b x \right )-{\mathrm e}^{b^{2} x^{2}}}{\sqrt {\pi }\, b}\)	\(29\)
meijerg	\(-\frac {-2+2 \,{\mathrm e}^{b^{2} x^{2}}-2 b x \sqrt {\pi }\, \operatorname {erfi}\left (b x \right )}{2 \sqrt {\pi }\, b}\)	\(32\)

[In]

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

[Out]

x*erfi(b*x)-exp(b^2*x^2)/b/Pi^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \text {erfi}(b x) \, dx=\frac {\pi b x \operatorname {erfi}\left (b x\right ) - \sqrt {\pi } e^{\left (b^{2} x^{2}\right )}}{\pi b} \]

[In]

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

[Out]

(pi*b*x*erfi(b*x) - sqrt(pi)*e^(b^2*x^2))/(pi*b)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \text {erfi}(b x) \, dx=\begin {cases} x \operatorname {erfi}{\left (b x \right )} - \frac {e^{b^{2} x^{2}}}{\sqrt {\pi } b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(erfi(b*x),x)

[Out]

Piecewise((x*erfi(b*x) - exp(b**2*x**2)/(sqrt(pi)*b), Ne(b, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \text {erfi}(b x) \, dx=\frac {b x \operatorname {erfi}\left (b x\right ) - \frac {e^{\left (b^{2} x^{2}\right )}}{\sqrt {\pi }}}{b} \]

[In]

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

[Out]

(b*x*erfi(b*x) - e^(b^2*x^2)/sqrt(pi))/b

Giac [F]

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

[In]

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

[Out]

integrate(erfi(b*x), x)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \text {erfi}(b x) \, dx=x\,\mathrm {erfi}\left (b\,x\right )-\frac {{\mathrm {e}}^{b^2\,x^2}}{b\,\sqrt {\pi }} \]

[In]

int(erfi(b*x),x)

[Out]

x*erfi(b*x) - exp(b^2*x^2)/(b*pi^(1/2))