\(\int e^{c+b^2 x^2} \text {erfi}(b x) \, dx\) [254]

Optimal result

Integrand size = 16, antiderivative size = 21 \[ \int e^{c+b^2 x^2} \text {erfi}(b x) \, dx=\frac {e^c \sqrt {\pi } \text {erfi}(b x)^2}{4 b} \]

[Out]

1/4*exp(c)*erfi(b*x)^2*Pi^(1/2)/b

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6510, 30} \[ \int e^{c+b^2 x^2} \text {erfi}(b x) \, dx=\frac {\sqrt {\pi } e^c \text {erfi}(b x)^2}{4 b} \]

[In]

Int[E^(c + b^2*x^2)*Erfi[b*x],x]

[Out]

(E^c*Sqrt[Pi]*Erfi[b*x]^2)/(4*b)

Rule 30

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

Rule 6510

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[E^c*(Sqrt[Pi]/(2*b)), Subst[Int[x^n, x]
, x, Erfi[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, b^2]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (e^c \sqrt {\pi }\right ) \text {Subst}(\int x \, dx,x,\text {erfi}(b x))}{2 b} \\ & = \frac {e^c \sqrt {\pi } \text {erfi}(b x)^2}{4 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int e^{c+b^2 x^2} \text {erfi}(b x) \, dx=\frac {e^c \sqrt {\pi } \text {erfi}(b x)^2}{4 b} \]

[In]

Integrate[E^(c + b^2*x^2)*Erfi[b*x],x]

[Out]

(E^c*Sqrt[Pi]*Erfi[b*x]^2)/(4*b)

Maple [F]

\[\int {\mathrm e}^{b^{2} x^{2}+c} \operatorname {erfi}\left (b x \right )d x\]

[In]

int(exp(b^2*x^2+c)*erfi(b*x),x)

[Out]

int(exp(b^2*x^2+c)*erfi(b*x),x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int e^{c+b^2 x^2} \text {erfi}(b x) \, dx=\frac {\sqrt {\pi } \operatorname {erfi}\left (b x\right )^{2} e^{c}}{4 \, b} \]

[In]

integrate(exp(b^2*x^2+c)*erfi(b*x),x, algorithm="fricas")

[Out]

1/4*sqrt(pi)*erfi(b*x)^2*e^c/b

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int e^{c+b^2 x^2} \text {erfi}(b x) \, dx=\begin {cases} \frac {\sqrt {\pi } e^{c} \operatorname {erfi}^{2}{\left (b x \right )}}{4 b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(exp(b**2*x**2+c)*erfi(b*x),x)

[Out]

Piecewise((sqrt(pi)*exp(c)*erfi(b*x)**2/(4*b), Ne(b, 0)), (0, True))

Maxima [F]

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

[In]

integrate(exp(b^2*x^2+c)*erfi(b*x),x, algorithm="maxima")

[Out]

integrate(erfi(b*x)*e^(b^2*x^2 + c), x)

Giac [F]

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

[In]

integrate(exp(b^2*x^2+c)*erfi(b*x),x, algorithm="giac")

[Out]

integrate(erfi(b*x)*e^(b^2*x^2 + c), x)

Mupad [B] (verification not implemented)

Time = 5.00 (sec) , antiderivative size = 91, normalized size of antiderivative = 4.33 \[ \int e^{c+b^2 x^2} \text {erfi}(b x) \, dx=\frac {\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b^2\,x}{\sqrt {b^2}}\right )\,{\mathrm {e}}^c\,\mathrm {erfi}\left (b\,x\right )}{2\,\sqrt {b^2}}-\frac {\sqrt {\pi }\,{\mathrm {e}}^c\,{\mathrm {erf}\left (x\,\sqrt {-b^2}\right )}^2}{4\,b}-\frac {b\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b^2\,x}{\sqrt {b^2}}\right )\,{\mathrm {e}}^c\,\mathrm {erf}\left (x\,\sqrt {-b^2}\right )}{2\,\sqrt {b^2}\,\sqrt {-b^2}} \]

[In]

int(exp(c + b^2*x^2)*erfi(b*x),x)

[Out]

(pi^(1/2)*erfi((b^2*x)/(b^2)^(1/2))*exp(c)*erfi(b*x))/(2*(b^2)^(1/2)) - (pi^(1/2)*exp(c)*erf(x*(-b^2)^(1/2))^2
)/(4*b) - (b*pi^(1/2)*erfi((b^2*x)/(b^2)^(1/2))*exp(c)*erf(x*(-b^2)^(1/2)))/(2*(b^2)^(1/2)*(-b^2)^(1/2))