\(\int e^{c+b^2 x^2} x \text {erfi}(b x) \, dx\) [285]

Optimal result

Integrand size = 17, antiderivative size = 47 \[ \int e^{c+b^2 x^2} x \text {erfi}(b x) \, dx=\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {e^c \text {erfi}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6519, 2235} \[ \int e^{c+b^2 x^2} x \text {erfi}(b x) \, dx=\frac {e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {e^c \text {erfi}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2} \]

[In]

Int[E^(c + b^2*x^2)*x*Erfi[b*x],x]

[Out]

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

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 6519

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.89 \[ \int e^{c+b^2 x^2} x \text {erfi}(b x) \, dx=\frac {e^c \left (2 e^{b^2 x^2} \text {erfi}(b x)-\sqrt {2} \text {erfi}\left (\sqrt {2} b x\right )\right )}{4 b^2} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.04 \[ \int e^{c+b^2 x^2} x \text {erfi}(b x) \, dx=\frac {2 \, b \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} + \sqrt {2} \sqrt {-b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {-b^{2}} x\right ) e^{c}}{4 \, b^{3}} \]

[In]

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

[Out]

1/4*(2*b*erfi(b*x)*e^(b^2*x^2 + c) + sqrt(2)*sqrt(-b^2)*erf(sqrt(2)*sqrt(-b^2)*x)*e^c)/b^3

Sympy [F]

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

[In]

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

[Out]

exp(c)*Integral(x*exp(b**2*x**2)*erfi(b*x), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.95 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.06 \[ \int e^{c+b^2 x^2} x \text {erfi}(b x) \, dx=\frac {{\mathrm {e}}^{b^2\,x^2}\,{\mathrm {e}}^c\,\mathrm {erfi}\left (b\,x\right )}{2\,b^2}-\frac {\sqrt {2}\,\mathrm {erf}\left (\sqrt {2}\,x\,\sqrt {-b^2}\right )\,{\mathrm {e}}^c}{4\,b\,\sqrt {-b^2}} \]

[In]

int(x*exp(c + b^2*x^2)*erfi(b*x),x)

[Out]

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