\(\int e^{c+d x^2} x \text {erfi}(b x) \, dx\) [261]

Optimal result

Integrand size = 15, antiderivative size = 53 \[ \int e^{c+d x^2} x \text {erfi}(b x) \, dx=\frac {e^{c+d x^2} \text {erfi}(b x)}{2 d}-\frac {b e^c \text {erfi}\left (\sqrt {b^2+d} x\right )}{2 d \sqrt {b^2+d}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6519, 2235} \[ \int e^{c+d x^2} x \text {erfi}(b x) \, dx=\frac {\text {erfi}(b x) e^{c+d x^2}}{2 d}-\frac {b e^c \text {erfi}\left (x \sqrt {b^2+d}\right )}{2 d \sqrt {b^2+d}} \]

[In]

Int[E^(c + d*x^2)*x*Erfi[b*x],x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.89 \[ \int e^{c+d x^2} x \text {erfi}(b x) \, dx=\frac {e^c \left (e^{d x^2} \text {erfi}(b x)-\frac {b \text {erfi}\left (\sqrt {b^2+d} x\right )}{\sqrt {b^2+d}}\right )}{2 d} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.15 \[ \int e^{c+d x^2} x \text {erfi}(b x) \, dx=\frac {\sqrt {-b^{2} - d} b \operatorname {erf}\left (\sqrt {-b^{2} - d} x\right ) e^{c} + {\left (b^{2} + d\right )} \operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{2 \, {\left (b^{2} d + d^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.85 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.96 \[ \int e^{c+d x^2} x \text {erfi}(b x) \, dx=\frac {{\mathrm {e}}^{d\,x^2}\,{\mathrm {e}}^c\,\mathrm {erfi}\left (b\,x\right )}{2\,d}-\frac {b\,{\mathrm {e}}^c\,\mathrm {erf}\left (x\,\sqrt {-b^2-d}\right )}{2\,d\,\sqrt {-b^2-d}} \]

[In]

int(x*exp(c + d*x^2)*erfi(b*x),x)

[Out]

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