\(\int e^{c+b^2 x^2} x^2 \text {erfi}(b x) \, dx\) [290]

Optimal result

Integrand size = 19, antiderivative size = 69 \[ \int e^{c+b^2 x^2} x^2 \text {erfi}(b x) \, dx=-\frac {e^{c+2 b^2 x^2}}{4 b^3 \sqrt {\pi }}+\frac {e^{c+b^2 x^2} x \text {erfi}(b x)}{2 b^2}-\frac {e^c \sqrt {\pi } \text {erfi}(b x)^2}{8 b^3} \]

[Out]

1/2*exp(b^2*x^2+c)*x*erfi(b*x)/b^2-1/4*exp(2*b^2*x^2+c)/b^3/Pi^(1/2)-1/8*exp(c)*erfi(b*x)^2*Pi^(1/2)/b^3

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6522, 6510, 30, 2240} \[ \int e^{c+b^2 x^2} x^2 \text {erfi}(b x) \, dx=-\frac {\sqrt {\pi } e^c \text {erfi}(b x)^2}{8 b^3}+\frac {x e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {e^{2 b^2 x^2+c}}{4 \sqrt {\pi } b^3} \]

[In]

Int[E^(c + b^2*x^2)*x^2*Erfi[b*x],x]

[Out]

-1/4*E^(c + 2*b^2*x^2)/(b^3*Sqrt[Pi]) + (E^(c + b^2*x^2)*x*Erfi[b*x])/(2*b^2) - (E^c*Sqrt[Pi]*Erfi[b*x]^2)/(8*
b^3)

Rule 30

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

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

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]

Rule 6522

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^(m - 1)*E^(c + d*x^2)*(Er
fi[a + b*x]/(2*d)), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erfi[a + b*x], x], x] - Dist[b/(d*S
qrt[Pi]), Int[x^(m - 1)*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1]

Rubi steps \begin{align*} \text {integral}& = \frac {e^{c+b^2 x^2} x \text {erfi}(b x)}{2 b^2}-\frac {\int e^{c+b^2 x^2} \text {erfi}(b x) \, dx}{2 b^2}-\frac {\int e^{c+2 b^2 x^2} x \, dx}{b \sqrt {\pi }} \\ & = -\frac {e^{c+2 b^2 x^2}}{4 b^3 \sqrt {\pi }}+\frac {e^{c+b^2 x^2} x \text {erfi}(b x)}{2 b^2}-\frac {\left (e^c \sqrt {\pi }\right ) \text {Subst}(\int x \, dx,x,\text {erfi}(b x))}{4 b^3} \\ & = -\frac {e^{c+2 b^2 x^2}}{4 b^3 \sqrt {\pi }}+\frac {e^{c+b^2 x^2} x \text {erfi}(b x)}{2 b^2}-\frac {e^c \sqrt {\pi } \text {erfi}(b x)^2}{8 b^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84 \[ \int e^{c+b^2 x^2} x^2 \text {erfi}(b x) \, dx=-\frac {e^c \left (2 e^{2 b^2 x^2}-4 b e^{b^2 x^2} \sqrt {\pi } x \text {erfi}(b x)+\pi \text {erfi}(b x)^2\right )}{8 b^3 \sqrt {\pi }} \]

[In]

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

[Out]

-1/8*(E^c*(2*E^(2*b^2*x^2) - 4*b*E^(b^2*x^2)*Sqrt[Pi]*x*Erfi[b*x] + Pi*Erfi[b*x]^2))/(b^3*Sqrt[Pi])

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.77 \[ \int e^{c+b^2 x^2} x^2 \text {erfi}(b x) \, dx=\frac {{\left (4 \, \pi b x \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - \sqrt {\pi } {\left (\pi \operatorname {erfi}\left (b x\right )^{2} + 2 \, e^{\left (2 \, b^{2} x^{2}\right )}\right )}\right )} e^{c}}{8 \, \pi b^{3}} \]

[In]

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

[Out]

1/8*(4*pi*b*x*erfi(b*x)*e^(b^2*x^2) - sqrt(pi)*(pi*erfi(b*x)^2 + 2*e^(2*b^2*x^2)))*e^c/(pi*b^3)

Sympy [A] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.99 \[ \int e^{c+b^2 x^2} x^2 \text {erfi}(b x) \, dx=\begin {cases} \frac {x e^{c} e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{2 b^{2}} - \frac {e^{c} e^{2 b^{2} x^{2}}}{4 \sqrt {\pi } b^{3}} - \frac {\sqrt {\pi } e^{c} \operatorname {erfi}^{2}{\left (b x \right )}}{8 b^{3}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x*exp(c)*exp(b**2*x**2)*erfi(b*x)/(2*b**2) - exp(c)*exp(2*b**2*x**2)/(4*sqrt(pi)*b**3) - sqrt(pi)*e
xp(c)*erfi(b*x)**2/(8*b**3), Ne(b, 0)), (0, True))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.25 \[ \int e^{c+b^2 x^2} x^2 \text {erfi}(b x) \, dx=\mathrm {erfi}\left (b\,x\right )\,\left (\frac {x\,{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^2}-\frac {\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b^2\,x}{\sqrt {b^2}}\right )\,{\mathrm {e}}^c}{4\,{\left (b^2\right )}^{3/2}}\right )-\frac {2\,{\mathrm {e}}^{2\,b^2\,x^2+c}-\pi \,{\mathrm {erfi}\left (\frac {b^2\,x}{\sqrt {b^2}}\right )}^2\,{\mathrm {e}}^c}{8\,b^3\,\sqrt {\pi }} \]

[In]

int(x^2*exp(c + b^2*x^2)*erfi(b*x),x)

[Out]

erfi(b*x)*((x*exp(c + b^2*x^2))/(2*b^2) - (pi^(1/2)*erfi((b^2*x)/(b^2)^(1/2))*exp(c))/(4*(b^2)^(3/2))) - (2*ex
p(c + 2*b^2*x^2) - pi*erfi((b^2*x)/(b^2)^(1/2))^2*exp(c))/(8*b^3*pi^(1/2))