\(\int e^{c+b^2 x^2} x^3 \text {erfi}(b x) \, dx\) [284]

Optimal result

Integrand size = 19, antiderivative size = 97 \[ \int e^{c+b^2 x^2} x^3 \text {erfi}(b x) \, dx=-\frac {e^{c+2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{2 b^4}+\frac {e^{c+b^2 x^2} x^2 \text {erfi}(b x)}{2 b^2}+\frac {5 e^c \text {erfi}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4} \]

[Out]

-1/2*exp(b^2*x^2+c)*erfi(b*x)/b^4+1/2*exp(b^2*x^2+c)*x^2*erfi(b*x)/b^2+5/16*exp(c)*erfi(b*x*2^(1/2))/b^4*2^(1/
2)-1/4*exp(2*b^2*x^2+c)*x/b^3/Pi^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6522, 6519, 2235, 2243} \[ \int e^{c+b^2 x^2} x^3 \text {erfi}(b x) \, dx=\frac {5 e^c \text {erfi}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4}+\frac {x^2 e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^4}-\frac {x e^{2 b^2 x^2+c}}{4 \sqrt {\pi } b^3} \]

[In]

Int[E^(c + b^2*x^2)*x^3*Erfi[b*x],x]

[Out]

-1/4*(E^(c + 2*b^2*x^2)*x)/(b^3*Sqrt[Pi]) - (E^(c + b^2*x^2)*Erfi[b*x])/(2*b^4) + (E^(c + b^2*x^2)*x^2*Erfi[b*
x])/(2*b^2) + (5*E^c*Erfi[Sqrt[2]*b*x])/(8*Sqrt[2]*b^4)

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 2243

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m
- n + 1)*(F^(a + b*(c + d*x)^n)/(b*d*n*Log[F])), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

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]

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^2 \text {erfi}(b x)}{2 b^2}-\frac {\int e^{c+b^2 x^2} x \text {erfi}(b x) \, dx}{b^2}-\frac {\int e^{c+2 b^2 x^2} x^2 \, dx}{b \sqrt {\pi }} \\ & = -\frac {e^{c+2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{2 b^4}+\frac {e^{c+b^2 x^2} x^2 \text {erfi}(b x)}{2 b^2}+\frac {\int e^{c+2 b^2 x^2} \, dx}{4 b^3 \sqrt {\pi }}+\frac {\int e^{c+2 b^2 x^2} \, dx}{b^3 \sqrt {\pi }} \\ & = -\frac {e^{c+2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{2 b^4}+\frac {e^{c+b^2 x^2} x^2 \text {erfi}(b x)}{2 b^2}+\frac {5 e^c \text {erfi}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int e^{c+b^2 x^2} x^3 \text {erfi}(b x) \, dx=\frac {e^c \left (-4 b e^{2 b^2 x^2} x+8 e^{b^2 x^2} \sqrt {\pi } \left (-1+b^2 x^2\right ) \text {erfi}(b x)+5 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} b x\right )\right )}{16 b^4 \sqrt {\pi }} \]

[In]

Integrate[E^(c + b^2*x^2)*x^3*Erfi[b*x],x]

[Out]

(E^c*(-4*b*E^(2*b^2*x^2)*x + 8*E^(b^2*x^2)*Sqrt[Pi]*(-1 + b^2*x^2)*Erfi[b*x] + 5*Sqrt[2*Pi]*Erfi[Sqrt[2]*b*x])
)/(16*b^4*Sqrt[Pi])

Maple [F]

\[\int {\mathrm e}^{b^{2} x^{2}+c} x^{3} \operatorname {erfi}\left (b x \right )d x\]

[In]

int(exp(b^2*x^2+c)*x^3*erfi(b*x),x)

[Out]

int(exp(b^2*x^2+c)*x^3*erfi(b*x),x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.89 \[ \int e^{c+b^2 x^2} x^3 \text {erfi}(b x) \, dx=-\frac {4 \, \sqrt {\pi } b^{2} x e^{\left (2 \, b^{2} x^{2} + c\right )} + 5 \, \sqrt {2} \pi \sqrt {-b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {-b^{2}} x\right ) e^{c} - 8 \, {\left (\pi b^{3} x^{2} - \pi b\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{16 \, \pi b^{5}} \]

[In]

integrate(exp(b^2*x^2+c)*x^3*erfi(b*x),x, algorithm="fricas")

[Out]

-1/16*(4*sqrt(pi)*b^2*x*e^(2*b^2*x^2 + c) + 5*sqrt(2)*pi*sqrt(-b^2)*erf(sqrt(2)*sqrt(-b^2)*x)*e^c - 8*(pi*b^3*
x^2 - pi*b)*erfi(b*x)*e^(b^2*x^2 + c))/(pi*b^5)

Sympy [F]

\[ \int e^{c+b^2 x^2} x^3 \text {erfi}(b x) \, dx=e^{c} \int x^{3} e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}\, dx \]

[In]

integrate(exp(b**2*x**2+c)*x**3*erfi(b*x),x)

[Out]

exp(c)*Integral(x**3*exp(b**2*x**2)*erfi(b*x), x)

Maxima [F]

\[ \int e^{c+b^2 x^2} x^3 \text {erfi}(b x) \, dx=\int { x^{3} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]

[In]

integrate(exp(b^2*x^2+c)*x^3*erfi(b*x),x, algorithm="maxima")

[Out]

integrate(x^3*erfi(b*x)*e^(b^2*x^2 + c), x)

Giac [F]

\[ \int e^{c+b^2 x^2} x^3 \text {erfi}(b x) \, dx=\int { x^{3} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]

[In]

integrate(exp(b^2*x^2+c)*x^3*erfi(b*x),x, algorithm="giac")

[Out]

integrate(x^3*erfi(b*x)*e^(b^2*x^2 + c), x)

Mupad [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.21 \[ \int e^{c+b^2 x^2} x^3 \text {erfi}(b x) \, dx=\frac {\sqrt {2}\,{\mathrm {e}}^c\,\mathrm {erfi}\left (\sqrt {2}\,x\,\sqrt {b^2}\right )}{16\,b\,{\left (b^2\right )}^{3/2}}-\frac {x\,{\mathrm {e}}^{2\,b^2\,x^2+c}}{4\,b^3\,\sqrt {\pi }}-\mathrm {erfi}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^4}-\frac {x^2\,{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^2}\right )-\frac {\sqrt {2}\,\mathrm {erf}\left (\sqrt {2}\,x\,\sqrt {-b^2}\right )\,{\mathrm {e}}^c}{4\,b\,{\left (-b^2\right )}^{3/2}} \]

[In]

int(x^3*exp(c + b^2*x^2)*erfi(b*x),x)

[Out]

(2^(1/2)*exp(c)*erfi(2^(1/2)*x*(b^2)^(1/2)))/(16*b*(b^2)^(3/2)) - (x*exp(c + 2*b^2*x^2))/(4*b^3*pi^(1/2)) - er
fi(b*x)*(exp(c + b^2*x^2)/(2*b^4) - (x^2*exp(c + b^2*x^2))/(2*b^2)) - (2^(1/2)*erf(2^(1/2)*x*(-b^2)^(1/2))*exp
(c))/(4*b*(-b^2)^(3/2))