\(\int x^3 \text {erfi}(b x) \, dx\) [208]

Optimal result

Integrand size = 8, antiderivative size = 69 \[ \int x^3 \text {erfi}(b x) \, dx=\frac {3 e^{b^2 x^2} x}{8 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^3}{4 b \sqrt {\pi }}-\frac {3 \text {erfi}(b x)}{16 b^4}+\frac {1}{4} x^4 \text {erfi}(b x) \]

[Out]

-3/16*erfi(b*x)/b^4+1/4*x^4*erfi(b*x)+3/8*exp(b^2*x^2)*x/b^3/Pi^(1/2)-1/4*exp(b^2*x^2)*x^3/b/Pi^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6498, 2243, 2235} \[ \int x^3 \text {erfi}(b x) \, dx=-\frac {3 \text {erfi}(b x)}{16 b^4}-\frac {x^3 e^{b^2 x^2}}{4 \sqrt {\pi } b}+\frac {3 x e^{b^2 x^2}}{8 \sqrt {\pi } b^3}+\frac {1}{4} x^4 \text {erfi}(b x) \]

[In]

Int[x^3*Erfi[b*x],x]

[Out]

(3*E^(b^2*x^2)*x)/(8*b^3*Sqrt[Pi]) - (E^(b^2*x^2)*x^3)/(4*b*Sqrt[Pi]) - (3*Erfi[b*x])/(16*b^4) + (x^4*Erfi[b*x
])/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 6498

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \text {erfi}(b x)-\frac {b \int e^{b^2 x^2} x^4 \, dx}{2 \sqrt {\pi }} \\ & = -\frac {e^{b^2 x^2} x^3}{4 b \sqrt {\pi }}+\frac {1}{4} x^4 \text {erfi}(b x)+\frac {3 \int e^{b^2 x^2} x^2 \, dx}{4 b \sqrt {\pi }} \\ & = \frac {3 e^{b^2 x^2} x}{8 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^3}{4 b \sqrt {\pi }}+\frac {1}{4} x^4 \text {erfi}(b x)-\frac {3 \int e^{b^2 x^2} \, dx}{8 b^3 \sqrt {\pi }} \\ & = \frac {3 e^{b^2 x^2} x}{8 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^3}{4 b \sqrt {\pi }}-\frac {3 \text {erfi}(b x)}{16 b^4}+\frac {1}{4} x^4 \text {erfi}(b x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.74 \[ \int x^3 \text {erfi}(b x) \, dx=\frac {-\frac {2 b e^{b^2 x^2} x \left (-3+2 b^2 x^2\right )}{\sqrt {\pi }}+\left (-3+4 b^4 x^4\right ) \text {erfi}(b x)}{16 b^4} \]

[In]

Integrate[x^3*Erfi[b*x],x]

[Out]

((-2*b*E^(b^2*x^2)*x*(-3 + 2*b^2*x^2))/Sqrt[Pi] + (-3 + 4*b^4*x^4)*Erfi[b*x])/(16*b^4)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.16 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.78

method	result	size
meijerg	\(-\frac {i \left (\frac {i x b \left (-10 b^{2} x^{2}+15\right ) {\mathrm e}^{b^{2} x^{2}}}{20}-\frac {i \left (-20 b^{4} x^{4}+15\right ) \operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{40}\right )}{2 b^{4} \sqrt {\pi }}\)	\(54\)
derivativedivides	\(\frac {\frac {b^{4} x^{4} \operatorname {erfi}\left (b x \right )}{4}-\frac {\frac {{\mathrm e}^{b^{2} x^{2}} b^{3} x^{3}}{2}-\frac {3 \,{\mathrm e}^{b^{2} x^{2}} b x}{4}+\frac {3 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{8}}{2 \sqrt {\pi }}}{b^{4}}\)	\(61\)
default	\(\frac {\frac {b^{4} x^{4} \operatorname {erfi}\left (b x \right )}{4}-\frac {\frac {{\mathrm e}^{b^{2} x^{2}} b^{3} x^{3}}{2}-\frac {3 \,{\mathrm e}^{b^{2} x^{2}} b x}{4}+\frac {3 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{8}}{2 \sqrt {\pi }}}{b^{4}}\)	\(61\)
parallelrisch	\(\frac {4 \,\operatorname {erfi}\left (b x \right ) x^{4} \sqrt {\pi }\, b^{4}-4 \,{\mathrm e}^{b^{2} x^{2}} b^{3} x^{3}+6 \,{\mathrm e}^{b^{2} x^{2}} b x -3 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{16 \sqrt {\pi }\, b^{4}}\)	\(62\)
parts	\(\frac {x^{4} \operatorname {erfi}\left (b x \right )}{4}-\frac {b \left (\frac {x^{3} {\mathrm e}^{b^{2} x^{2}}}{2 b^{2}}-\frac {3 \left (\frac {x \,{\mathrm e}^{b^{2} x^{2}}}{2 b^{2}}+\frac {i \sqrt {\pi }\, \operatorname {erf}\left (i b x \right )}{4 b^{3}}\right )}{2 b^{2}}\right )}{2 \sqrt {\pi }}\)	\(69\)

[In]

int(x^3*erfi(b*x),x,method=_RETURNVERBOSE)

[Out]

-1/2*I/b^4/Pi^(1/2)*(1/20*I*x*b*(-10*b^2*x^2+15)*exp(b^2*x^2)-1/40*I*(-20*b^4*x^4+15)*erfi(b*x)*Pi^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.77 \[ \int x^3 \text {erfi}(b x) \, dx=-\frac {2 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} - 3 \, b x\right )} e^{\left (b^{2} x^{2}\right )} + {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erfi}\left (b x\right )}{16 \, \pi b^{4}} \]

[In]

integrate(x^3*erfi(b*x),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.94 \[ \int x^3 \text {erfi}(b x) \, dx=\begin {cases} \frac {x^{4} \operatorname {erfi}{\left (b x \right )}}{4} - \frac {x^{3} e^{b^{2} x^{2}}}{4 \sqrt {\pi } b} + \frac {3 x e^{b^{2} x^{2}}}{8 \sqrt {\pi } b^{3}} - \frac {3 \operatorname {erfi}{\left (b x \right )}}{16 b^{4}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(x**3*erfi(b*x),x)

[Out]

Piecewise((x**4*erfi(b*x)/4 - x**3*exp(b**2*x**2)/(4*sqrt(pi)*b) + 3*x*exp(b**2*x**2)/(8*sqrt(pi)*b**3) - 3*er
fi(b*x)/(16*b**4), Ne(b, 0)), (0, True))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.80 \[ \int x^3 \text {erfi}(b x) \, dx=\frac {1}{4} \, x^{4} \operatorname {erfi}\left (b x\right ) - \frac {b {\left (\frac {2 \, {\left (2 \, b^{2} x^{3} - 3 \, x\right )} e^{\left (b^{2} x^{2}\right )}}{b^{4}} - \frac {3 i \, \sqrt {\pi } \operatorname {erf}\left (i \, b x\right )}{b^{5}}\right )}}{16 \, \sqrt {\pi }} \]

[In]

integrate(x^3*erfi(b*x),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(x^3*erfi(b*x),x, algorithm="giac")

[Out]

integrate(x^3*erfi(b*x), x)

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.29 \[ \int x^3 \text {erfi}(b x) \, dx=\frac {x^4\,\mathrm {erfi}\left (b\,x\right )}{4}-\frac {3\,b\,x^5}{16\,{\left (-b^2\,x^2\right )}^{5/2}}-\frac {x^3\,{\mathrm {e}}^{b^2\,x^2}}{4\,b\,\sqrt {\pi }}+\frac {3\,x\,{\mathrm {e}}^{b^2\,x^2}}{8\,b^3\,\sqrt {\pi }}+\frac {3\,b\,x^5\,\mathrm {erfc}\left (\sqrt {-b^2\,x^2}\right )}{16\,{\left (-b^2\,x^2\right )}^{5/2}} \]

[In]

int(x^3*erfi(b*x),x)

[Out]

(x^4*erfi(b*x))/4 - (3*b*x^5)/(16*(-b^2*x^2)^(5/2)) - (x^3*exp(b^2*x^2))/(4*b*pi^(1/2)) + (3*x*exp(b^2*x^2))/(
8*b^3*pi^(1/2)) + (3*b*x^5*erfc((-b^2*x^2)^(1/2)))/(16*(-b^2*x^2)^(5/2))