\(\int x^5 \text {erfi}(b x)^2 \, dx\) [228]

Optimal result

Integrand size = 10, antiderivative size = 175 \[ \int x^5 \text {erfi}(b x)^2 \, dx=\frac {11 e^{2 b^2 x^2}}{12 b^6 \pi }-\frac {7 e^{2 b^2 x^2} x^2}{12 b^4 \pi }+\frac {e^{2 b^2 x^2} x^4}{6 b^2 \pi }-\frac {5 e^{b^2 x^2} x \text {erfi}(b x)}{4 b^5 \sqrt {\pi }}+\frac {5 e^{b^2 x^2} x^3 \text {erfi}(b x)}{6 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^5 \text {erfi}(b x)}{3 b \sqrt {\pi }}+\frac {5 \text {erfi}(b x)^2}{16 b^6}+\frac {1}{6} x^6 \text {erfi}(b x)^2 \]

[Out]

11/12*exp(2*b^2*x^2)/b^6/Pi-7/12*exp(2*b^2*x^2)*x^2/b^4/Pi+1/6*exp(2*b^2*x^2)*x^4/b^2/Pi+5/16*erfi(b*x)^2/b^6+
1/6*x^6*erfi(b*x)^2-5/4*exp(b^2*x^2)*x*erfi(b*x)/b^5/Pi^(1/2)+5/6*exp(b^2*x^2)*x^3*erfi(b*x)/b^3/Pi^(1/2)-1/3*
exp(b^2*x^2)*x^5*erfi(b*x)/b/Pi^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6501, 6522, 6510, 30, 2240, 2243} \[ \int x^5 \text {erfi}(b x)^2 \, dx=\frac {5 \text {erfi}(b x)^2}{16 b^6}-\frac {x^5 e^{b^2 x^2} \text {erfi}(b x)}{3 \sqrt {\pi } b}+\frac {x^4 e^{2 b^2 x^2}}{6 \pi b^2}+\frac {11 e^{2 b^2 x^2}}{12 \pi b^6}-\frac {5 x e^{b^2 x^2} \text {erfi}(b x)}{4 \sqrt {\pi } b^5}-\frac {7 x^2 e^{2 b^2 x^2}}{12 \pi b^4}+\frac {5 x^3 e^{b^2 x^2} \text {erfi}(b x)}{6 \sqrt {\pi } b^3}+\frac {1}{6} x^6 \text {erfi}(b x)^2 \]

[In]

Int[x^5*Erfi[b*x]^2,x]

[Out]

(11*E^(2*b^2*x^2))/(12*b^6*Pi) - (7*E^(2*b^2*x^2)*x^2)/(12*b^4*Pi) + (E^(2*b^2*x^2)*x^4)/(6*b^2*Pi) - (5*E^(b^
2*x^2)*x*Erfi[b*x])/(4*b^5*Sqrt[Pi]) + (5*E^(b^2*x^2)*x^3*Erfi[b*x])/(6*b^3*Sqrt[Pi]) - (E^(b^2*x^2)*x^5*Erfi[
b*x])/(3*b*Sqrt[Pi]) + (5*Erfi[b*x]^2)/(16*b^6) + (x^6*Erfi[b*x]^2)/6

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 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 6501

Int[Erfi[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erfi[b*x]^2/(m + 1)), x] - Dist[4*(b/(Sqrt[Pi]
*(m + 1))), Int[x^(m + 1)*E^(b^2*x^2)*Erfi[b*x], x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 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 {1}{6} x^6 \text {erfi}(b x)^2-\frac {(2 b) \int e^{b^2 x^2} x^6 \text {erfi}(b x) \, dx}{3 \sqrt {\pi }} \\ & = -\frac {e^{b^2 x^2} x^5 \text {erfi}(b x)}{3 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfi}(b x)^2+\frac {2 \int e^{2 b^2 x^2} x^5 \, dx}{3 \pi }+\frac {5 \int e^{b^2 x^2} x^4 \text {erfi}(b x) \, dx}{3 b \sqrt {\pi }} \\ & = \frac {e^{2 b^2 x^2} x^4}{6 b^2 \pi }+\frac {5 e^{b^2 x^2} x^3 \text {erfi}(b x)}{6 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^5 \text {erfi}(b x)}{3 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfi}(b x)^2-\frac {2 \int e^{2 b^2 x^2} x^3 \, dx}{3 b^2 \pi }-\frac {5 \int e^{2 b^2 x^2} x^3 \, dx}{3 b^2 \pi }-\frac {5 \int e^{b^2 x^2} x^2 \text {erfi}(b x) \, dx}{2 b^3 \sqrt {\pi }} \\ & = -\frac {7 e^{2 b^2 x^2} x^2}{12 b^4 \pi }+\frac {e^{2 b^2 x^2} x^4}{6 b^2 \pi }-\frac {5 e^{b^2 x^2} x \text {erfi}(b x)}{4 b^5 \sqrt {\pi }}+\frac {5 e^{b^2 x^2} x^3 \text {erfi}(b x)}{6 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^5 \text {erfi}(b x)}{3 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfi}(b x)^2+\frac {\int e^{2 b^2 x^2} x \, dx}{3 b^4 \pi }+\frac {5 \int e^{2 b^2 x^2} x \, dx}{6 b^4 \pi }+\frac {5 \int e^{2 b^2 x^2} x \, dx}{2 b^4 \pi }+\frac {5 \int e^{b^2 x^2} \text {erfi}(b x) \, dx}{4 b^5 \sqrt {\pi }} \\ & = \frac {11 e^{2 b^2 x^2}}{12 b^6 \pi }-\frac {7 e^{2 b^2 x^2} x^2}{12 b^4 \pi }+\frac {e^{2 b^2 x^2} x^4}{6 b^2 \pi }-\frac {5 e^{b^2 x^2} x \text {erfi}(b x)}{4 b^5 \sqrt {\pi }}+\frac {5 e^{b^2 x^2} x^3 \text {erfi}(b x)}{6 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^5 \text {erfi}(b x)}{3 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfi}(b x)^2+\frac {5 \text {Subst}(\int x \, dx,x,\text {erfi}(b x))}{8 b^6} \\ & = \frac {11 e^{2 b^2 x^2}}{12 b^6 \pi }-\frac {7 e^{2 b^2 x^2} x^2}{12 b^4 \pi }+\frac {e^{2 b^2 x^2} x^4}{6 b^2 \pi }-\frac {5 e^{b^2 x^2} x \text {erfi}(b x)}{4 b^5 \sqrt {\pi }}+\frac {5 e^{b^2 x^2} x^3 \text {erfi}(b x)}{6 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^5 \text {erfi}(b x)}{3 b \sqrt {\pi }}+\frac {5 \text {erfi}(b x)^2}{16 b^6}+\frac {1}{6} x^6 \text {erfi}(b x)^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.57 \[ \int x^5 \text {erfi}(b x)^2 \, dx=\frac {4 e^{2 b^2 x^2} \left (11-7 b^2 x^2+2 b^4 x^4\right )-4 b e^{b^2 x^2} \sqrt {\pi } x \left (15-10 b^2 x^2+4 b^4 x^4\right ) \text {erfi}(b x)+\pi \left (15+8 b^6 x^6\right ) \text {erfi}(b x)^2}{48 b^6 \pi } \]

[In]

Integrate[x^5*Erfi[b*x]^2,x]

[Out]

(4*E^(2*b^2*x^2)*(11 - 7*b^2*x^2 + 2*b^4*x^4) - 4*b*E^(b^2*x^2)*Sqrt[Pi]*x*(15 - 10*b^2*x^2 + 4*b^4*x^4)*Erfi[
b*x] + Pi*(15 + 8*b^6*x^6)*Erfi[b*x]^2)/(48*b^6*Pi)

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.88

method	result	size
parallelrisch	\(\frac {8 \operatorname {erfi}\left (b x \right )^{2} x^{6} b^{6} \pi ^{\frac {3}{2}}-16 \,\operatorname {erfi}\left (b x \right ) {\mathrm e}^{b^{2} x^{2}} x^{5} b^{5} \pi +8 \,{\mathrm e}^{2 b^{2} x^{2}} x^{4} b^{4} \sqrt {\pi }+40 \,\operatorname {erfi}\left (b x \right ) {\mathrm e}^{b^{2} x^{2}} x^{3} b^{3} \pi -28 \,{\mathrm e}^{2 b^{2} x^{2}} x^{2} b^{2} \sqrt {\pi }-60 \,\operatorname {erfi}\left (b x \right ) x \,{\mathrm e}^{b^{2} x^{2}} b \pi +15 \operatorname {erfi}\left (b x \right )^{2} \pi ^{\frac {3}{2}}+44 \,{\mathrm e}^{2 b^{2} x^{2}} \sqrt {\pi }}{48 b^{6} \pi ^{\frac {3}{2}}}\)	\(154\)

[In]

int(x^5*erfi(b*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/48*(8*erfi(b*x)^2*x^6*b^6*Pi^(3/2)-16*erfi(b*x)*exp(b^2*x^2)*x^5*b^5*Pi+8*exp(b^2*x^2)^2*x^4*b^4*Pi^(1/2)+40
*erfi(b*x)*exp(b^2*x^2)*x^3*b^3*Pi-28*exp(b^2*x^2)^2*x^2*b^2*Pi^(1/2)-60*erfi(b*x)*x*exp(b^2*x^2)*b*Pi+15*erfi
(b*x)^2*Pi^(3/2)+44*exp(b^2*x^2)^2*Pi^(1/2))/b^6/Pi^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.55 \[ \int x^5 \text {erfi}(b x)^2 \, dx=-\frac {4 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} - 10 \, b^{3} x^{3} + 15 \, b x\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - {\left (15 \, \pi + 8 \, \pi b^{6} x^{6}\right )} \operatorname {erfi}\left (b x\right )^{2} - 4 \, {\left (2 \, b^{4} x^{4} - 7 \, b^{2} x^{2} + 11\right )} e^{\left (2 \, b^{2} x^{2}\right )}}{48 \, \pi b^{6}} \]

[In]

integrate(x^5*erfi(b*x)^2,x, algorithm="fricas")

[Out]

-1/48*(4*sqrt(pi)*(4*b^5*x^5 - 10*b^3*x^3 + 15*b*x)*erfi(b*x)*e^(b^2*x^2) - (15*pi + 8*pi*b^6*x^6)*erfi(b*x)^2
- 4*(2*b^4*x^4 - 7*b^2*x^2 + 11)*e^(2*b^2*x^2))/(pi*b^6)

Sympy [A] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.96 \[ \int x^5 \text {erfi}(b x)^2 \, dx=\begin {cases} \frac {x^{6} \operatorname {erfi}^{2}{\left (b x \right )}}{6} - \frac {x^{5} e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{3 \sqrt {\pi } b} + \frac {x^{4} e^{2 b^{2} x^{2}}}{6 \pi b^{2}} + \frac {5 x^{3} e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{6 \sqrt {\pi } b^{3}} - \frac {7 x^{2} e^{2 b^{2} x^{2}}}{12 \pi b^{4}} - \frac {5 x e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{4 \sqrt {\pi } b^{5}} + \frac {11 e^{2 b^{2} x^{2}}}{12 \pi b^{6}} + \frac {5 \operatorname {erfi}^{2}{\left (b x \right )}}{16 b^{6}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(x**5*erfi(b*x)**2,x)

[Out]

Piecewise((x**6*erfi(b*x)**2/6 - x**5*exp(b**2*x**2)*erfi(b*x)/(3*sqrt(pi)*b) + x**4*exp(2*b**2*x**2)/(6*pi*b*
*2) + 5*x**3*exp(b**2*x**2)*erfi(b*x)/(6*sqrt(pi)*b**3) - 7*x**2*exp(2*b**2*x**2)/(12*pi*b**4) - 5*x*exp(b**2*
x**2)*erfi(b*x)/(4*sqrt(pi)*b**5) + 11*exp(2*b**2*x**2)/(12*pi*b**6) + 5*erfi(b*x)**2/(16*b**6), Ne(b, 0)), (0
, True))

Maxima [F]

\[ \int x^5 \text {erfi}(b x)^2 \, dx=\int { x^{5} \operatorname {erfi}\left (b x\right )^{2} \,d x } \]

[In]

integrate(x^5*erfi(b*x)^2,x, algorithm="maxima")

[Out]

integrate(x^5*erfi(b*x)^2, x)

Giac [F]

\[ \int x^5 \text {erfi}(b x)^2 \, dx=\int { x^{5} \operatorname {erfi}\left (b x\right )^{2} \,d x } \]

[In]

integrate(x^5*erfi(b*x)^2,x, algorithm="giac")

[Out]

integrate(x^5*erfi(b*x)^2, x)

Mupad [B] (verification not implemented)

Time = 5.27 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.79 \[ \int x^5 \text {erfi}(b x)^2 \, dx=\frac {x^6\,{\mathrm {erfi}\left (b\,x\right )}^2}{6}+\frac {\frac {11\,{\mathrm {e}}^{2\,b^2\,x^2}}{12}+\frac {5\,\pi \,{\mathrm {erfi}\left (b\,x\right )}^2}{16}-\frac {7\,b^2\,x^2\,{\mathrm {e}}^{2\,b^2\,x^2}}{12}+\frac {b^4\,x^4\,{\mathrm {e}}^{2\,b^2\,x^2}}{6}+\frac {5\,b^3\,x^3\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{6}-\frac {b^5\,x^5\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{3}-\frac {5\,b\,x\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{4}}{b^6\,\pi } \]

[In]

int(x^5*erfi(b*x)^2,x)

[Out]

(x^6*erfi(b*x)^2)/6 + ((11*exp(2*b^2*x^2))/12 + (5*pi*erfi(b*x)^2)/16 - (7*b^2*x^2*exp(2*b^2*x^2))/12 + (b^4*x
^4*exp(2*b^2*x^2))/6 + (5*b^3*x^3*pi^(1/2)*exp(b^2*x^2)*erfi(b*x))/6 - (b^5*x^5*pi^(1/2)*exp(b^2*x^2)*erfi(b*x
))/3 - (5*b*x*pi^(1/2)*exp(b^2*x^2)*erfi(b*x))/4)/(b^6*pi)