\(\int x^5 \text {erfi}(b x) \, dx\) [207]

Optimal result

Integrand size = 8, antiderivative size = 93 \[ \int x^5 \text {erfi}(b x) \, dx=-\frac {5 e^{b^2 x^2} x}{8 b^5 \sqrt {\pi }}+\frac {5 e^{b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {5 \text {erfi}(b x)}{16 b^6}+\frac {1}{6} x^6 \text {erfi}(b x) \]

[Out]

5/16*erfi(b*x)/b^6+1/6*x^6*erfi(b*x)-5/8*exp(b^2*x^2)*x/b^5/Pi^(1/2)+5/12*exp(b^2*x^2)*x^3/b^3/Pi^(1/2)-1/6*ex
p(b^2*x^2)*x^5/b/Pi^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6498, 2243, 2235} \[ \int x^5 \text {erfi}(b x) \, dx=\frac {5 \text {erfi}(b x)}{16 b^6}-\frac {x^5 e^{b^2 x^2}}{6 \sqrt {\pi } b}-\frac {5 x e^{b^2 x^2}}{8 \sqrt {\pi } b^5}+\frac {5 x^3 e^{b^2 x^2}}{12 \sqrt {\pi } b^3}+\frac {1}{6} x^6 \text {erfi}(b x) \]

[In]

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

[Out]

(-5*E^(b^2*x^2)*x)/(8*b^5*Sqrt[Pi]) + (5*E^(b^2*x^2)*x^3)/(12*b^3*Sqrt[Pi]) - (E^(b^2*x^2)*x^5)/(6*b*Sqrt[Pi])
+ (5*Erfi[b*x])/(16*b^6) + (x^6*Erfi[b*x])/6

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}{6} x^6 \text {erfi}(b x)-\frac {b \int e^{b^2 x^2} x^6 \, dx}{3 \sqrt {\pi }} \\ & = -\frac {e^{b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfi}(b x)+\frac {5 \int e^{b^2 x^2} x^4 \, dx}{6 b \sqrt {\pi }} \\ & = \frac {5 e^{b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfi}(b x)-\frac {5 \int e^{b^2 x^2} x^2 \, dx}{4 b^3 \sqrt {\pi }} \\ & = -\frac {5 e^{b^2 x^2} x}{8 b^5 \sqrt {\pi }}+\frac {5 e^{b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfi}(b x)+\frac {5 \int e^{b^2 x^2} \, dx}{8 b^5 \sqrt {\pi }} \\ & = -\frac {5 e^{b^2 x^2} x}{8 b^5 \sqrt {\pi }}+\frac {5 e^{b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {5 \text {erfi}(b x)}{16 b^6}+\frac {1}{6} x^6 \text {erfi}(b x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.69 \[ \int x^5 \text {erfi}(b x) \, dx=\frac {-2 b e^{b^2 x^2} x \left (15-10 b^2 x^2+4 b^4 x^4\right )+\sqrt {\pi } \left (15+8 b^6 x^6\right ) \text {erfi}(b x)}{48 b^6 \sqrt {\pi }} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.67

method	result	size
meijerg	\(\frac {i \left (\frac {i x b \left (28 b^{4} x^{4}-70 b^{2} x^{2}+105\right ) {\mathrm e}^{b^{2} x^{2}}}{84}-\frac {i \left (56 b^{6} x^{6}+105\right ) \operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{168}\right )}{2 b^{6} \sqrt {\pi }}\)	\(62\)
derivativedivides	\(\frac {\frac {b^{6} x^{6} \operatorname {erfi}\left (b x \right )}{6}-\frac {\frac {{\mathrm e}^{b^{2} x^{2}} b^{5} x^{5}}{2}-\frac {5 \,{\mathrm e}^{b^{2} x^{2}} b^{3} x^{3}}{4}+\frac {15 \,{\mathrm e}^{b^{2} x^{2}} b x}{8}-\frac {15 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{16}}{3 \sqrt {\pi }}}{b^{6}}\)	\(77\)
default	\(\frac {\frac {b^{6} x^{6} \operatorname {erfi}\left (b x \right )}{6}-\frac {\frac {{\mathrm e}^{b^{2} x^{2}} b^{5} x^{5}}{2}-\frac {5 \,{\mathrm e}^{b^{2} x^{2}} b^{3} x^{3}}{4}+\frac {15 \,{\mathrm e}^{b^{2} x^{2}} b x}{8}-\frac {15 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{16}}{3 \sqrt {\pi }}}{b^{6}}\)	\(77\)
parallelrisch	\(\frac {8 \,\operatorname {erfi}\left (b x \right ) x^{6} b^{6} \sqrt {\pi }-8 \,{\mathrm e}^{b^{2} x^{2}} b^{5} x^{5}+20 \,{\mathrm e}^{b^{2} x^{2}} b^{3} x^{3}-30 \,{\mathrm e}^{b^{2} x^{2}} b x +15 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{48 b^{6} \sqrt {\pi }}\)	\(78\)
parts	\(\frac {x^{6} \operatorname {erfi}\left (b x \right )}{6}-\frac {b \left (\frac {x^{5} {\mathrm e}^{b^{2} x^{2}}}{2 b^{2}}-\frac {5 \left (\frac {x^{3} {\mathrm e}^{b^{2} x^{2}}}{2 b^{2}}-\frac {3 \left (\frac {x \,{\mathrm e}^{b^{2} x^{2}}}{2 b^{2}}+\frac {i \sqrt {\pi }\, \operatorname {erf}\left (i b x \right )}{4 b^{3}}\right )}{2 b^{2}}\right )}{2 b^{2}}\right )}{3 \sqrt {\pi }}\)	\(91\)

[In]

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

[Out]

1/2*I/b^6/Pi^(1/2)*(1/84*I*x*b*(28*b^4*x^4-70*b^2*x^2+105)*exp(b^2*x^2)-1/168*I*(56*b^6*x^6+105)*erfi(b*x)*Pi^
(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.67 \[ \int x^5 \text {erfi}(b x) \, dx=-\frac {2 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} - 10 \, b^{3} x^{3} + 15 \, b x\right )} e^{\left (b^{2} x^{2}\right )} - {\left (15 \, \pi + 8 \, \pi b^{6} x^{6}\right )} \operatorname {erfi}\left (b x\right )}{48 \, \pi b^{6}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.95 \[ \int x^5 \text {erfi}(b x) \, dx=\begin {cases} \frac {x^{6} \operatorname {erfi}{\left (b x \right )}}{6} - \frac {x^{5} e^{b^{2} x^{2}}}{6 \sqrt {\pi } b} + \frac {5 x^{3} e^{b^{2} x^{2}}}{12 \sqrt {\pi } b^{3}} - \frac {5 x e^{b^{2} x^{2}}}{8 \sqrt {\pi } b^{5}} + \frac {5 \operatorname {erfi}{\left (b x \right )}}{16 b^{6}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x**6*erfi(b*x)/6 - x**5*exp(b**2*x**2)/(6*sqrt(pi)*b) + 5*x**3*exp(b**2*x**2)/(12*sqrt(pi)*b**3) -
5*x*exp(b**2*x**2)/(8*sqrt(pi)*b**5) + 5*erfi(b*x)/(16*b**6), Ne(b, 0)), (0, True))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.19 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.68 \[ \int x^5 \text {erfi}(b x) \, dx=\frac {1}{6} \, x^{6} \operatorname {erfi}\left (b x\right ) - \frac {b {\left (\frac {2 \, {\left (4 \, b^{4} x^{5} - 10 \, b^{2} x^{3} + 15 \, x\right )} e^{\left (b^{2} x^{2}\right )}}{b^{6}} + \frac {15 i \, \sqrt {\pi } \operatorname {erf}\left (i \, b x\right )}{b^{7}}\right )}}{48 \, \sqrt {\pi }} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(x^5*erfi(b*x), x)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.16 \[ \int x^5 \text {erfi}(b x) \, dx=\frac {x^6\,\mathrm {erfi}\left (b\,x\right )}{6}-\frac {5\,b\,x^7}{16\,{\left (-b^2\,x^2\right )}^{7/2}}-\frac {x^5\,{\mathrm {e}}^{b^2\,x^2}}{6\,b\,\sqrt {\pi }}+\frac {5\,x^3\,{\mathrm {e}}^{b^2\,x^2}}{12\,b^3\,\sqrt {\pi }}-\frac {5\,x\,{\mathrm {e}}^{b^2\,x^2}}{8\,b^5\,\sqrt {\pi }}+\frac {5\,b\,x^7\,\mathrm {erfc}\left (\sqrt {-b^2\,x^2}\right )}{16\,{\left (-b^2\,x^2\right )}^{7/2}} \]

[In]

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

[Out]

(x^6*erfi(b*x))/6 - (5*b*x^7)/(16*(-b^2*x^2)^(7/2)) - (x^5*exp(b^2*x^2))/(6*b*pi^(1/2)) + (5*x^3*exp(b^2*x^2))
/(12*b^3*pi^(1/2)) - (5*x*exp(b^2*x^2))/(8*b^5*pi^(1/2)) + (5*b*x^7*erfc((-b^2*x^2)^(1/2)))/(16*(-b^2*x^2)^(7/
2))