\(\int x^6 \text {erfi}(b x) \, dx\) [214]

Optimal result

Integrand size = 8, antiderivative size = 105 \[ \int x^6 \text {erfi}(b x) \, dx=\frac {6 e^{b^2 x^2}}{7 b^7 \sqrt {\pi }}-\frac {6 e^{b^2 x^2} x^2}{7 b^5 \sqrt {\pi }}+\frac {3 e^{b^2 x^2} x^4}{7 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^6}{7 b \sqrt {\pi }}+\frac {1}{7} x^7 \text {erfi}(b x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 105, 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, 2240} \[ \int x^6 \text {erfi}(b x) \, dx=-\frac {x^6 e^{b^2 x^2}}{7 \sqrt {\pi } b}+\frac {6 e^{b^2 x^2}}{7 \sqrt {\pi } b^7}-\frac {6 x^2 e^{b^2 x^2}}{7 \sqrt {\pi } b^5}+\frac {3 x^4 e^{b^2 x^2}}{7 \sqrt {\pi } b^3}+\frac {1}{7} x^7 \text {erfi}(b x) \]

[In]

Int[x^6*Erfi[b*x],x]

[Out]

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

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 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}{7} x^7 \text {erfi}(b x)-\frac {(2 b) \int e^{b^2 x^2} x^7 \, dx}{7 \sqrt {\pi }} \\ & = -\frac {e^{b^2 x^2} x^6}{7 b \sqrt {\pi }}+\frac {1}{7} x^7 \text {erfi}(b x)+\frac {6 \int e^{b^2 x^2} x^5 \, dx}{7 b \sqrt {\pi }} \\ & = \frac {3 e^{b^2 x^2} x^4}{7 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^6}{7 b \sqrt {\pi }}+\frac {1}{7} x^7 \text {erfi}(b x)-\frac {12 \int e^{b^2 x^2} x^3 \, dx}{7 b^3 \sqrt {\pi }} \\ & = -\frac {6 e^{b^2 x^2} x^2}{7 b^5 \sqrt {\pi }}+\frac {3 e^{b^2 x^2} x^4}{7 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^6}{7 b \sqrt {\pi }}+\frac {1}{7} x^7 \text {erfi}(b x)+\frac {12 \int e^{b^2 x^2} x \, dx}{7 b^5 \sqrt {\pi }} \\ & = \frac {6 e^{b^2 x^2}}{7 b^7 \sqrt {\pi }}-\frac {6 e^{b^2 x^2} x^2}{7 b^5 \sqrt {\pi }}+\frac {3 e^{b^2 x^2} x^4}{7 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^6}{7 b \sqrt {\pi }}+\frac {1}{7} x^7 \text {erfi}(b x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.54 \[ \int x^6 \text {erfi}(b x) \, dx=\frac {1}{7} \left (\frac {e^{b^2 x^2} \left (6-6 b^2 x^2+3 b^4 x^4-b^6 x^6\right )}{b^7 \sqrt {\pi }}+x^7 \text {erfi}(b x)\right ) \]

[In]

Integrate[x^6*Erfi[b*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.59

method	result	size
meijerg	\(\frac {-\frac {12}{7}+\frac {\left (-4 b^{6} x^{6}+12 b^{4} x^{4}-24 b^{2} x^{2}+24\right ) {\mathrm e}^{b^{2} x^{2}}}{14}+\frac {2 x^{7} b^{7} \sqrt {\pi }\, \operatorname {erfi}\left (b x \right )}{7}}{2 b^{7} \sqrt {\pi }}\)	\(62\)
derivativedivides	\(\frac {\frac {b^{7} x^{7} \operatorname {erfi}\left (b x \right )}{7}-\frac {2 \left (\frac {b^{6} x^{6} {\mathrm e}^{b^{2} x^{2}}}{2}-\frac {3 \,{\mathrm e}^{b^{2} x^{2}} b^{4} x^{4}}{2}+3 b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}-3 \,{\mathrm e}^{b^{2} x^{2}}\right )}{7 \sqrt {\pi }}}{b^{7}}\)	\(82\)
default	\(\frac {\frac {b^{7} x^{7} \operatorname {erfi}\left (b x \right )}{7}-\frac {2 \left (\frac {b^{6} x^{6} {\mathrm e}^{b^{2} x^{2}}}{2}-\frac {3 \,{\mathrm e}^{b^{2} x^{2}} b^{4} x^{4}}{2}+3 b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}-3 \,{\mathrm e}^{b^{2} x^{2}}\right )}{7 \sqrt {\pi }}}{b^{7}}\)	\(82\)
parallelrisch	\(\frac {x^{7} b^{7} \sqrt {\pi }\, \operatorname {erfi}\left (b x \right )-b^{6} x^{6} {\mathrm e}^{b^{2} x^{2}}+3 \,{\mathrm e}^{b^{2} x^{2}} b^{4} x^{4}-6 b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}+6 \,{\mathrm e}^{b^{2} x^{2}}}{7 b^{7} \sqrt {\pi }}\)	\(82\)
parts	\(\frac {x^{7} \operatorname {erfi}\left (b x \right )}{7}-\frac {2 b \left (\frac {x^{6} {\mathrm e}^{b^{2} x^{2}}}{2 b^{2}}-\frac {3 \left (\frac {x^{4} {\mathrm e}^{b^{2} x^{2}}}{2 b^{2}}-\frac {2 \left (\frac {x^{2} {\mathrm e}^{b^{2} x^{2}}}{2 b^{2}}-\frac {{\mathrm e}^{b^{2} x^{2}}}{2 b^{4}}\right )}{b^{2}}\right )}{b^{2}}\right )}{7 \sqrt {\pi }}\)	\(91\)

[In]

int(x^6*erfi(b*x),x,method=_RETURNVERBOSE)

[Out]

1/2/b^7/Pi^(1/2)*(-12/7+1/14*(-4*b^6*x^6+12*b^4*x^4-24*b^2*x^2+24)*exp(b^2*x^2)+2/7*x^7*b^7*Pi^(1/2)*erfi(b*x)
)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.56 \[ \int x^6 \text {erfi}(b x) \, dx=\frac {\pi b^{7} x^{7} \operatorname {erfi}\left (b x\right ) - \sqrt {\pi } {\left (b^{6} x^{6} - 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} - 6\right )} e^{\left (b^{2} x^{2}\right )}}{7 \, \pi b^{7}} \]

[In]

integrate(x^6*erfi(b*x),x, algorithm="fricas")

[Out]

1/7*(pi*b^7*x^7*erfi(b*x) - sqrt(pi)*(b^6*x^6 - 3*b^4*x^4 + 6*b^2*x^2 - 6)*e^(b^2*x^2))/(pi*b^7)

Sympy [A] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.94 \[ \int x^6 \text {erfi}(b x) \, dx=\begin {cases} \frac {x^{7} \operatorname {erfi}{\left (b x \right )}}{7} - \frac {x^{6} e^{b^{2} x^{2}}}{7 \sqrt {\pi } b} + \frac {3 x^{4} e^{b^{2} x^{2}}}{7 \sqrt {\pi } b^{3}} - \frac {6 x^{2} e^{b^{2} x^{2}}}{7 \sqrt {\pi } b^{5}} + \frac {6 e^{b^{2} x^{2}}}{7 \sqrt {\pi } b^{7}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(x**6*erfi(b*x),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.49 \[ \int x^6 \text {erfi}(b x) \, dx=\frac {1}{7} \, x^{7} \operatorname {erfi}\left (b x\right ) - \frac {{\left (b^{6} x^{6} - 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} - 6\right )} e^{\left (b^{2} x^{2}\right )}}{7 \, \sqrt {\pi } b^{7}} \]

[In]

integrate(x^6*erfi(b*x),x, algorithm="maxima")

[Out]

1/7*x^7*erfi(b*x) - 1/7*(b^6*x^6 - 3*b^4*x^4 + 6*b^2*x^2 - 6)*e^(b^2*x^2)/(sqrt(pi)*b^7)

Giac [F]

\[ \int x^6 \text {erfi}(b x) \, dx=\int { x^{6} \operatorname {erfi}\left (b x\right ) \,d x } \]

[In]

integrate(x^6*erfi(b*x),x, algorithm="giac")

[Out]

integrate(x^6*erfi(b*x), x)

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.49 \[ \int x^6 \text {erfi}(b x) \, dx=\frac {x^7\,\mathrm {erfi}\left (b\,x\right )}{7}-\frac {{\mathrm {e}}^{b^2\,x^2}\,\left (b^6\,x^6-3\,b^4\,x^4+6\,b^2\,x^2-6\right )}{7\,b^7\,\sqrt {\pi }} \]

[In]

int(x^6*erfi(b*x),x)

[Out]

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