Integrand size = 8, antiderivative size = 105 \[ \int x^6 \text {erfi}(b x) \, dx=\frac {6 e^{b^2 x^2}}{7 b^7 \sqrt {\pi }}-\frac {6 e^{b^2 x^2} x^2}{7 b^5 \sqrt {\pi }}+\frac {3 e^{b^2 x^2} x^4}{7 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^6}{7 b \sqrt {\pi }}+\frac {1}{7} x^7 \text {erfi}(b x) \] Output:
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)+1/7*x^7*erfi(b*x)
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.54 \[ \int x^6 \text {erfi}(b x) \, dx=\frac {1}{7} \left (\frac {e^{b^2 x^2} \left (6-6 b^2 x^2+3 b^4 x^4-b^6 x^6\right )}{b^7 \sqrt {\pi }}+x^7 \text {erfi}(b x)\right ) \] Input:
Integrate[x^6*Erfi[b*x],x]
Output:
((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
Time = 0.38 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6917, 2641, 2641, 2641, 2638}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^6 \text {erfi}(b x) \, dx\) |
\(\Big \downarrow \) 6917 |
\(\displaystyle \frac {1}{7} x^7 \text {erfi}(b x)-\frac {2 b \int e^{b^2 x^2} x^7dx}{7 \sqrt {\pi }}\) |
\(\Big \downarrow \) 2641 |
\(\displaystyle \frac {1}{7} x^7 \text {erfi}(b x)-\frac {2 b \left (\frac {x^6 e^{b^2 x^2}}{2 b^2}-\frac {3 \int e^{b^2 x^2} x^5dx}{b^2}\right )}{7 \sqrt {\pi }}\) |
\(\Big \downarrow \) 2641 |
\(\displaystyle \frac {1}{7} x^7 \text {erfi}(b x)-\frac {2 b \left (\frac {x^6 e^{b^2 x^2}}{2 b^2}-\frac {3 \left (\frac {x^4 e^{b^2 x^2}}{2 b^2}-\frac {2 \int e^{b^2 x^2} x^3dx}{b^2}\right )}{b^2}\right )}{7 \sqrt {\pi }}\) |
\(\Big \downarrow \) 2641 |
\(\displaystyle \frac {1}{7} x^7 \text {erfi}(b x)-\frac {2 b \left (\frac {x^6 e^{b^2 x^2}}{2 b^2}-\frac {3 \left (\frac {x^4 e^{b^2 x^2}}{2 b^2}-\frac {2 \left (\frac {x^2 e^{b^2 x^2}}{2 b^2}-\frac {\int e^{b^2 x^2} xdx}{b^2}\right )}{b^2}\right )}{b^2}\right )}{7 \sqrt {\pi }}\) |
\(\Big \downarrow \) 2638 |
\(\displaystyle \frac {1}{7} x^7 \text {erfi}(b x)-\frac {2 b \left (\frac {x^6 e^{b^2 x^2}}{2 b^2}-\frac {3 \left (\frac {x^4 e^{b^2 x^2}}{2 b^2}-\frac {2 \left (\frac {x^2 e^{b^2 x^2}}{2 b^2}-\frac {e^{b^2 x^2}}{2 b^4}\right )}{b^2}\right )}{b^2}\right )}{7 \sqrt {\pi }}\) |
Input:
Int[x^6*Erfi[b*x],x]
Output:
(-2*b*((E^(b^2*x^2)*x^6)/(2*b^2) - (3*((E^(b^2*x^2)*x^4)/(2*b^2) - (2*(-1/ 2*E^(b^2*x^2)/b^4 + (E^(b^2*x^2)*x^2)/(2*b^2)))/b^2))/b^2))/(7*Sqrt[Pi]) + (x^7*Erfi[b*x])/7
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]
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*L og[F])), x] - Simp[(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])
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] - Simp[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]
Time = 0.41 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.59
method | result | size |
meijerg | \(\frac {-\frac {12}{7}+\frac {\left (-4 b^{6} x^{6}+12 b^{4} x^{4}-24 b^{2} x^{2}+24\right ) {\mathrm e}^{b^{2} x^{2}}}{14}+\frac {2 b^{7} x^{7} \sqrt {\pi }\, \operatorname {erfi}\left (b x \right )}{7}}{2 b^{7} \sqrt {\pi }}\) | \(62\) |
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 | \(\frac {b^{7} x^{7} \sqrt {\pi }\, \operatorname {erfi}\left (b x \right )-b^{6} x^{6} {\mathrm e}^{b^{2} x^{2}}+3 \,{\mathrm e}^{b^{2} x^{2}} b^{4} x^{4}-6 b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}+6 \,{\mathrm e}^{b^{2} x^{2}}}{7 b^{7} \sqrt {\pi }}\) | \(82\) |
parts | \(\frac {x^{7} \operatorname {erfi}\left (b x \right )}{7}-\frac {2 b \left (\frac {x^{6} {\mathrm e}^{b^{2} x^{2}}}{2 b^{2}}-\frac {3 \left (\frac {x^{4} {\mathrm e}^{b^{2} x^{2}}}{2 b^{2}}-\frac {2 \left (\frac {x^{2} {\mathrm e}^{b^{2} x^{2}}}{2 b^{2}}-\frac {{\mathrm e}^{b^{2} x^{2}}}{2 b^{4}}\right )}{b^{2}}\right )}{b^{2}}\right )}{7 \sqrt {\pi }}\) | \(91\) |
Input:
int(x^6*erfi(b*x),x,method=_RETURNVERBOSE)
Output:
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*b^7*x^7*Pi^(1/2)*erfi(b*x))
Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.56 \[ \int x^6 \text {erfi}(b x) \, dx=\frac {\pi b^{7} x^{7} \operatorname {erfi}\left (b x\right ) - \sqrt {\pi } {\left (b^{6} x^{6} - 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} - 6\right )} e^{\left (b^{2} x^{2}\right )}}{7 \, \pi b^{7}} \] Input:
integrate(x^6*erfi(b*x),x, algorithm="fricas")
Output:
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)
Time = 0.59 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.94 \[ \int x^6 \text {erfi}(b x) \, dx=\begin {cases} \frac {x^{7} \operatorname {erfi}{\left (b x \right )}}{7} - \frac {x^{6} e^{b^{2} x^{2}}}{7 \sqrt {\pi } b} + \frac {3 x^{4} e^{b^{2} x^{2}}}{7 \sqrt {\pi } b^{3}} - \frac {6 x^{2} e^{b^{2} x^{2}}}{7 \sqrt {\pi } b^{5}} + \frac {6 e^{b^{2} x^{2}}}{7 \sqrt {\pi } b^{7}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x**6*erfi(b*x),x)
Output:
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))
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.49 \[ \int x^6 \text {erfi}(b x) \, dx=\frac {1}{7} \, x^{7} \operatorname {erfi}\left (b x\right ) - \frac {{\left (b^{6} x^{6} - 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} - 6\right )} e^{\left (b^{2} x^{2}\right )}}{7 \, \sqrt {\pi } b^{7}} \] Input:
integrate(x^6*erfi(b*x),x, algorithm="maxima")
Output:
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)
\[ \int x^6 \text {erfi}(b x) \, dx=\int { x^{6} \operatorname {erfi}\left (b x\right ) \,d x } \] Input:
integrate(x^6*erfi(b*x),x, algorithm="giac")
Output:
integrate(x^6*erfi(b*x), x)
Time = 0.11 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.49 \[ \int x^6 \text {erfi}(b x) \, dx=\frac {x^7\,\mathrm {erfi}\left (b\,x\right )}{7}-\frac {{\mathrm {e}}^{b^2\,x^2}\,\left (b^6\,x^6-3\,b^4\,x^4+6\,b^2\,x^2-6\right )}{7\,b^7\,\sqrt {\pi }} \] Input:
int(x^6*erfi(b*x),x)
Output:
(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))
Time = 0.16 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int x^6 \text {erfi}(b x) \, dx=\frac {-\mathrm {erf}\left (b i x \right ) b^{7} i \pi \,x^{7}-\sqrt {\pi }\, e^{b^{2} x^{2}} b^{6} x^{6}+3 \sqrt {\pi }\, e^{b^{2} x^{2}} b^{4} x^{4}-6 \sqrt {\pi }\, e^{b^{2} x^{2}} b^{2} x^{2}+6 \sqrt {\pi }\, e^{b^{2} x^{2}}}{7 b^{7} \pi } \] Input:
int(x^6*erfi(b*x),x)
Output:
( - erf(b*i*x)*b**7*i*pi*x**7 - sqrt(pi)*e**(b**2*x**2)*b**6*x**6 + 3*sqrt (pi)*e**(b**2*x**2)*b**4*x**4 - 6*sqrt(pi)*e**(b**2*x**2)*b**2*x**2 + 6*sq rt(pi)*e**(b**2*x**2))/(7*b**7*pi)