Integrand size = 8, antiderivative size = 109 \[ \int x^6 \text {erf}(b x) \, dx=\frac {6 e^{-b^2 x^2}}{7 b^7 \sqrt {\pi }}+\frac {6 e^{-b^2 x^2} x^2}{7 b^5 \sqrt {\pi }}+\frac {3 e^{-b^2 x^2} x^4}{7 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^6}{7 b \sqrt {\pi }}+\frac {1}{7} x^7 \text {erf}(b x) \] Output:
6/7/b^7/exp(b^2*x^2)/Pi^(1/2)+6/7*x^2/b^5/exp(b^2*x^2)/Pi^(1/2)+3/7*x^4/b^ 3/exp(b^2*x^2)/Pi^(1/2)+1/7*x^6/b/exp(b^2*x^2)/Pi^(1/2)+1/7*x^7*erf(b*x)
Time = 0.01 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.66 \[ \int x^6 \text {erf}(b x) \, dx=\frac {e^{-b^2 x^2} \left (6+6 b^2 x^2+3 b^4 x^4+b^6 x^6+b^7 e^{b^2 x^2} \sqrt {\pi } x^7 \text {erf}(b x)\right )}{7 b^7 \sqrt {\pi }} \] Input:
Integrate[x^6*Erf[b*x],x]
Output:
(6 + 6*b^2*x^2 + 3*b^4*x^4 + b^6*x^6 + b^7*E^(b^2*x^2)*Sqrt[Pi]*x^7*Erf[b* x])/(7*b^7*E^(b^2*x^2)*Sqrt[Pi])
Time = 0.40 (sec) , antiderivative size = 112, 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 = {6915, 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.
\(\displaystyle \int x^6 \text {erf}(b x) \, dx\) |
\(\Big \downarrow \) 6915 |
\(\displaystyle \frac {1}{7} x^7 \text {erf}(b x)-\frac {2 b \int e^{-b^2 x^2} x^7dx}{7 \sqrt {\pi }}\) |
\(\Big \downarrow \) 2641 |
\(\displaystyle \frac {1}{7} x^7 \text {erf}(b x)-\frac {2 b \left (\frac {3 \int e^{-b^2 x^2} x^5dx}{b^2}-\frac {x^6 e^{-b^2 x^2}}{2 b^2}\right )}{7 \sqrt {\pi }}\) |
\(\Big \downarrow \) 2641 |
\(\displaystyle \frac {1}{7} x^7 \text {erf}(b x)-\frac {2 b \left (\frac {3 \left (\frac {2 \int e^{-b^2 x^2} x^3dx}{b^2}-\frac {x^4 e^{-b^2 x^2}}{2 b^2}\right )}{b^2}-\frac {x^6 e^{-b^2 x^2}}{2 b^2}\right )}{7 \sqrt {\pi }}\) |
\(\Big \downarrow \) 2641 |
\(\displaystyle \frac {1}{7} x^7 \text {erf}(b x)-\frac {2 b \left (\frac {3 \left (\frac {2 \left (\frac {\int e^{-b^2 x^2} xdx}{b^2}-\frac {x^2 e^{-b^2 x^2}}{2 b^2}\right )}{b^2}-\frac {x^4 e^{-b^2 x^2}}{2 b^2}\right )}{b^2}-\frac {x^6 e^{-b^2 x^2}}{2 b^2}\right )}{7 \sqrt {\pi }}\) |
\(\Big \downarrow \) 2638 |
\(\displaystyle \frac {1}{7} x^7 \text {erf}(b x)-\frac {2 b \left (\frac {3 \left (\frac {2 \left (-\frac {x^2 e^{-b^2 x^2}}{2 b^2}-\frac {e^{-b^2 x^2}}{2 b^4}\right )}{b^2}-\frac {x^4 e^{-b^2 x^2}}{2 b^2}\right )}{b^2}-\frac {x^6 e^{-b^2 x^2}}{2 b^2}\right )}{7 \sqrt {\pi }}\) |
Input:
Int[x^6*Erf[b*x],x]
Output:
(-2*b*(-1/2*x^6/(b^2*E^(b^2*x^2)) + (3*(-1/2*x^4/(b^2*E^(b^2*x^2)) + (2*(- 1/2*1/(b^4*E^(b^2*x^2)) - x^2/(2*b^2*E^(b^2*x^2))))/b^2))/b^2))/(7*Sqrt[Pi ]) + (x^7*Erf[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[Erf[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[( c + d*x)^(m + 1)*(Erf[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 = 0.61 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.58
method | result | size |
meijerg | \(\frac {-\frac {12}{7}+\frac {\left (4 b^{6} x^{6}+12 b^{4} x^{4}+24 b^{2} x^{2}+24\right ) {\mathrm e}^{-b^{2} x^{2}}}{14}+\frac {2 x^{7} b^{7} \operatorname {erf}\left (b x \right ) \sqrt {\pi }}{7}}{2 b^{7} \sqrt {\pi }}\) | \(63\) |
parallelrisch | \(\frac {x^{7} b^{7} \operatorname {erf}\left (b x \right ) \sqrt {\pi }+{\mathrm e}^{-b^{2} x^{2}} x^{6} b^{6}+3 \,{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}+6 \,{\mathrm e}^{-b^{2} x^{2}} x^{2} b^{2}+6 \,{\mathrm e}^{-b^{2} x^{2}}}{7 b^{7} \sqrt {\pi }}\) | \(85\) |
derivativedivides | \(\frac {\frac {\operatorname {erf}\left (b x \right ) b^{7} x^{7}}{7}-\frac {2 \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{6} b^{6}}{2}-\frac {3 \,{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{2}-3 \,{\mathrm e}^{-b^{2} x^{2}} x^{2} b^{2}-3 \,{\mathrm e}^{-b^{2} x^{2}}\right )}{7 \sqrt {\pi }}}{b^{7}}\) | \(90\) |
default | \(\frac {\frac {\operatorname {erf}\left (b x \right ) b^{7} x^{7}}{7}-\frac {2 \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{6} b^{6}}{2}-\frac {3 \,{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{2}-3 \,{\mathrm e}^{-b^{2} x^{2}} x^{2} b^{2}-3 \,{\mathrm e}^{-b^{2} x^{2}}\right )}{7 \sqrt {\pi }}}{b^{7}}\) | \(90\) |
parts | \(\frac {x^{7} \operatorname {erf}\left (b x \right )}{7}-\frac {2 b \left (-\frac {x^{6} {\mathrm e}^{-b^{2} x^{2}}}{2 b^{2}}+\frac {-\frac {3 x^{4} {\mathrm e}^{-b^{2} x^{2}}}{2 b^{2}}+\frac {3 \left (-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}}}{b^{2}}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{b^{4}}\right )}{b^{2}}}{b^{2}}\right )}{7 \sqrt {\pi }}\) | \(95\) |
orering | \(\frac {\left (x^{8} b^{8}-3 b^{6} x^{6}-9 b^{4} x^{4}-18 b^{2} x^{2}-18\right ) \operatorname {erf}\left (b x \right )}{7 b^{8} x}+\frac {\left (b^{6} x^{6}+3 b^{4} x^{4}+6 b^{2} x^{2}+6\right ) \left (6 x^{5} \operatorname {erf}\left (b x \right )+\frac {2 x^{6} {\mathrm e}^{-b^{2} x^{2}} b}{\sqrt {\pi }}\right )}{14 b^{8} x^{6}}\) | \(108\) |
Input:
int(x^6*erf(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*x^7*b^7*erf(b*x)*Pi^(1/2))
Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.54 \[ \int x^6 \text {erf}(b x) \, dx=\frac {\pi b^{7} x^{7} \operatorname {erf}\left (b x\right ) + \sqrt {\pi } {\left (b^{6} x^{6} + 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} + 6\right )} e^{\left (-b^{2} x^{2}\right )}}{7 \, \pi b^{7}} \] Input:
integrate(x^6*erf(b*x),x, algorithm="fricas")
Output:
1/7*(pi*b^7*x^7*erf(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 = 0.65 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.91 \[ \int x^6 \text {erf}(b x) \, dx=\begin {cases} \frac {x^{7} \operatorname {erf}{\left (b x \right )}}{7} + \frac {x^{6} e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b} + \frac {3 x^{4} e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b^{3}} + \frac {6 x^{2} e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b^{5}} + \frac {6 e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b^{7}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x**6*erf(b*x),x)
Output:
Piecewise((x**7*erf(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 = 52, normalized size of antiderivative = 0.48 \[ \int x^6 \text {erf}(b x) \, dx=\frac {1}{7} \, x^{7} \operatorname {erf}\left (b x\right ) + \frac {{\left (b^{6} x^{6} + 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} + 6\right )} e^{\left (-b^{2} x^{2}\right )}}{7 \, \sqrt {\pi } b^{7}} \] Input:
integrate(x^6*erf(b*x),x, algorithm="maxima")
Output:
1/7*x^7*erf(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)
Time = 0.11 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.48 \[ \int x^6 \text {erf}(b x) \, dx=\frac {1}{7} \, x^{7} \operatorname {erf}\left (b x\right ) + \frac {{\left (b^{6} x^{6} + 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} + 6\right )} e^{\left (-b^{2} x^{2}\right )}}{7 \, \sqrt {\pi } b^{7}} \] Input:
integrate(x^6*erf(b*x),x, algorithm="giac")
Output:
1/7*x^7*erf(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)
Time = 3.81 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.48 \[ \int x^6 \text {erf}(b x) \, dx=\frac {x^7\,\mathrm {erf}\left (b\,x\right )}{7}+\frac {{\mathrm {e}}^{-b^2\,x^2}\,\left (b^6\,x^6+3\,b^4\,x^4+6\,b^2\,x^2+6\right )}{7\,b^7\,\sqrt {\pi }} \] Input:
int(x^6*erf(b*x),x)
Output:
(x^7*erf(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 = 0.18 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.68 \[ \int x^6 \text {erf}(b x) \, dx=\frac {e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) b^{7} \pi \,x^{7}+\sqrt {\pi }\, b^{6} x^{6}+3 \sqrt {\pi }\, b^{4} x^{4}+6 \sqrt {\pi }\, b^{2} x^{2}+6 \sqrt {\pi }}{7 e^{b^{2} x^{2}} b^{7} \pi } \] Input:
int(x^6*erf(b*x),x)
Output:
(e**(b**2*x**2)*erf(b*x)*b**7*pi*x**7 + sqrt(pi)*b**6*x**6 + 3*sqrt(pi)*b* *4*x**4 + 6*sqrt(pi)*b**2*x**2 + 6*sqrt(pi))/(7*e**(b**2*x**2)*b**7*pi)