Integrand size = 8, antiderivative size = 71 \[ \int x \text {erf}(b x)^2 \, dx=\frac {e^{-2 b^2 x^2}}{2 b^2 \pi }+\frac {e^{-b^2 x^2} x \text {erf}(b x)}{b \sqrt {\pi }}-\frac {\text {erf}(b x)^2}{4 b^2}+\frac {1}{2} x^2 \text {erf}(b x)^2 \] Output:
1/2/b^2/exp(2*b^2*x^2)/Pi+x*erf(b*x)/b/exp(b^2*x^2)/Pi^(1/2)-1/4*erf(b*x)^ 2/b^2+1/2*x^2*erf(b*x)^2
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.90 \[ \int x \text {erf}(b x)^2 \, dx=\frac {2 e^{-2 b^2 x^2}+4 b e^{-b^2 x^2} \sqrt {\pi } x \text {erf}(b x)+\pi \left (-1+2 b^2 x^2\right ) \text {erf}(b x)^2}{4 b^2 \pi } \] Input:
Integrate[x*Erf[b*x]^2,x]
Output:
(2/E^(2*b^2*x^2) + (4*b*Sqrt[Pi]*x*Erf[b*x])/E^(b^2*x^2) + Pi*(-1 + 2*b^2* x^2)*Erf[b*x]^2)/(4*b^2*Pi)
Time = 0.42 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.20, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6918, 6939, 2638, 6927, 15}
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.
\(\displaystyle \int x \text {erf}(b x)^2 \, dx\) |
\(\Big \downarrow \) 6918 |
\(\displaystyle \frac {1}{2} x^2 \text {erf}(b x)^2-\frac {2 b \int e^{-b^2 x^2} x^2 \text {erf}(b x)dx}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 6939 |
\(\displaystyle \frac {1}{2} x^2 \text {erf}(b x)^2-\frac {2 b \left (\frac {\int e^{-b^2 x^2} \text {erf}(b x)dx}{2 b^2}+\frac {\int e^{-2 b^2 x^2} xdx}{\sqrt {\pi } b}-\frac {x e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\right )}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 2638 |
\(\displaystyle \frac {1}{2} x^2 \text {erf}(b x)^2-\frac {2 b \left (\frac {\int e^{-b^2 x^2} \text {erf}(b x)dx}{2 b^2}-\frac {x e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 6927 |
\(\displaystyle \frac {1}{2} x^2 \text {erf}(b x)^2-\frac {2 b \left (\frac {\sqrt {\pi } \int \text {erf}(b x)d\text {erf}(b x)}{4 b^3}-\frac {x e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {1}{2} x^2 \text {erf}(b x)^2-\frac {2 b \left (\frac {\sqrt {\pi } \text {erf}(b x)^2}{8 b^3}-\frac {x e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{\sqrt {\pi }}\) |
Input:
Int[x*Erf[b*x]^2,x]
Output:
(x^2*Erf[b*x]^2)/2 - (2*b*(-1/4*1/(b^3*E^(2*b^2*x^2)*Sqrt[Pi]) - (x*Erf[b* x])/(2*b^2*E^(b^2*x^2)) + (Sqrt[Pi]*Erf[b*x]^2)/(8*b^3)))/Sqrt[Pi]
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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[Erf[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erf[b*x]^2/( m + 1)), x] - Simp[4*(b/(Sqrt[Pi]*(m + 1))) Int[(x^(m + 1)*Erf[b*x])/E^(b ^2*x^2), x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])
Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[E^c*( Sqrt[Pi]/(2*b)) Subst[Int[x^n, x], x, Erf[b*x]], x] /; FreeQ[{b, c, d, n} , x] && EqQ[d, -b^2]
Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] : > Simp[x^(m - 1)*E^(c + d*x^2)*(Erf[a + b*x]/(2*d)), x] + (-Simp[(m - 1)/(2 *d) Int[x^(m - 2)*E^(c + d*x^2)*Erf[a + b*x], x], x] - Simp[b/(d*Sqrt[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]
Time = 0.38 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.01
method | result | size |
parallelrisch | \(\frac {2 \operatorname {erf}\left (b x \right )^{2} x^{2} \pi ^{\frac {3}{2}} b^{2}+4 \,{\mathrm e}^{-b^{2} x^{2}} x \,\operatorname {erf}\left (b x \right ) b \pi -\operatorname {erf}\left (b x \right )^{2} \pi ^{\frac {3}{2}}+2 \,{\mathrm e}^{-2 b^{2} x^{2}} \sqrt {\pi }}{4 \pi ^{\frac {3}{2}} b^{2}}\) | \(72\) |
orering | \(\frac {\left (4 b^{4} x^{4}-5 b^{2} x^{2}+1\right ) \operatorname {erf}\left (b x \right )^{2}}{8 b^{4} x^{2}}+\frac {\left (3 b^{2} x^{2}-1\right ) \left (\operatorname {erf}\left (b x \right )^{2}+\frac {4 x \,\operatorname {erf}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}} b}{\sqrt {\pi }}\right )}{8 b^{4} x^{2}}+\frac {\frac {8 \,\operatorname {erf}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}} b}{\sqrt {\pi }}+\frac {8 x \,{\mathrm e}^{-2 b^{2} x^{2}} b^{2}}{\pi }-\frac {8 x^{2} \operatorname {erf}\left (b x \right ) b^{3} {\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }}}{16 b^{4} x}\) | \(151\) |
Input:
int(x*erf(b*x)^2,x,method=_RETURNVERBOSE)
Output:
1/4*(2*erf(b*x)^2*x^2*Pi^(3/2)*b^2+4*exp(-b^2*x^2)*x*erf(b*x)*b*Pi-erf(b*x )^2*Pi^(3/2)+2*exp(-b^2*x^2)^2*Pi^(1/2))/Pi^(3/2)/b^2
Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.83 \[ \int x \text {erf}(b x)^2 \, dx=\frac {4 \, \sqrt {\pi } b x \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - {\left (\pi - 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right )^{2} + 2 \, e^{\left (-2 \, b^{2} x^{2}\right )}}{4 \, \pi b^{2}} \] Input:
integrate(x*erf(b*x)^2,x, algorithm="fricas")
Output:
1/4*(4*sqrt(pi)*b*x*erf(b*x)*e^(-b^2*x^2) - (pi - 2*pi*b^2*x^2)*erf(b*x)^2 + 2*e^(-2*b^2*x^2))/(pi*b^2)
Time = 0.21 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.92 \[ \int x \text {erf}(b x)^2 \, dx=\begin {cases} \frac {x^{2} \operatorname {erf}^{2}{\left (b x \right )}}{2} + \frac {x e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{\sqrt {\pi } b} - \frac {\operatorname {erf}^{2}{\left (b x \right )}}{4 b^{2}} + \frac {e^{- 2 b^{2} x^{2}}}{2 \pi b^{2}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x*erf(b*x)**2,x)
Output:
Piecewise((x**2*erf(b*x)**2/2 + x*exp(-b**2*x**2)*erf(b*x)/(sqrt(pi)*b) - erf(b*x)**2/(4*b**2) + exp(-2*b**2*x**2)/(2*pi*b**2), Ne(b, 0)), (0, True) )
\[ \int x \text {erf}(b x)^2 \, dx=\int { x \operatorname {erf}\left (b x\right )^{2} \,d x } \] Input:
integrate(x*erf(b*x)^2,x, algorithm="maxima")
Output:
-2*integrate(x*e^(-2*b^2*x^2), x)/pi + 1/4*(4*b*x*erf(b*x)*e^(-b^2*x^2) + (2*sqrt(pi)*b^2*x^2 - sqrt(pi))*erf(b*x)^2)/(sqrt(pi)*b^2)
\[ \int x \text {erf}(b x)^2 \, dx=\int { x \operatorname {erf}\left (b x\right )^{2} \,d x } \] Input:
integrate(x*erf(b*x)^2,x, algorithm="giac")
Output:
integrate(x*erf(b*x)^2, x)
Time = 0.15 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.94 \[ \int x \text {erf}(b x)^2 \, dx=\frac {\frac {{\mathrm {e}}^{-2\,b^2\,x^2}}{2}+b\,x\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{b^2\,\pi }-\frac {\frac {{\mathrm {erf}\left (b\,x\right )}^2}{4}-\frac {b^2\,x^2\,{\mathrm {erf}\left (b\,x\right )}^2}{2}}{b^2} \] Input:
int(x*erf(b*x)^2,x)
Output:
(exp(-2*b^2*x^2)/2 + b*x*pi^(1/2)*exp(-b^2*x^2)*erf(b*x))/(b^2*pi) - (erf( b*x)^2/4 - (b^2*x^2*erf(b*x)^2)/2)/b^2
\[ \int x \text {erf}(b x)^2 \, dx=\int \mathrm {erf}\left (b x \right )^{2} x d x \] Input:
int(x*erf(b*x)^2,x)
Output:
int(erf(b*x)**2*x,x)