Integrand size = 8, antiderivative size = 71 \[ \int x \text {erfi}(b x)^2 \, dx=\frac {e^{2 b^2 x^2}}{2 b^2 \pi }-\frac {e^{b^2 x^2} x \text {erfi}(b x)}{b \sqrt {\pi }}+\frac {\text {erfi}(b x)^2}{4 b^2}+\frac {1}{2} x^2 \text {erfi}(b x)^2 \] Output:
1/2*exp(2*b^2*x^2)/b^2/Pi-exp(b^2*x^2)*x*erfi(b*x)/b/Pi^(1/2)+1/4*erfi(b*x )^2/b^2+1/2*x^2*erfi(b*x)^2
Time = 0.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89 \[ \int x \text {erfi}(b x)^2 \, dx=\frac {2 e^{2 b^2 x^2}-4 b e^{b^2 x^2} \sqrt {\pi } x \text {erfi}(b x)+\left (\pi +2 b^2 \pi x^2\right ) \text {erfi}(b x)^2}{4 b^2 \pi } \] Input:
Integrate[x*Erfi[b*x]^2,x]
Output:
(2*E^(2*b^2*x^2) - 4*b*E^(b^2*x^2)*Sqrt[Pi]*x*Erfi[b*x] + (Pi + 2*b^2*Pi*x ^2)*Erfi[b*x]^2)/(4*b^2*Pi)
Time = 0.43 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.18, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6920, 6941, 2638, 6929, 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.
\(\displaystyle \int x \text {erfi}(b x)^2 \, dx\) |
\(\Big \downarrow \) 6920 |
\(\displaystyle \frac {1}{2} x^2 \text {erfi}(b x)^2-\frac {2 b \int e^{b^2 x^2} x^2 \text {erfi}(b x)dx}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 6941 |
\(\displaystyle \frac {1}{2} x^2 \text {erfi}(b x)^2-\frac {2 b \left (-\frac {\int e^{b^2 x^2} \text {erfi}(b x)dx}{2 b^2}-\frac {\int e^{2 b^2 x^2} xdx}{\sqrt {\pi } b}+\frac {x e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}\right )}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 2638 |
\(\displaystyle \frac {1}{2} x^2 \text {erfi}(b x)^2-\frac {2 b \left (-\frac {\int e^{b^2 x^2} \text {erfi}(b x)dx}{2 b^2}+\frac {x e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {e^{2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 6929 |
\(\displaystyle \frac {1}{2} x^2 \text {erfi}(b x)^2-\frac {2 b \left (-\frac {\sqrt {\pi } \int \text {erfi}(b x)d\text {erfi}(b x)}{4 b^3}+\frac {x e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {e^{2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {1}{2} x^2 \text {erfi}(b x)^2-\frac {2 b \left (-\frac {\sqrt {\pi } \text {erfi}(b x)^2}{8 b^3}+\frac {x e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {e^{2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{\sqrt {\pi }}\) |
Input:
Int[x*Erfi[b*x]^2,x]
Output:
(x^2*Erfi[b*x]^2)/2 - (2*b*(-1/4*E^(2*b^2*x^2)/(b^3*Sqrt[Pi]) + (E^(b^2*x^ 2)*x*Erfi[b*x])/(2*b^2) - (Sqrt[Pi]*Erfi[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[Erfi[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erfi[b*x]^2 /(m + 1)), x] - Simp[4*(b/(Sqrt[Pi]*(m + 1))) Int[x^(m + 1)*E^(b^2*x^2)*E rfi[b*x], x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])
Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[E^c* (Sqrt[Pi]/(2*b)) Subst[Int[x^n, x], x, Erfi[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, b^2]
Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^(m - 1)*E^(c + d*x^2)*(Erfi[a + b*x]/(2*d)), x] + (-Simp[(m - 1)/ (2*d) Int[x^(m - 2)*E^(c + d*x^2)*Erfi[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]) /; Free Q[{a, b, c, d}, x] && IGtQ[m, 1]
Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.97
method | result | size |
parallelrisch | \(\frac {2 \operatorname {erfi}\left (b x \right )^{2} x^{2} \pi ^{\frac {3}{2}} b^{2}-4 \,\operatorname {erfi}\left (b x \right ) x \,{\mathrm e}^{b^{2} x^{2}} b \pi +\operatorname {erfi}\left (b x \right )^{2} \pi ^{\frac {3}{2}}+2 \,{\mathrm e}^{2 b^{2} x^{2}} \sqrt {\pi }}{4 \pi ^{\frac {3}{2}} b^{2}}\) | \(69\) |
Input:
int(x*erfi(b*x)^2,x,method=_RETURNVERBOSE)
Output:
1/4*(2*erfi(b*x)^2*x^2*Pi^(3/2)*b^2-4*erfi(b*x)*x*exp(b^2*x^2)*b*Pi+erfi(b *x)^2*Pi^(3/2)+2*exp(b^2*x^2)^2*Pi^(1/2))/Pi^(3/2)/b^2
Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.82 \[ \int x \text {erfi}(b x)^2 \, dx=-\frac {4 \, \sqrt {\pi } b x \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - {\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname {erfi}\left (b x\right )^{2} - 2 \, e^{\left (2 \, b^{2} x^{2}\right )}}{4 \, \pi b^{2}} \] Input:
integrate(x*erfi(b*x)^2,x, algorithm="fricas")
Output:
-1/4*(4*sqrt(pi)*b*x*erfi(b*x)*e^(b^2*x^2) - (pi + 2*pi*b^2*x^2)*erfi(b*x) ^2 - 2*e^(2*b^2*x^2))/(pi*b^2)
Time = 0.22 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89 \[ \int x \text {erfi}(b x)^2 \, dx=\begin {cases} \frac {x^{2} \operatorname {erfi}^{2}{\left (b x \right )}}{2} - \frac {x e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{\sqrt {\pi } b} + \frac {e^{2 b^{2} x^{2}}}{2 \pi b^{2}} + \frac {\operatorname {erfi}^{2}{\left (b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x*erfi(b*x)**2,x)
Output:
Piecewise((x**2*erfi(b*x)**2/2 - x*exp(b**2*x**2)*erfi(b*x)/(sqrt(pi)*b) + exp(2*b**2*x**2)/(2*pi*b**2) + erfi(b*x)**2/(4*b**2), Ne(b, 0)), (0, True ))
\[ \int x \text {erfi}(b x)^2 \, dx=\int { x \operatorname {erfi}\left (b x\right )^{2} \,d x } \] Input:
integrate(x*erfi(b*x)^2,x, algorithm="maxima")
Output:
integrate(x*erfi(b*x)^2, x)
\[ \int x \text {erfi}(b x)^2 \, dx=\int { x \operatorname {erfi}\left (b x\right )^{2} \,d x } \] Input:
integrate(x*erfi(b*x)^2,x, algorithm="giac")
Output:
integrate(x*erfi(b*x)^2, x)
Time = 3.78 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.93 \[ \int x \text {erfi}(b x)^2 \, dx=\frac {\frac {b^2\,x^2\,{\mathrm {erfi}\left (b\,x\right )}^2}{2}+\frac {{\mathrm {erfi}\left (b\,x\right )}^2}{4}}{b^2}+\frac {\frac {{\mathrm {e}}^{2\,b^2\,x^2}}{2}-b\,x\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{b^2\,\pi } \] Input:
int(x*erfi(b*x)^2,x)
Output:
(erfi(b*x)^2/4 + (b^2*x^2*erfi(b*x)^2)/2)/b^2 + (exp(2*b^2*x^2)/2 - b*x*pi ^(1/2)*exp(b^2*x^2)*erfi(b*x))/(b^2*pi)
\[ \int x \text {erfi}(b x)^2 \, dx=-\left (\int \mathrm {erf}\left (b i x \right )^{2} x d x \right ) \] Input:
int(x*erfi(b*x)^2,x)
Output:
- int(erf(b*i*x)**2*x,x)