Integrand size = 18, antiderivative size = 107 \[ \int e^{-b^2 x^2} x^5 \text {erfi}(b x) \, dx=\frac {2 x}{b^5 \sqrt {\pi }}+\frac {2 x^3}{3 b^3 \sqrt {\pi }}+\frac {x^5}{5 b \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{b^6}-\frac {e^{-b^2 x^2} x^2 \text {erfi}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erfi}(b x)}{2 b^2} \]
-erfi(b*x)/b^6/exp(b^2*x^2)-x^2*erfi(b*x)/b^4/exp(b^2*x^2)-1/2*x^4*erfi(b* x)/b^2/exp(b^2*x^2)+2*x/b^5/Pi^(1/2)+2/3*x^3/b^3/Pi^(1/2)+1/5*x^5/b/Pi^(1/ 2)
Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.64 \[ \int e^{-b^2 x^2} x^5 \text {erfi}(b x) \, dx=\frac {\frac {60 b x+20 b^3 x^3+6 b^5 x^5}{\sqrt {\pi }}-15 e^{-b^2 x^2} \left (2+2 b^2 x^2+b^4 x^4\right ) \text {erfi}(b x)}{30 b^6} \]
((60*b*x + 20*b^3*x^3 + 6*b^5*x^5)/Sqrt[Pi] - (15*(2 + 2*b^2*x^2 + b^4*x^4 )*Erfi[b*x])/E^(b^2*x^2))/(30*b^6)
Time = 0.43 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6941, 15, 6941, 15, 6938, 24}
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.
\(\displaystyle \int x^5 e^{-b^2 x^2} \text {erfi}(b x) \, dx\) |
\(\Big \downarrow \) 6941 |
\(\displaystyle \frac {2 \int e^{-b^2 x^2} x^3 \text {erfi}(b x)dx}{b^2}+\frac {\int x^4dx}{\sqrt {\pi } b}-\frac {x^4 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {2 \int e^{-b^2 x^2} x^3 \text {erfi}(b x)dx}{b^2}-\frac {x^4 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {x^5}{5 \sqrt {\pi } b}\) |
\(\Big \downarrow \) 6941 |
\(\displaystyle \frac {2 \left (\frac {\int e^{-b^2 x^2} x \text {erfi}(b x)dx}{b^2}+\frac {\int x^2dx}{\sqrt {\pi } b}-\frac {x^2 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}\right )}{b^2}-\frac {x^4 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {x^5}{5 \sqrt {\pi } b}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {2 \left (\frac {\int e^{-b^2 x^2} x \text {erfi}(b x)dx}{b^2}-\frac {x^2 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {x^3}{3 \sqrt {\pi } b}\right )}{b^2}-\frac {x^4 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {x^5}{5 \sqrt {\pi } b}\) |
\(\Big \downarrow \) 6938 |
\(\displaystyle \frac {2 \left (\frac {\frac {\int 1dx}{\sqrt {\pi } b}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}}{b^2}-\frac {x^2 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {x^3}{3 \sqrt {\pi } b}\right )}{b^2}-\frac {x^4 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {x^5}{5 \sqrt {\pi } b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {x^4 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {2 \left (-\frac {x^2 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {\frac {x}{\sqrt {\pi } b}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}}{b^2}+\frac {x^3}{3 \sqrt {\pi } b}\right )}{b^2}+\frac {x^5}{5 \sqrt {\pi } b}\) |
x^5/(5*b*Sqrt[Pi]) - (x^4*Erfi[b*x])/(2*b^2*E^(b^2*x^2)) + (2*(x^3/(3*b*Sq rt[Pi]) - (x^2*Erfi[b*x])/(2*b^2*E^(b^2*x^2)) + (x/(b*Sqrt[Pi]) - Erfi[b*x ]/(2*b^2*E^(b^2*x^2)))/b^2))/b^2
3.3.70.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Si mp[E^(c + d*x^2)*(Erfi[a + b*x]/(2*d)), x] - Simp[b/(d*Sqrt[Pi]) Int[E^(a ^2 + c + 2*a*b*x + (b^2 + d)*x^2), x], x] /; FreeQ[{a, b, c, d}, x]
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 = 0.40 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {\left (6 \,{\mathrm e}^{b^{2} x^{2}} b^{5} x^{5}-15 \,\operatorname {erfi}\left (b x \right ) x^{4} \sqrt {\pi }\, b^{4}+20 \,{\mathrm e}^{b^{2} x^{2}} b^{3} x^{3}-30 x^{2} \operatorname {erfi}\left (b x \right ) \sqrt {\pi }\, b^{2}+60 \,{\mathrm e}^{b^{2} x^{2}} b x -30 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }\right ) {\mathrm e}^{-b^{2} x^{2}}}{30 b^{6} \sqrt {\pi }}\) | \(103\) |
parallelrisch | \(\frac {\left (6 \,{\mathrm e}^{b^{2} x^{2}} b^{5} x^{5}-15 \,\operatorname {erfi}\left (b x \right ) x^{4} \sqrt {\pi }\, b^{4}+20 \,{\mathrm e}^{b^{2} x^{2}} b^{3} x^{3}-30 x^{2} \operatorname {erfi}\left (b x \right ) \sqrt {\pi }\, b^{2}+60 \,{\mathrm e}^{b^{2} x^{2}} b x -30 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }\right ) {\mathrm e}^{-b^{2} x^{2}}}{30 b^{6} \sqrt {\pi }}\) | \(103\) |
1/30*(6*exp(b^2*x^2)*b^5*x^5-15*erfi(b*x)*x^4*Pi^(1/2)*b^4+20*exp(b^2*x^2) *b^3*x^3-30*x^2*erfi(b*x)*Pi^(1/2)*b^2+60*exp(b^2*x^2)*b*x-30*erfi(b*x)*Pi ^(1/2))/b^6/Pi^(1/2)/exp(b^2*x^2)
Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.74 \[ \int e^{-b^2 x^2} x^5 \text {erfi}(b x) \, dx=\frac {{\left (2 \, \sqrt {\pi } {\left (3 \, b^{5} x^{5} + 10 \, b^{3} x^{3} + 30 \, b x\right )} e^{\left (b^{2} x^{2}\right )} - 15 \, {\left (2 \, \pi + \pi b^{4} x^{4} + 2 \, \pi b^{2} x^{2}\right )} \operatorname {erfi}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{30 \, \pi b^{6}} \]
1/30*(2*sqrt(pi)*(3*b^5*x^5 + 10*b^3*x^3 + 30*b*x)*e^(b^2*x^2) - 15*(2*pi + pi*b^4*x^4 + 2*pi*b^2*x^2)*erfi(b*x))*e^(-b^2*x^2)/(pi*b^6)
Time = 104.88 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.93 \[ \int e^{-b^2 x^2} x^5 \text {erfi}(b x) \, dx=\begin {cases} \frac {x^{5}}{5 \sqrt {\pi } b} - \frac {x^{4} e^{- b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{2 b^{2}} + \frac {2 x^{3}}{3 \sqrt {\pi } b^{3}} - \frac {x^{2} e^{- b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{b^{4}} + \frac {2 x}{\sqrt {\pi } b^{5}} - \frac {e^{- b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{b^{6}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((x**5/(5*sqrt(pi)*b) - x**4*exp(-b**2*x**2)*erfi(b*x)/(2*b**2) + 2*x**3/(3*sqrt(pi)*b**3) - x**2*exp(-b**2*x**2)*erfi(b*x)/b**4 + 2*x/(sqr t(pi)*b**5) - exp(-b**2*x**2)*erfi(b*x)/b**6, Ne(b, 0)), (0, True))
\[ \int e^{-b^2 x^2} x^5 \text {erfi}(b x) \, dx=\int { x^{5} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]
\[ \int e^{-b^2 x^2} x^5 \text {erfi}(b x) \, dx=\int { x^{5} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]
Time = 0.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.77 \[ \int e^{-b^2 x^2} x^5 \text {erfi}(b x) \, dx=\frac {3\,b^4\,x^5+10\,b^2\,x^3+30\,x}{15\,b^5\,\sqrt {\pi }}-\mathrm {erfi}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{-b^2\,x^2}}{b^6}+\frac {x^4\,{\mathrm {e}}^{-b^2\,x^2}}{2\,b^2}+\frac {x^2\,{\mathrm {e}}^{-b^2\,x^2}}{b^4}\right ) \]