Integrand size = 19, antiderivative size = 121 \[ \int e^{c+b^2 x^2} x^4 \text {erfi}(b x) \, dx=\frac {e^{c+2 b^2 x^2}}{2 b^5 \sqrt {\pi }}-\frac {e^{c+2 b^2 x^2} x^2}{4 b^3 \sqrt {\pi }}-\frac {3 e^{c+b^2 x^2} x \text {erfi}(b x)}{4 b^4}+\frac {e^{c+b^2 x^2} x^3 \text {erfi}(b x)}{2 b^2}+\frac {3 e^c \sqrt {\pi } \text {erfi}(b x)^2}{16 b^5} \]
-3/4*exp(b^2*x^2+c)*x*erfi(b*x)/b^4+1/2*exp(b^2*x^2+c)*x^3*erfi(b*x)/b^2+1 /2*exp(2*b^2*x^2+c)/b^5/Pi^(1/2)-1/4*exp(2*b^2*x^2+c)*x^2/b^3/Pi^(1/2)+3/1 6*exp(c)*erfi(b*x)^2*Pi^(1/2)/b^5
Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.64 \[ \int e^{c+b^2 x^2} x^4 \text {erfi}(b x) \, dx=\frac {e^c \left (-4 e^{2 b^2 x^2} \left (-2+b^2 x^2\right )+4 b e^{b^2 x^2} \sqrt {\pi } x \left (-3+2 b^2 x^2\right ) \text {erfi}(b x)+3 \pi \text {erfi}(b x)^2\right )}{16 b^5 \sqrt {\pi }} \]
(E^c*(-4*E^(2*b^2*x^2)*(-2 + b^2*x^2) + 4*b*E^(b^2*x^2)*Sqrt[Pi]*x*(-3 + 2 *b^2*x^2)*Erfi[b*x] + 3*Pi*Erfi[b*x]^2))/(16*b^5*Sqrt[Pi])
Time = 0.71 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.27, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6941, 2641, 2638, 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.
\(\displaystyle \int x^4 e^{b^2 x^2+c} \text {erfi}(b x) \, dx\) |
\(\Big \downarrow \) 6941 |
\(\displaystyle -\frac {3 \int e^{b^2 x^2+c} x^2 \text {erfi}(b x)dx}{2 b^2}-\frac {\int e^{2 b^2 x^2+c} x^3dx}{\sqrt {\pi } b}+\frac {x^3 e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}\) |
\(\Big \downarrow \) 2641 |
\(\displaystyle -\frac {3 \int e^{b^2 x^2+c} x^2 \text {erfi}(b x)dx}{2 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2+c}}{4 b^2}-\frac {\int e^{2 b^2 x^2+c} xdx}{2 b^2}}{\sqrt {\pi } b}+\frac {x^3 e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}\) |
\(\Big \downarrow \) 2638 |
\(\displaystyle -\frac {3 \int e^{b^2 x^2+c} x^2 \text {erfi}(b x)dx}{2 b^2}+\frac {x^3 e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2+c}}{4 b^2}-\frac {e^{2 b^2 x^2+c}}{8 b^4}}{\sqrt {\pi } b}\) |
\(\Big \downarrow \) 6941 |
\(\displaystyle -\frac {3 \left (-\frac {\int e^{b^2 x^2+c} \text {erfi}(b x)dx}{2 b^2}-\frac {\int e^{2 b^2 x^2+c} xdx}{\sqrt {\pi } b}+\frac {x e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}\right )}{2 b^2}+\frac {x^3 e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2+c}}{4 b^2}-\frac {e^{2 b^2 x^2+c}}{8 b^4}}{\sqrt {\pi } b}\) |
\(\Big \downarrow \) 2638 |
\(\displaystyle -\frac {3 \left (-\frac {\int e^{b^2 x^2+c} \text {erfi}(b x)dx}{2 b^2}+\frac {x e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {e^{2 b^2 x^2+c}}{4 \sqrt {\pi } b^3}\right )}{2 b^2}+\frac {x^3 e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2+c}}{4 b^2}-\frac {e^{2 b^2 x^2+c}}{8 b^4}}{\sqrt {\pi } b}\) |
\(\Big \downarrow \) 6929 |
\(\displaystyle -\frac {3 \left (-\frac {\sqrt {\pi } e^c \int \text {erfi}(b x)d\text {erfi}(b x)}{4 b^3}+\frac {x e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {e^{2 b^2 x^2+c}}{4 \sqrt {\pi } b^3}\right )}{2 b^2}+\frac {x^3 e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2+c}}{4 b^2}-\frac {e^{2 b^2 x^2+c}}{8 b^4}}{\sqrt {\pi } b}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {x^3 e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2+c}}{4 b^2}-\frac {e^{2 b^2 x^2+c}}{8 b^4}}{\sqrt {\pi } b}-\frac {3 \left (-\frac {\sqrt {\pi } e^c \text {erfi}(b x)^2}{8 b^3}+\frac {x e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {e^{2 b^2 x^2+c}}{4 \sqrt {\pi } b^3}\right )}{2 b^2}\) |
-((-1/8*E^(c + 2*b^2*x^2)/b^4 + (E^(c + 2*b^2*x^2)*x^2)/(4*b^2))/(b*Sqrt[P i])) + (E^(c + b^2*x^2)*x^3*Erfi[b*x])/(2*b^2) - (3*(-1/4*E^(c + 2*b^2*x^2 )/(b^3*Sqrt[Pi]) + (E^(c + b^2*x^2)*x*Erfi[b*x])/(2*b^2) - (E^c*Sqrt[Pi]*E rfi[b*x]^2)/(8*b^3)))/(2*b^2)
3.3.89.3.1 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[(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[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]
\[\int {\mathrm e}^{b^{2} x^{2}+c} x^{4} \operatorname {erfi}\left (b x \right )d x\]
Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.61 \[ \int e^{c+b^2 x^2} x^4 \text {erfi}(b x) \, dx=\frac {{\left (4 \, {\left (2 \, \pi b^{3} x^{3} - 3 \, \pi b x\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} + \sqrt {\pi } {\left (3 \, \pi \operatorname {erfi}\left (b x\right )^{2} - 4 \, {\left (b^{2} x^{2} - 2\right )} e^{\left (2 \, b^{2} x^{2}\right )}\right )}\right )} e^{c}}{16 \, \pi b^{5}} \]
1/16*(4*(2*pi*b^3*x^3 - 3*pi*b*x)*erfi(b*x)*e^(b^2*x^2) + sqrt(pi)*(3*pi*e rfi(b*x)^2 - 4*(b^2*x^2 - 2)*e^(2*b^2*x^2)))*e^c/(pi*b^5)
Time = 1.42 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02 \[ \int e^{c+b^2 x^2} x^4 \text {erfi}(b x) \, dx=\begin {cases} \frac {x^{3} e^{c} e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{2 b^{2}} - \frac {x^{2} e^{c} e^{2 b^{2} x^{2}}}{4 \sqrt {\pi } b^{3}} - \frac {3 x e^{c} e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{4 b^{4}} + \frac {e^{c} e^{2 b^{2} x^{2}}}{2 \sqrt {\pi } b^{5}} + \frac {3 \sqrt {\pi } e^{c} \operatorname {erfi}^{2}{\left (b x \right )}}{16 b^{5}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((x**3*exp(c)*exp(b**2*x**2)*erfi(b*x)/(2*b**2) - x**2*exp(c)*exp (2*b**2*x**2)/(4*sqrt(pi)*b**3) - 3*x*exp(c)*exp(b**2*x**2)*erfi(b*x)/(4*b **4) + exp(c)*exp(2*b**2*x**2)/(2*sqrt(pi)*b**5) + 3*sqrt(pi)*exp(c)*erfi( b*x)**2/(16*b**5), Ne(b, 0)), (0, True))
\[ \int e^{c+b^2 x^2} x^4 \text {erfi}(b x) \, dx=\int { x^{4} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]
\[ \int e^{c+b^2 x^2} x^4 \text {erfi}(b x) \, dx=\int { x^{4} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]
Time = 5.15 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.04 \[ \int e^{c+b^2 x^2} x^4 \text {erfi}(b x) \, dx=\mathrm {erfi}\left (b\,x\right )\,\left (\frac {x^3\,{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^2}-\frac {3\,x\,{\mathrm {e}}^{b^2\,x^2+c}}{4\,b^4}+\frac {3\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b^2\,x}{\sqrt {b^2}}\right )\,{\mathrm {e}}^c}{8\,{\left (b^2\right )}^{5/2}}\right )+\frac {8\,{\mathrm {e}}^{2\,b^2\,x^2+c}-3\,\pi \,{\mathrm {erfi}\left (\frac {b^2\,x}{\sqrt {b^2}}\right )}^2\,{\mathrm {e}}^c}{16\,b^5\,\sqrt {\pi }}-\frac {x^2\,{\mathrm {e}}^{2\,b^2\,x^2+c}}{4\,b^3\,\sqrt {\pi }} \]