Integrand size = 17, antiderivative size = 257 \[ \int e^{c+d x^2} x^5 \text {erfi}(b x) \, dx=\frac {3 b e^{c+\left (b^2+d\right ) x^2} x}{4 d \left (b^2+d\right )^2 \sqrt {\pi }}+\frac {b e^{c+\left (b^2+d\right ) x^2} x}{d^2 \left (b^2+d\right ) \sqrt {\pi }}-\frac {b e^{c+\left (b^2+d\right ) x^2} x^3}{2 d \left (b^2+d\right ) \sqrt {\pi }}+\frac {e^{c+d x^2} \text {erfi}(b x)}{d^3}-\frac {e^{c+d x^2} x^2 \text {erfi}(b x)}{d^2}+\frac {e^{c+d x^2} x^4 \text {erfi}(b x)}{2 d}-\frac {3 b e^c \text {erfi}\left (\sqrt {b^2+d} x\right )}{8 d \left (b^2+d\right )^{5/2}}-\frac {b e^c \text {erfi}\left (\sqrt {b^2+d} x\right )}{2 d^2 \left (b^2+d\right )^{3/2}}-\frac {b e^c \text {erfi}\left (\sqrt {b^2+d} x\right )}{d^3 \sqrt {b^2+d}} \]
exp(d*x^2+c)*erfi(b*x)/d^3-exp(d*x^2+c)*x^2*erfi(b*x)/d^2+1/2*exp(d*x^2+c) *x^4*erfi(b*x)/d-3/8*b*exp(c)*erfi(x*(b^2+d)^(1/2))/d/(b^2+d)^(5/2)-1/2*b* exp(c)*erfi(x*(b^2+d)^(1/2))/d^2/(b^2+d)^(3/2)-b*exp(c)*erfi(x*(b^2+d)^(1/ 2))/d^3/(b^2+d)^(1/2)+3/4*b*exp(c+(b^2+d)*x^2)*x/d/(b^2+d)^2/Pi^(1/2)+b*ex p(c+(b^2+d)*x^2)*x/d^2/(b^2+d)/Pi^(1/2)-1/2*b*exp(c+(b^2+d)*x^2)*x^3/d/(b^ 2+d)/Pi^(1/2)
Time = 0.20 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.51 \[ \int e^{c+d x^2} x^5 \text {erfi}(b x) \, dx=\frac {e^c \left (-\frac {2 b d e^{\left (b^2+d\right ) x^2} x \left (2 b^2 \left (-2+d x^2\right )+d \left (-7+2 d x^2\right )\right )}{\left (b^2+d\right )^2 \sqrt {\pi }}+4 e^{d x^2} \left (2-2 d x^2+d^2 x^4\right ) \text {erfi}(b x)-\frac {b \left (8 b^4+20 b^2 d+15 d^2\right ) \text {erfi}\left (\sqrt {b^2+d} x\right )}{\left (b^2+d\right )^{5/2}}\right )}{8 d^3} \]
(E^c*((-2*b*d*E^((b^2 + d)*x^2)*x*(2*b^2*(-2 + d*x^2) + d*(-7 + 2*d*x^2))) /((b^2 + d)^2*Sqrt[Pi]) + 4*E^(d*x^2)*(2 - 2*d*x^2 + d^2*x^4)*Erfi[b*x] - (b*(8*b^4 + 20*b^2*d + 15*d^2)*Erfi[Sqrt[b^2 + d]*x])/(b^2 + d)^(5/2)))/(8 *d^3)
Time = 1.26 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {6941, 2641, 2641, 2633, 6941, 2641, 2633, 6938, 2633}
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.
| \(\displaystyle \int x^5 \text {erfi}(b x) e^{c+d x^2} \, dx\) |
| \(\Big \downarrow \) 6941 |
| \(\displaystyle -\frac {b \int e^{\left (b^2+d\right ) x^2+c} x^4dx}{\sqrt {\pi } d}-\frac {2 \int e^{d x^2+c} x^3 \text {erfi}(b x)dx}{d}+\frac {x^4 \text {erfi}(b x) e^{c+d x^2}}{2 d}\) |
| \(\Big \downarrow \) 2641 |
| \(\displaystyle -\frac {b \left (\frac {x^3 e^{x^2 \left (b^2+d\right )+c}}{2 \left (b^2+d\right )}-\frac {3 \int e^{\left (b^2+d\right ) x^2+c} x^2dx}{2 \left (b^2+d\right )}\right )}{\sqrt {\pi } d}-\frac {2 \int e^{d x^2+c} x^3 \text {erfi}(b x)dx}{d}+\frac {x^4 \text {erfi}(b x) e^{c+d x^2}}{2 d}\) |
| \(\Big \downarrow \) 2641 |
| \(\displaystyle -\frac {b \left (\frac {x^3 e^{x^2 \left (b^2+d\right )+c}}{2 \left (b^2+d\right )}-\frac {3 \left (\frac {x e^{x^2 \left (b^2+d\right )+c}}{2 \left (b^2+d\right )}-\frac {\int e^{\left (b^2+d\right ) x^2+c}dx}{2 \left (b^2+d\right )}\right )}{2 \left (b^2+d\right )}\right )}{\sqrt {\pi } d}-\frac {2 \int e^{d x^2+c} x^3 \text {erfi}(b x)dx}{d}+\frac {x^4 \text {erfi}(b x) e^{c+d x^2}}{2 d}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle -\frac {2 \int e^{d x^2+c} x^3 \text {erfi}(b x)dx}{d}-\frac {b \left (\frac {x^3 e^{x^2 \left (b^2+d\right )+c}}{2 \left (b^2+d\right )}-\frac {3 \left (\frac {x e^{x^2 \left (b^2+d\right )+c}}{2 \left (b^2+d\right )}-\frac {\sqrt {\pi } e^c \text {erfi}\left (x \sqrt {b^2+d}\right )}{4 \left (b^2+d\right )^{3/2}}\right )}{2 \left (b^2+d\right )}\right )}{\sqrt {\pi } d}+\frac {x^4 \text {erfi}(b x) e^{c+d x^2}}{2 d}\) |
| \(\Big \downarrow \) 6941 |
| \(\displaystyle -\frac {2 \left (-\frac {b \int e^{\left (b^2+d\right ) x^2+c} x^2dx}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} x \text {erfi}(b x)dx}{d}+\frac {x^2 \text {erfi}(b x) e^{c+d x^2}}{2 d}\right )}{d}-\frac {b \left (\frac {x^3 e^{x^2 \left (b^2+d\right )+c}}{2 \left (b^2+d\right )}-\frac {3 \left (\frac {x e^{x^2 \left (b^2+d\right )+c}}{2 \left (b^2+d\right )}-\frac {\sqrt {\pi } e^c \text {erfi}\left (x \sqrt {b^2+d}\right )}{4 \left (b^2+d\right )^{3/2}}\right )}{2 \left (b^2+d\right )}\right )}{\sqrt {\pi } d}+\frac {x^4 \text {erfi}(b x) e^{c+d x^2}}{2 d}\) |
| \(\Big \downarrow \) 2641 |
| \(\displaystyle -\frac {2 \left (-\frac {b \left (\frac {x e^{x^2 \left (b^2+d\right )+c}}{2 \left (b^2+d\right )}-\frac {\int e^{\left (b^2+d\right ) x^2+c}dx}{2 \left (b^2+d\right )}\right )}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} x \text {erfi}(b x)dx}{d}+\frac {x^2 \text {erfi}(b x) e^{c+d x^2}}{2 d}\right )}{d}-\frac {b \left (\frac {x^3 e^{x^2 \left (b^2+d\right )+c}}{2 \left (b^2+d\right )}-\frac {3 \left (\frac {x e^{x^2 \left (b^2+d\right )+c}}{2 \left (b^2+d\right )}-\frac {\sqrt {\pi } e^c \text {erfi}\left (x \sqrt {b^2+d}\right )}{4 \left (b^2+d\right )^{3/2}}\right )}{2 \left (b^2+d\right )}\right )}{\sqrt {\pi } d}+\frac {x^4 \text {erfi}(b x) e^{c+d x^2}}{2 d}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle -\frac {2 \left (-\frac {\int e^{d x^2+c} x \text {erfi}(b x)dx}{d}-\frac {b \left (\frac {x e^{x^2 \left (b^2+d\right )+c}}{2 \left (b^2+d\right )}-\frac {\sqrt {\pi } e^c \text {erfi}\left (x \sqrt {b^2+d}\right )}{4 \left (b^2+d\right )^{3/2}}\right )}{\sqrt {\pi } d}+\frac {x^2 \text {erfi}(b x) e^{c+d x^2}}{2 d}\right )}{d}-\frac {b \left (\frac {x^3 e^{x^2 \left (b^2+d\right )+c}}{2 \left (b^2+d\right )}-\frac {3 \left (\frac {x e^{x^2 \left (b^2+d\right )+c}}{2 \left (b^2+d\right )}-\frac {\sqrt {\pi } e^c \text {erfi}\left (x \sqrt {b^2+d}\right )}{4 \left (b^2+d\right )^{3/2}}\right )}{2 \left (b^2+d\right )}\right )}{\sqrt {\pi } d}+\frac {x^4 \text {erfi}(b x) e^{c+d x^2}}{2 d}\) |
| \(\Big \downarrow \) 6938 |
| \(\displaystyle -\frac {2 \left (-\frac {\frac {\text {erfi}(b x) e^{c+d x^2}}{2 d}-\frac {b \int e^{\left (b^2+d\right ) x^2+c}dx}{\sqrt {\pi } d}}{d}-\frac {b \left (\frac {x e^{x^2 \left (b^2+d\right )+c}}{2 \left (b^2+d\right )}-\frac {\sqrt {\pi } e^c \text {erfi}\left (x \sqrt {b^2+d}\right )}{4 \left (b^2+d\right )^{3/2}}\right )}{\sqrt {\pi } d}+\frac {x^2 \text {erfi}(b x) e^{c+d x^2}}{2 d}\right )}{d}-\frac {b \left (\frac {x^3 e^{x^2 \left (b^2+d\right )+c}}{2 \left (b^2+d\right )}-\frac {3 \left (\frac {x e^{x^2 \left (b^2+d\right )+c}}{2 \left (b^2+d\right )}-\frac {\sqrt {\pi } e^c \text {erfi}\left (x \sqrt {b^2+d}\right )}{4 \left (b^2+d\right )^{3/2}}\right )}{2 \left (b^2+d\right )}\right )}{\sqrt {\pi } d}+\frac {x^4 \text {erfi}(b x) e^{c+d x^2}}{2 d}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle -\frac {2 \left (-\frac {\frac {\text {erfi}(b x) e^{c+d x^2}}{2 d}-\frac {b e^c \text {erfi}\left (x \sqrt {b^2+d}\right )}{2 d \sqrt {b^2+d}}}{d}-\frac {b \left (\frac {x e^{x^2 \left (b^2+d\right )+c}}{2 \left (b^2+d\right )}-\frac {\sqrt {\pi } e^c \text {erfi}\left (x \sqrt {b^2+d}\right )}{4 \left (b^2+d\right )^{3/2}}\right )}{\sqrt {\pi } d}+\frac {x^2 \text {erfi}(b x) e^{c+d x^2}}{2 d}\right )}{d}-\frac {b \left (\frac {x^3 e^{x^2 \left (b^2+d\right )+c}}{2 \left (b^2+d\right )}-\frac {3 \left (\frac {x e^{x^2 \left (b^2+d\right )+c}}{2 \left (b^2+d\right )}-\frac {\sqrt {\pi } e^c \text {erfi}\left (x \sqrt {b^2+d}\right )}{4 \left (b^2+d\right )^{3/2}}\right )}{2 \left (b^2+d\right )}\right )}{\sqrt {\pi } d}+\frac {x^4 \text {erfi}(b x) e^{c+d x^2}}{2 d}\) |
(E^(c + d*x^2)*x^4*Erfi[b*x])/(2*d) - (b*((E^(c + (b^2 + d)*x^2)*x^3)/(2*( b^2 + d)) - (3*((E^(c + (b^2 + d)*x^2)*x)/(2*(b^2 + d)) - (E^c*Sqrt[Pi]*Er fi[Sqrt[b^2 + d]*x])/(4*(b^2 + d)^(3/2))))/(2*(b^2 + d))))/(d*Sqrt[Pi]) - (2*((E^(c + d*x^2)*x^2*Erfi[b*x])/(2*d) - ((E^(c + d*x^2)*Erfi[b*x])/(2*d) - (b*E^c*Erfi[Sqrt[b^2 + d]*x])/(2*d*Sqrt[b^2 + d]))/d - (b*((E^(c + (b^2 + d)*x^2)*x)/(2*(b^2 + d)) - (E^c*Sqrt[Pi]*Erfi[Sqrt[b^2 + d]*x])/(4*(b^2 + d)^(3/2))))/(d*Sqrt[Pi])))/d
3.3.59.3.1 Defintions of rubi rules used
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
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[(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]
\[\int {\mathrm e}^{d \,x^{2}+c} x^{5} \operatorname {erfi}\left (b x \right )d x\]
Time = 0.27 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.99 \[ \int e^{c+d x^2} x^5 \text {erfi}(b x) \, dx=\frac {\pi {\left (8 \, b^{5} + 20 \, b^{3} d + 15 \, b d^{2}\right )} \sqrt {-b^{2} - d} \operatorname {erf}\left (\sqrt {-b^{2} - d} x\right ) e^{c} + 4 \, {\left (\pi {\left (b^{6} d^{2} + 3 \, b^{4} d^{3} + 3 \, b^{2} d^{4} + d^{5}\right )} x^{4} - 2 \, \pi {\left (b^{6} d + 3 \, b^{4} d^{2} + 3 \, b^{2} d^{3} + d^{4}\right )} x^{2} + 2 \, \pi {\left (b^{6} + 3 \, b^{4} d + 3 \, b^{2} d^{2} + d^{3}\right )}\right )} \operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )} - 2 \, \sqrt {\pi } {\left (2 \, {\left (b^{5} d^{2} + 2 \, b^{3} d^{3} + b d^{4}\right )} x^{3} - {\left (4 \, b^{5} d + 11 \, b^{3} d^{2} + 7 \, b d^{3}\right )} x\right )} e^{\left (b^{2} x^{2} + d x^{2} + c\right )}}{8 \, \pi {\left (b^{6} d^{3} + 3 \, b^{4} d^{4} + 3 \, b^{2} d^{5} + d^{6}\right )}} \]
1/8*(pi*(8*b^5 + 20*b^3*d + 15*b*d^2)*sqrt(-b^2 - d)*erf(sqrt(-b^2 - d)*x) *e^c + 4*(pi*(b^6*d^2 + 3*b^4*d^3 + 3*b^2*d^4 + d^5)*x^4 - 2*pi*(b^6*d + 3 *b^4*d^2 + 3*b^2*d^3 + d^4)*x^2 + 2*pi*(b^6 + 3*b^4*d + 3*b^2*d^2 + d^3))* erfi(b*x)*e^(d*x^2 + c) - 2*sqrt(pi)*(2*(b^5*d^2 + 2*b^3*d^3 + b*d^4)*x^3 - (4*b^5*d + 11*b^3*d^2 + 7*b*d^3)*x)*e^(b^2*x^2 + d*x^2 + c))/(pi*(b^6*d^ 3 + 3*b^4*d^4 + 3*b^2*d^5 + d^6))
\[ \int e^{c+d x^2} x^5 \text {erfi}(b x) \, dx=e^{c} \int x^{5} e^{d x^{2}} \operatorname {erfi}{\left (b x \right )}\, dx \]
\[ \int e^{c+d x^2} x^5 \text {erfi}(b x) \, dx=\int { x^{5} \operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]
\[ \int e^{c+d x^2} x^5 \text {erfi}(b x) \, dx=\int { x^{5} \operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]
Time = 5.30 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.90 \[ \int e^{c+d x^2} x^5 \text {erfi}(b x) \, dx=\mathrm {erfi}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{d\,x^2+c}}{d^3}-\frac {x^2\,{\mathrm {e}}^{d\,x^2+c}}{d^2}+\frac {x^4\,{\mathrm {e}}^{d\,x^2+c}}{2\,d}\right )-\frac {b\,\mathrm {erfi}\left (x\,\sqrt {b^2+d}\right )\,{\mathrm {e}}^c}{2\,d^2\,{\left (b^2+d\right )}^{3/2}}-\frac {b\,{\mathrm {e}}^c\,\mathrm {erf}\left (x\,\sqrt {-b^2-d}\right )}{d^3\,\sqrt {-b^2-d}}+\frac {b\,x\,{\mathrm {e}}^{b^2\,x^2+d\,x^2+c}}{d^2\,\sqrt {\pi }\,\left (b^2+d\right )}+\frac {b\,x^5\,{\mathrm {e}}^c\,\left (\frac {3\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-x^2\,\left (b^2+d\right )}\right )}{4}+{\mathrm {e}}^{b^2\,x^2+d\,x^2}\,\left (\frac {3\,\sqrt {-x^2\,\left (b^2+d\right )}}{2}+{\left (-x^2\,\left (b^2+d\right )\right )}^{3/2}\right )-\frac {3\,\sqrt {\pi }}{4}\right )}{2\,d\,\sqrt {\pi }\,{\left (-x^2\,\left (b^2+d\right )\right )}^{5/2}} \]
erfi(b*x)*(exp(c + d*x^2)/d^3 - (x^2*exp(c + d*x^2))/d^2 + (x^4*exp(c + d* x^2))/(2*d)) - (b*erfi(x*(d + b^2)^(1/2))*exp(c))/(2*d^2*(d + b^2)^(3/2)) - (b*exp(c)*erf(x*(- d - b^2)^(1/2)))/(d^3*(- d - b^2)^(1/2)) + (b*x*exp(c + d*x^2 + b^2*x^2))/(d^2*pi^(1/2)*(d + b^2)) + (b*x^5*exp(c)*((3*pi^(1/2) *erfc((-x^2*(d + b^2))^(1/2)))/4 + exp(d*x^2 + b^2*x^2)*((3*(-x^2*(d + b^2 ))^(1/2))/2 + (-x^2*(d + b^2))^(3/2)) - (3*pi^(1/2))/4))/(2*d*pi^(1/2)*(-x ^2*(d + b^2))^(5/2))