Integrand size = 19, antiderivative size = 69 \[ \int e^{c+b^2 x^2} x^2 \text {erfi}(b x) \, dx=-\frac {e^{c+2 b^2 x^2}}{4 b^3 \sqrt {\pi }}+\frac {e^{c+b^2 x^2} x \text {erfi}(b x)}{2 b^2}-\frac {e^c \sqrt {\pi } \text {erfi}(b x)^2}{8 b^3} \] Output:
-1/4*exp(2*b^2*x^2+c)/b^3/Pi^(1/2)+1/2*exp(b^2*x^2+c)*x*erfi(b*x)/b^2-1/8* exp(c)*Pi^(1/2)*erfi(b*x)^2/b^3
Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84 \[ \int e^{c+b^2 x^2} x^2 \text {erfi}(b x) \, dx=-\frac {e^c \left (2 e^{2 b^2 x^2}-4 b e^{b^2 x^2} \sqrt {\pi } x \text {erfi}(b x)+\pi \text {erfi}(b x)^2\right )}{8 b^3 \sqrt {\pi }} \] Input:
Integrate[E^(c + b^2*x^2)*x^2*Erfi[b*x],x]
Output:
-1/8*(E^c*(2*E^(2*b^2*x^2) - 4*b*E^(b^2*x^2)*Sqrt[Pi]*x*Erfi[b*x] + Pi*Erf i[b*x]^2))/(b^3*Sqrt[Pi])
Time = 0.37 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {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.
\(\displaystyle \int x^2 e^{b^2 x^2+c} \text {erfi}(b x) \, dx\) |
\(\Big \downarrow \) 6941 |
\(\displaystyle -\frac {\int e^{b^2 x^2+c} \text {erfi}(b x)dx}{2 b^2}-\frac {\int e^{2 b^2 x^2+c} xdx}{\sqrt {\pi } b}+\frac {x e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}\) |
\(\Big \downarrow \) 2638 |
\(\displaystyle -\frac {\int e^{b^2 x^2+c} \text {erfi}(b x)dx}{2 b^2}+\frac {x e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {e^{2 b^2 x^2+c}}{4 \sqrt {\pi } b^3}\) |
\(\Big \downarrow \) 6929 |
\(\displaystyle -\frac {\sqrt {\pi } e^c \int \text {erfi}(b x)d\text {erfi}(b x)}{4 b^3}+\frac {x e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {e^{2 b^2 x^2+c}}{4 \sqrt {\pi } b^3}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\sqrt {\pi } e^c \text {erfi}(b x)^2}{8 b^3}+\frac {x e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {e^{2 b^2 x^2+c}}{4 \sqrt {\pi } b^3}\) |
Input:
Int[E^(c + b^2*x^2)*x^2*Erfi[b*x],x]
Output:
-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]*Erfi[b*x]^2)/(8*b^3)
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[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]
\[\int {\mathrm e}^{b^{2} x^{2}+c} x^{2} \operatorname {erfi}\left (b x \right )d x\]
Input:
int(exp(b^2*x^2+c)*x^2*erfi(b*x),x)
Output:
int(exp(b^2*x^2+c)*x^2*erfi(b*x),x)
Time = 0.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.77 \[ \int e^{c+b^2 x^2} x^2 \text {erfi}(b x) \, dx=\frac {{\left (4 \, \pi b x \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - \sqrt {\pi } {\left (\pi \operatorname {erfi}\left (b x\right )^{2} + 2 \, e^{\left (2 \, b^{2} x^{2}\right )}\right )}\right )} e^{c}}{8 \, \pi b^{3}} \] Input:
integrate(exp(b^2*x^2+c)*x^2*erfi(b*x),x, algorithm="fricas")
Output:
1/8*(4*pi*b*x*erfi(b*x)*e^(b^2*x^2) - sqrt(pi)*(pi*erfi(b*x)^2 + 2*e^(2*b^ 2*x^2)))*e^c/(pi*b^3)
Time = 0.52 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.99 \[ \int e^{c+b^2 x^2} x^2 \text {erfi}(b x) \, dx=\begin {cases} \frac {x e^{c} e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{2 b^{2}} - \frac {e^{c} e^{2 b^{2} x^{2}}}{4 \sqrt {\pi } b^{3}} - \frac {\sqrt {\pi } e^{c} \operatorname {erfi}^{2}{\left (b x \right )}}{8 b^{3}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(exp(b**2*x**2+c)*x**2*erfi(b*x),x)
Output:
Piecewise((x*exp(c)*exp(b**2*x**2)*erfi(b*x)/(2*b**2) - exp(c)*exp(2*b**2* x**2)/(4*sqrt(pi)*b**3) - sqrt(pi)*exp(c)*erfi(b*x)**2/(8*b**3), Ne(b, 0)) , (0, True))
\[ \int e^{c+b^2 x^2} x^2 \text {erfi}(b x) \, dx=\int { x^{2} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*x^2*erfi(b*x),x, algorithm="maxima")
Output:
integrate(x^2*erfi(b*x)*e^(b^2*x^2 + c), x)
\[ \int e^{c+b^2 x^2} x^2 \text {erfi}(b x) \, dx=\int { x^{2} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*x^2*erfi(b*x),x, algorithm="giac")
Output:
integrate(x^2*erfi(b*x)*e^(b^2*x^2 + c), x)
Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.25 \[ \int e^{c+b^2 x^2} x^2 \text {erfi}(b x) \, dx=\mathrm {erfi}\left (b\,x\right )\,\left (\frac {x\,{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^2}-\frac {\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b^2\,x}{\sqrt {b^2}}\right )\,{\mathrm {e}}^c}{4\,{\left (b^2\right )}^{3/2}}\right )-\frac {2\,{\mathrm {e}}^{2\,b^2\,x^2+c}-\pi \,{\mathrm {erfi}\left (\frac {b^2\,x}{\sqrt {b^2}}\right )}^2\,{\mathrm {e}}^c}{8\,b^3\,\sqrt {\pi }} \] Input:
int(x^2*exp(c + b^2*x^2)*erfi(b*x),x)
Output:
erfi(b*x)*((x*exp(c + b^2*x^2))/(2*b^2) - (pi^(1/2)*erfi((b^2*x)/(b^2)^(1/ 2))*exp(c))/(4*(b^2)^(3/2))) - (2*exp(c + 2*b^2*x^2) - pi*erfi((b^2*x)/(b^ 2)^(1/2))^2*exp(c))/(8*b^3*pi^(1/2))
Time = 0.16 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int e^{c+b^2 x^2} x^2 \text {erfi}(b x) \, dx=\frac {e^{c} \left (\sqrt {\pi }\, \mathrm {erf}\left (b i x \right )^{2} \pi -4 e^{b^{2} x^{2}} \mathrm {erf}\left (b i x \right ) b i \pi x -2 \sqrt {\pi }\, e^{2 b^{2} x^{2}}\right )}{8 b^{3} \pi } \] Input:
int(exp(b^2*x^2+c)*x^2*erfi(b*x),x)
Output:
(e**c*(sqrt(pi)*erf(b*i*x)**2*pi - 4*e**(b**2*x**2)*erf(b*i*x)*b*i*pi*x - 2*sqrt(pi)*e**(2*b**2*x**2)))/(8*b**3*pi)