Integrand size = 19, antiderivative size = 79 \[ \int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx=\frac {e^c x}{b^3 \sqrt {\pi }}-\frac {e^c x^3}{3 b \sqrt {\pi }}-\frac {e^{c+b^2 x^2} \text {erf}(b x)}{2 b^4}+\frac {e^{c+b^2 x^2} x^2 \text {erf}(b x)}{2 b^2} \] Output:
exp(c)*x/b^3/Pi^(1/2)-1/3*exp(c)*x^3/b/Pi^(1/2)-1/2*exp(b^2*x^2+c)*erf(b*x )/b^4+1/2*exp(b^2*x^2+c)*x^2*erf(b*x)/b^2
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.72 \[ \int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx=\frac {e^c \left (6 b x-2 b^3 x^3+3 e^{b^2 x^2} \sqrt {\pi } \left (-1+b^2 x^2\right ) \text {erf}(b x)\right )}{6 b^4 \sqrt {\pi }} \] Input:
Integrate[E^(c + b^2*x^2)*x^3*Erf[b*x],x]
Output:
(E^c*(6*b*x - 2*b^3*x^3 + 3*E^(b^2*x^2)*Sqrt[Pi]*(-1 + b^2*x^2)*Erf[b*x])) /(6*b^4*Sqrt[Pi])
Time = 0.34 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6939, 15, 6936, 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.
\(\displaystyle \int x^3 e^{b^2 x^2+c} \text {erf}(b x) \, dx\) |
\(\Big \downarrow \) 6939 |
\(\displaystyle -\frac {\int e^{b^2 x^2+c} x \text {erf}(b x)dx}{b^2}-\frac {\int e^c x^2dx}{\sqrt {\pi } b}+\frac {x^2 e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\int e^{b^2 x^2+c} x \text {erf}(b x)dx}{b^2}+\frac {x^2 e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {e^c x^3}{3 \sqrt {\pi } b}\) |
\(\Big \downarrow \) 6936 |
\(\displaystyle -\frac {\frac {e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {\int e^cdx}{\sqrt {\pi } b}}{b^2}+\frac {x^2 e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {e^c x^3}{3 \sqrt {\pi } b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {x^2 e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {\frac {e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {e^c x}{\sqrt {\pi } b}}{b^2}-\frac {e^c x^3}{3 \sqrt {\pi } b}\) |
Input:
Int[E^(c + b^2*x^2)*x^3*Erf[b*x],x]
Output:
-1/3*(E^c*x^3)/(b*Sqrt[Pi]) + (E^(c + b^2*x^2)*x^2*Erf[b*x])/(2*b^2) - (-( (E^c*x)/(b*Sqrt[Pi])) + (E^(c + b^2*x^2)*Erf[b*x])/(2*b^2))/b^2
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)*Erf[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Sim p[E^(c + d*x^2)*(Erf[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)*Erf[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] : > Simp[x^(m - 1)*E^(c + d*x^2)*(Erf[a + b*x]/(2*d)), x] + (-Simp[(m - 1)/(2 *d) Int[x^(m - 2)*E^(c + d*x^2)*Erf[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]) /; FreeQ[ {a, b, c, d}, x] && IGtQ[m, 1]
Time = 1.31 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {\frac {\operatorname {erf}\left (b x \right ) {\mathrm e}^{c} \left (\frac {b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}}{2}-\frac {{\mathrm e}^{b^{2} x^{2}}}{2}\right )}{b^{3}}-\frac {{\mathrm e}^{c} \left (\frac {1}{3} b^{3} x^{3}-b x \right )}{\sqrt {\pi }\, b^{3}}}{b}\) | \(66\) |
parallelrisch | \(\frac {-2 \,{\mathrm e}^{b^{2} x^{2}+c} {\mathrm e}^{-b^{2} x^{2}} x^{3} b^{3}+3 \,{\mathrm e}^{b^{2} x^{2}+c} x^{2} \operatorname {erf}\left (b x \right ) b^{2} \sqrt {\pi }+6 \,{\mathrm e}^{b^{2} x^{2}+c} x \,{\mathrm e}^{-b^{2} x^{2}} b -3 \,\operatorname {erf}\left (b x \right ) {\mathrm e}^{b^{2} x^{2}+c} \sqrt {\pi }}{6 \sqrt {\pi }\, b^{4}}\) | \(104\) |
orering | \(\frac {\left (b^{4} x^{4}-6\right ) {\mathrm e}^{b^{2} x^{2}+c} \operatorname {erf}\left (b x \right )}{3 b^{4}}-\frac {\left (b^{2} x^{2}-3\right ) \left (2 b^{2} x^{4} {\mathrm e}^{b^{2} x^{2}+c} \operatorname {erf}\left (b x \right )+3 \,{\mathrm e}^{b^{2} x^{2}+c} x^{2} \operatorname {erf}\left (b x \right )+\frac {2 \,{\mathrm e}^{b^{2} x^{2}+c} x^{3} {\mathrm e}^{-b^{2} x^{2}} b}{\sqrt {\pi }}\right )}{6 x^{2} b^{4}}\) | \(117\) |
Input:
int(exp(b^2*x^2+c)*x^3*erf(b*x),x,method=_RETURNVERBOSE)
Output:
(erf(b*x)/b^3*exp(c)*(1/2*b^2*x^2*exp(b^2*x^2)-1/2*exp(b^2*x^2))-1/Pi^(1/2 )/b^3*exp(c)*(1/3*b^3*x^3-b*x))/b
Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx=-\frac {3 \, {\left (\pi - \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} + 2 \, \sqrt {\pi } {\left (b^{3} x^{3} - 3 \, b x\right )} e^{c}}{6 \, \pi b^{4}} \] Input:
integrate(exp(b^2*x^2+c)*x^3*erf(b*x),x, algorithm="fricas")
Output:
-1/6*(3*(pi - pi*b^2*x^2)*erf(b*x)*e^(b^2*x^2 + c) + 2*sqrt(pi)*(b^3*x^3 - 3*b*x)*e^c)/(pi*b^4)
Time = 6.96 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.96 \[ \int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx=\begin {cases} - \frac {x^{3} e^{c}}{3 \sqrt {\pi } b} + \frac {x^{2} e^{c} e^{b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 b^{2}} + \frac {x e^{c}}{\sqrt {\pi } b^{3}} - \frac {e^{c} e^{b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 b^{4}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(exp(b**2*x**2+c)*x**3*erf(b*x),x)
Output:
Piecewise((-x**3*exp(c)/(3*sqrt(pi)*b) + x**2*exp(c)*exp(b**2*x**2)*erf(b* x)/(2*b**2) + x*exp(c)/(sqrt(pi)*b**3) - exp(c)*exp(b**2*x**2)*erf(b*x)/(2 *b**4), Ne(b, 0)), (0, True))
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.75 \[ \int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx=-\frac {2 \, b^{3} x^{3} e^{c} - 3 \, {\left (\sqrt {\pi } b^{2} x^{2} e^{c} - \sqrt {\pi } e^{c}\right )} \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - 6 \, b x e^{c}}{6 \, \sqrt {\pi } b^{4}} \] Input:
integrate(exp(b^2*x^2+c)*x^3*erf(b*x),x, algorithm="maxima")
Output:
-1/6*(2*b^3*x^3*e^c - 3*(sqrt(pi)*b^2*x^2*e^c - sqrt(pi)*e^c)*erf(b*x)*e^( b^2*x^2) - 6*b*x*e^c)/(sqrt(pi)*b^4)
Time = 0.12 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90 \[ \int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx=\frac {1}{2} \, {\left (\frac {{\left (b^{2} x^{2} + c - 1\right )} e^{\left (b^{2} x^{2} + c\right )}}{b^{4}} - \frac {c e^{\left (b^{2} x^{2} + c\right )}}{b^{4}}\right )} \operatorname {erf}\left (b x\right ) - \frac {b^{2} x^{3} e^{c} - 3 \, x e^{c}}{3 \, \sqrt {\pi } b^{3}} \] Input:
integrate(exp(b^2*x^2+c)*x^3*erf(b*x),x, algorithm="giac")
Output:
1/2*((b^2*x^2 + c - 1)*e^(b^2*x^2 + c)/b^4 - c*e^(b^2*x^2 + c)/b^4)*erf(b* x) - 1/3*(b^2*x^3*e^c - 3*x*e^c)/(sqrt(pi)*b^3)
Time = 0.21 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.82 \[ \int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx=\frac {3\,x\,{\mathrm {e}}^c-b^2\,x^3\,{\mathrm {e}}^c}{3\,b^3\,\sqrt {\pi }}-\mathrm {erf}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^4}-\frac {x^2\,{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^2}\right ) \] Input:
int(x^3*exp(c + b^2*x^2)*erf(b*x),x)
Output:
(3*x*exp(c) - b^2*x^3*exp(c))/(3*b^3*pi^(1/2)) - erf(b*x)*(exp(c + b^2*x^2 )/(2*b^4) - (x^2*exp(c + b^2*x^2))/(2*b^2))
Time = 0.15 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.84 \[ \int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx=\frac {e^{c} \left (3 e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) b^{2} \pi \,x^{2}-3 e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) \pi -2 \sqrt {\pi }\, b^{3} x^{3}+6 \sqrt {\pi }\, b x \right )}{6 b^{4} \pi } \] Input:
int(exp(b^2*x^2+c)*x^3*erf(b*x),x)
Output:
(e**c*(3*e**(b**2*x**2)*erf(b*x)*b**2*pi*x**2 - 3*e**(b**2*x**2)*erf(b*x)* pi - 2*sqrt(pi)*b**3*x**3 + 6*sqrt(pi)*b*x))/(6*b**4*pi)