Integrand size = 18, antiderivative size = 112 \[ \int e^{-b^2 x^2} x^4 \text {erf}(b x) \, dx=-\frac {e^{-2 b^2 x^2}}{2 b^5 \sqrt {\pi }}-\frac {e^{-2 b^2 x^2} x^2}{4 b^3 \sqrt {\pi }}-\frac {3 e^{-b^2 x^2} x \text {erf}(b x)}{4 b^4}-\frac {e^{-b^2 x^2} x^3 \text {erf}(b x)}{2 b^2}+\frac {3 \sqrt {\pi } \text {erf}(b x)^2}{16 b^5} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6520, 6508, 30, 2240, 2243} \[ \int e^{-b^2 x^2} x^4 \text {erf}(b x) \, dx=\frac {3 \sqrt {\pi } \text {erf}(b x)^2}{16 b^5}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-2 b^2 x^2}}{2 \sqrt {\pi } b^5}-\frac {3 x e^{-b^2 x^2} \text {erf}(b x)}{4 b^4}-\frac {x^2 e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3} \]
[In]
[Out]
Rule 30
Rule 2240
Rule 2243
Rule 6508
Rule 6520
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-b^2 x^2} x^3 \text {erf}(b x)}{2 b^2}+\frac {3 \int e^{-b^2 x^2} x^2 \text {erf}(b x) \, dx}{2 b^2}+\frac {\int e^{-2 b^2 x^2} x^3 \, dx}{b \sqrt {\pi }} \\ & = -\frac {e^{-2 b^2 x^2} x^2}{4 b^3 \sqrt {\pi }}-\frac {3 e^{-b^2 x^2} x \text {erf}(b x)}{4 b^4}-\frac {e^{-b^2 x^2} x^3 \text {erf}(b x)}{2 b^2}+\frac {3 \int e^{-b^2 x^2} \text {erf}(b x) \, dx}{4 b^4}+\frac {\int e^{-2 b^2 x^2} x \, dx}{2 b^3 \sqrt {\pi }}+\frac {3 \int e^{-2 b^2 x^2} x \, dx}{2 b^3 \sqrt {\pi }} \\ & = -\frac {e^{-2 b^2 x^2}}{2 b^5 \sqrt {\pi }}-\frac {e^{-2 b^2 x^2} x^2}{4 b^3 \sqrt {\pi }}-\frac {3 e^{-b^2 x^2} x \text {erf}(b x)}{4 b^4}-\frac {e^{-b^2 x^2} x^3 \text {erf}(b x)}{2 b^2}+\frac {\left (3 \sqrt {\pi }\right ) \text {Subst}(\int x \, dx,x,\text {erf}(b x))}{8 b^5} \\ & = -\frac {e^{-2 b^2 x^2}}{2 b^5 \sqrt {\pi }}-\frac {e^{-2 b^2 x^2} x^2}{4 b^3 \sqrt {\pi }}-\frac {3 e^{-b^2 x^2} x \text {erf}(b x)}{4 b^4}-\frac {e^{-b^2 x^2} x^3 \text {erf}(b x)}{2 b^2}+\frac {3 \sqrt {\pi } \text {erf}(b x)^2}{16 b^5} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.76 \[ \int e^{-b^2 x^2} x^4 \text {erf}(b x) \, dx=\frac {e^{-2 b^2 x^2} \left (-4 \left (2+b^2 x^2\right )-4 b e^{b^2 x^2} \sqrt {\pi } x \left (3+2 b^2 x^2\right ) \text {erf}(b x)+3 e^{2 b^2 x^2} \pi \text {erf}(b x)^2\right )}{16 b^5 \sqrt {\pi }} \]
[In]
[Out]
\[\int x^{4} \operatorname {erf}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}d x\]
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.66 \[ \int e^{-b^2 x^2} x^4 \text {erf}(b x) \, dx=-\frac {4 \, {\left (2 \, \pi b^{3} x^{3} + 3 \, \pi b x\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - \sqrt {\pi } {\left (3 \, \pi \operatorname {erf}\left (b x\right )^{2} - 4 \, {\left (b^{2} x^{2} + 2\right )} e^{\left (-2 \, b^{2} x^{2}\right )}\right )}}{16 \, \pi b^{5}} \]
[In]
[Out]
Time = 6.61 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.97 \[ \int e^{-b^2 x^2} x^4 \text {erf}(b x) \, dx=\begin {cases} - \frac {x^{3} e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 b^{2}} - \frac {x^{2} e^{- 2 b^{2} x^{2}}}{4 \sqrt {\pi } b^{3}} - \frac {3 x e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{4 b^{4}} + \frac {3 \sqrt {\pi } \operatorname {erf}^{2}{\left (b x \right )}}{16 b^{5}} - \frac {e^{- 2 b^{2} x^{2}}}{2 \sqrt {\pi } b^{5}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
[In]
[Out]
\[ \int e^{-b^2 x^2} x^4 \text {erf}(b x) \, dx=\int { x^{4} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]
[In]
[Out]
\[ \int e^{-b^2 x^2} x^4 \text {erf}(b x) \, dx=\int { x^{4} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]
[In]
[Out]
Time = 5.70 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.80 \[ \int e^{-b^2 x^2} x^4 \text {erf}(b x) \, dx=-\frac {8\,{\mathrm {e}}^{-2\,b^2\,x^2}-3\,\pi \,{\mathrm {erf}\left (b\,x\right )}^2}{16\,b^5\,\sqrt {\pi }}-\frac {x^2\,{\mathrm {e}}^{-2\,b^2\,x^2}}{4\,b^3\,\sqrt {\pi }}-\frac {3\,x\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{4\,b^4}-\frac {x^3\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{2\,b^2} \]
[In]
[Out]